Symbolic Expressions#
RELATIONAL EXPRESSIONS:
We create a relational expression:
sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.subs(x == 5)
16 <= 18
Notice that squaring the relation squares both sides.
sage: eqn^2
(x - 1)^4 <= (x^2 - 2*x + 3)^2
sage: eqn.expand()
x^2 - 2*x + 1 <= x^2 - 2*x + 3
This can transform a true relation into a false one:
sage: eqn = SR(-5) < SR(-3); eqn
-5 < -3
sage: bool(eqn)
True
sage: eqn^2
25 < 9
sage: bool(eqn^2)
False
We can do arithmetic with relations:
sage: e = x+1 <= x-2
sage: e + 2
x + 3 <= x
sage: e - 1
x <= x - 3
sage: e*(-1)
-x - 1 <= -x + 2
sage: (-2)*e
-2*x - 2 <= -2*x + 4
sage: e*5
5*x + 5 <= 5*x - 10
sage: e/5
1/5*x + 1/5 <= 1/5*x - 2/5
sage: 5/e
5/(x + 1) <= 5/(x - 2)
sage: e/(-2)
-1/2*x - 1/2 <= -1/2*x + 1
sage: -2/e
-2/(x + 1) <= -2/(x - 2)
We can even add together two relations, as long as the operators are the same:
sage: (x^3 + x <= x - 17) + (-x <= x - 10)
x^3 <= 2*x - 27
Here they are not:
sage: (x^3 + x <= x - 17) + (-x >= x - 10)
Traceback (most recent call last):
...
TypeError: incompatible relations
ARBITRARY SAGE ELEMENTS:
You can work symbolically with any Sage data type. This can lead to nonsense if the data type is strange, e.g., an element of a finite field (at present).
We mix Singular variables with symbolic variables:
sage: R.<u,v> = QQ[]
sage: var('a,b,c')
(a, b, c)
sage: expand((u + v + a + b + c)^2)
a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2 + 2*a*u + 2*b*u + 2*c*u + u^2 + 2*a*v + 2*b*v + 2*c*v + 2*u*v + v^2
- class sage.symbolic.expression.E#
Bases:
ExpressionDummy class to represent base of the natural logarithm.
The base of the natural logarithm
eis not a constant in GiNaC/Sage. It is represented byexp(1).This class provides a dummy object that behaves well under addition, multiplication, etc. and on exponentiation calls the function
exp.EXAMPLES:
The constant defined at the top level is just
exp(1):sage: e.operator() exp sage: e.operands() [1]
Arithmetic works:
sage: e + 2 e + 2 sage: 2 + e e + 2 sage: 2*e 2*e sage: e*2 2*e sage: x*e x*e sage: var('a,b') (a, b) sage: t = e^(a+b); t e^(a + b) sage: t.operands() [a + b]
Numeric evaluation, conversion to other systems, and pickling works as expected. Note that these are properties of the
exp()function, not this class:sage: RR(e) 2.71828182845905 sage: R = RealField(200); R Real Field with 200 bits of precision sage: R(e) 2.7182818284590452353602874713526624977572470936999595749670 sage: em = 1 + e^(1-e); em e^(-e + 1) + 1 sage: R(em) 1.1793740787340171819619895873183164984596816017589156131574 sage: maxima(e).float() 2.718281828459045 sage: t = mathematica(e) # optional - mathematica sage: t # optional - mathematica E sage: float(t) # optional - mathematica 2.718281828459045... sage: loads(dumps(e)) e sage: float(e) 2.718281828459045... sage: e.__float__() 2.718281828459045... sage: e._mpfr_(RealField(100)) 2.7182818284590452353602874714 sage: e._real_double_(RDF) # abs tol 5e-16 2.718281828459045 sage: import sympy # needs sympy sage: sympy.E == e # indirect doctest # needs sympy True
- class sage.symbolic.expression.Expression#
Bases:
ExpressionNearly all expressions are created by calling new_Expression_from_*, but we need to make sure this at least does not leave self._gobj uninitialized and segfault.
- Order(hold=False)#
Return the order of the expression, as in big oh notation.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: n = var('n') sage: t = (17*n^3).Order(); t Order(n^3) sage: t.derivative(n) Order(n^2)
To prevent automatic evaluation use the
holdargument:sage: (17*n^3).Order(hold=True) Order(17*n^3)
- WZ_certificate(n, k)#
Return the Wilf-Zeilberger certificate for this hypergeometric summand in
n,k.To prove the identity \(\sum_k F(n,k)=\textrm{const}\) it suffices to show that \(F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k),\) with \(G=RF\) and \(R\) the WZ certificate.
EXAMPLES:
To show that \(\sum_k \binom{n}{k} = 2^n\) do:
sage: _ = var('k n') sage: F(n,k) = binomial(n,k) / 2^n sage: c = F(n,k).WZ_certificate(n,k); c 1/2*k/(k - n - 1) sage: G(n,k) = c * F(n,k); G (n, k) |--> 1/2*k*binomial(n, k)/(2^n*(k - n - 1)) sage: (F(n+1,k) - F(n,k) - G(n,k+1) + G(n,k)).simplify_full() 0
- abs(hold=False)#
Return the absolute value of this expression.
EXAMPLES:
sage: var('x, y') (x, y) sage: (x+y).abs() abs(x + y)
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(-5).abs(hold=True) abs(-5)
To then evaluate again, we use
unhold():sage: a = SR(-5).abs(hold=True); a.unhold() 5
- add(hold=False, *args)#
Return the sum of the current expression and the given arguments.
To prevent automatic evaluation use the
holdargument.EXAMPLES:
sage: x.add(x) 2*x sage: x.add(x, hold=True) x + x sage: x.add(x, (2+x), hold=True) (x + 2) + x + x sage: x.add(x, (2+x), x, hold=True) (x + 2) + x + x + x sage: x.add(x, (2+x), x, 2*x, hold=True) (x + 2) + 2*x + x + x + x
To then evaluate again, we use
unhold():sage: a = x.add(x, hold=True); a.unhold() 2*x
- add_to_both_sides(x)#
Return a relation obtained by adding
xto both sides of this relation.EXAMPLES:
sage: var('x y z') (x, y, z) sage: eqn = x^2 + y^2 + z^2 <= 1 sage: eqn.add_to_both_sides(-z^2) x^2 + y^2 <= -z^2 + 1 sage: eqn.add_to_both_sides(I) x^2 + y^2 + z^2 + I <= (I + 1)
- arccos(hold=False)#
Return the arc cosine of self.
EXAMPLES:
sage: x.arccos() arccos(x) sage: SR(1).arccos() 0 sage: SR(1/2).arccos() 1/3*pi sage: SR(0.4).arccos() 1.15927948072741 sage: plot(lambda x: SR(x).arccos(), -1,1) # needs sage.plot Graphics object consisting of 1 graphics primitive
To prevent automatic evaluation use the
holdargument:sage: SR(1).arccos(hold=True) arccos(1)
This also works using functional notation:
sage: arccos(1,hold=True) arccos(1) sage: arccos(1) 0
To then evaluate again, we use
unhold():sage: a = SR(1).arccos(hold=True); a.unhold() 0
- arccosh(hold=False)#
Return the inverse hyperbolic cosine of
self.EXAMPLES:
sage: x.arccosh() arccosh(x) sage: SR(0).arccosh() 1/2*I*pi sage: SR(1/2).arccosh() arccosh(1/2) sage: SR(CDF(1/2)).arccosh() # rel tol 1e-15 1.0471975511965976*I sage: z = maxima('acosh(0.5)') sage: z.real(), z.imag() # abs tol 1e-15 (0.0, 1.047197551196598)
To prevent automatic evaluation use the
holdargument:sage: SR(-1).arccosh() I*pi sage: SR(-1).arccosh(hold=True) arccosh(-1)
This also works using functional notation:
sage: arccosh(-1,hold=True) arccosh(-1) sage: arccosh(-1) I*pi
To then evaluate again, we use
unhold():sage: a = SR(-1).arccosh(hold=True); a.unhold() I*pi
- arcsin(hold=False)#
Return the arcsin of x, i.e., the number y between -pi and pi such that sin(y) == x.
EXAMPLES:
sage: x.arcsin() arcsin(x) sage: SR(0.5).arcsin() 1/6*pi sage: SR(0.999).arcsin() 1.52607123962616 sage: SR(1/3).arcsin() arcsin(1/3) sage: SR(-1/3).arcsin() -arcsin(1/3)
To prevent automatic evaluation use the
holdargument:sage: SR(0).arcsin() 0 sage: SR(0).arcsin(hold=True) arcsin(0)
This also works using functional notation:
sage: arcsin(0,hold=True) arcsin(0) sage: arcsin(0) 0
To then evaluate again, we use
unhold():sage: a = SR(0).arcsin(hold=True); a.unhold() 0
- arcsinh(hold=False)#
Return the inverse hyperbolic sine of self.
EXAMPLES:
sage: x.arcsinh() arcsinh(x) sage: SR(0).arcsinh() 0 sage: SR(1).arcsinh() arcsinh(1) sage: SR(1.0).arcsinh() 0.881373587019543 sage: maxima('asinh(2.0)') 1.4436354751788...
Sage automatically applies certain identities:
sage: SR(3/2).arcsinh().cosh() 1/2*sqrt(13)
To prevent automatic evaluation use the
holdargument:sage: SR(-2).arcsinh() -arcsinh(2) sage: SR(-2).arcsinh(hold=True) arcsinh(-2)
This also works using functional notation:
sage: arcsinh(-2,hold=True) arcsinh(-2) sage: arcsinh(-2) -arcsinh(2)
To then evaluate again, we use
unhold():sage: a = SR(-2).arcsinh(hold=True); a.unhold() -arcsinh(2)
- arctan(hold=False)#
Return the arc tangent of self.
EXAMPLES:
sage: x = var('x') sage: x.arctan() arctan(x) sage: SR(1).arctan() 1/4*pi sage: SR(1/2).arctan() arctan(1/2) sage: SR(0.5).arctan() 0.463647609000806 sage: plot(lambda x: SR(x).arctan(), -20,20) # needs sage.plot Graphics object consisting of 1 graphics primitive
To prevent automatic evaluation use the
holdargument:sage: SR(1).arctan(hold=True) arctan(1)
This also works using functional notation:
sage: arctan(1,hold=True) arctan(1) sage: arctan(1) 1/4*pi
To then evaluate again, we use
unhold():sage: a = SR(1).arctan(hold=True); a.unhold() 1/4*pi
- arctan2(x, hold=False)#
Return the inverse of the 2-variable tan function on self and x.
EXAMPLES:
sage: var('x,y') (x, y) sage: x.arctan2(y) arctan2(x, y) sage: SR(1/2).arctan2(1/2) 1/4*pi sage: maxima.eval('atan2(1/2,1/2)') '%pi/4' sage: SR(-0.7).arctan2(SR(-0.6)) -2.27942259892257
To prevent automatic evaluation use the
holdargument:sage: SR(1/2).arctan2(1/2, hold=True) arctan2(1/2, 1/2)
This also works using functional notation:
sage: arctan2(1,2,hold=True) arctan2(1, 2) sage: arctan2(1,2) arctan(1/2)
To then evaluate again, we use
unhold():sage: a = SR(1/2).arctan2(1/2, hold=True); a.unhold() 1/4*pi
- arctanh(hold=False)#
Return the inverse hyperbolic tangent of self.
EXAMPLES:
sage: x.arctanh() arctanh(x) sage: SR(0).arctanh() 0 sage: SR(1/2).arctanh() 1/2*log(3) sage: SR(0.5).arctanh() 0.549306144334055 sage: SR(0.5).arctanh().tanh() 0.500000000000000 sage: maxima('atanh(0.5)') # abs tol 2e-16 0.5493061443340548
To prevent automatic evaluation use the
holdargument:sage: SR(-1/2).arctanh() -1/2*log(3) sage: SR(-1/2).arctanh(hold=True) arctanh(-1/2)
This also works using functional notation:
sage: arctanh(-1/2,hold=True) arctanh(-1/2) sage: arctanh(-1/2) -1/2*log(3)
To then evaluate again, we use
unhold():sage: a = SR(-1/2).arctanh(hold=True); a.unhold() -1/2*log(3)
- args()#
EXAMPLES:
sage: x,y = var('x,y') sage: f = x + y sage: f.arguments() (x, y) sage: g = f.function(x) sage: g.arguments() (x,)
- arguments()#
EXAMPLES:
sage: x,y = var('x,y') sage: f = x + y sage: f.arguments() (x, y) sage: g = f.function(x) sage: g.arguments() (x,)
- assume()#
Assume that this equation holds. This is relevant for symbolic integration, among other things.
EXAMPLES: We call the assume method to assume that \(x>2\):
sage: (x > 2).assume()
Bool returns True below if the inequality is definitely known to be True.
sage: bool(x > 0) True sage: bool(x < 0) False
This may or may not be True, so bool returns False:
sage: bool(x > 3) False
If you make inconsistent or meaningless assumptions, Sage will let you know:
sage: forget() sage: assume(x<0) sage: assume(x>0) Traceback (most recent call last): ... ValueError: Assumption is inconsistent sage: assumptions() [x < 0] sage: forget()
- binomial(k, hold=False)#
Return binomial coefficient “self choose k”.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: var('x, y') (x, y) sage: SR(5).binomial(SR(3)) 10 sage: x.binomial(SR(3)) 1/6*(x - 1)*(x - 2)*x sage: x.binomial(y) binomial(x, y)
To prevent automatic evaluation use the
holdargument:sage: x.binomial(3, hold=True) binomial(x, 3) sage: SR(5).binomial(3, hold=True) binomial(5, 3)
To then evaluate again, we use
unhold():sage: a = SR(5).binomial(3, hold=True); a.unhold() 10
The
holdparameter is also supported in functional notation:sage: binomial(5,3, hold=True) binomial(5, 3)
- canonicalize_radical()#
Choose a canonical branch of the given expression.
The square root, cube root, natural log, etc. functions are multi-valued. The
canonicalize_radical()method will choose one of these values based on a heuristic.For example,
sqrt(x^2)has two values:x, and-x. Thecanonicalize_radical()function will choose one of them, consistently, based on the behavior of the expression asxtends to positive infinity. The solution chosen is the one which exhibits this same behavior. Sincesqrt(x^2)approaches positive infinity asxdoes, the solution chosen isx(which also tends to positive infinity).Warning
As shown in the examples below, a canonical form is not always returned, i.e., two mathematically identical expressions might be converted to different expressions.
Assumptions are not taken into account during the transformation. This may result in a branch choice inconsistent with your assumptions.
ALGORITHM:
This uses the Maxima
radcan()command. From the Maxima documentation:Simplifies an expression, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, radcan produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by radcan to zero.
For some expressions radcan is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial fraction expansions of exponents.
EXAMPLES:
canonicalize_radical()can perform some of the same manipulations aslog_expand():sage: y = SR.symbol('y') sage: f = log(x*y) sage: f.log_expand() log(x) + log(y) sage: f.canonicalize_radical() log(x) + log(y)
And also handles some exponential functions:
sage: f = (e^x-1)/(1+e^(x/2)) sage: f.canonicalize_radical() e^(1/2*x) - 1
It can also be used to change the base of a logarithm when the arguments to
log()are positive real numbers:sage: f = log(8)/log(2) sage: f.canonicalize_radical() 3
sage: a = SR.symbol('a') sage: f = (log(x+x^2)-log(x))^a/log(1+x)^(a/2) sage: f.canonicalize_radical() log(x + 1)^(1/2*a)
The simplest example of counter-intuitive behavior is what happens when we take the square root of a square:
sage: sqrt(x^2).canonicalize_radical() x
If you don’t want this kind of “simplification,” don’t use
canonicalize_radical().This behavior can also be triggered when the expression under the radical is not given explicitly as a square:
sage: sqrt(x^2 - 2*x + 1).canonicalize_radical() x - 1
Another place where this can become confusing is with logarithms of complex numbers. Suppose
xis complex withx == r*e^(I*t)(rreal). Thenlog(x)islog(r) + I*(t + 2*k*pi)for some integerk.Calling
canonicalize_radical()will choose a branch, eliminating the solutions for all choices ofkbut one. Simplified by hand, the expression below is(1/2)*log(2) + I*pi*kfor integerk. However,canonicalize_radical()will take each log expression, and choose one particular solution, dropping the other. When the results are subtracted, we’re left with no imaginary part:sage: f = (1/2)*log(2*x) + (1/2)*log(1/x) sage: f.canonicalize_radical() 1/2*log(2)
Naturally the result is wrong for some choices of
x:sage: f(x = -1) I*pi + 1/2*log(2)
The example below shows two expressions e1 and e2 which are “simplified” to different expressions, while their difference is “simplified” to zero; thus
canonicalize_radical()does not return a canonical form:sage: e1 = 1/(sqrt(5)+sqrt(2)) sage: e2 = (sqrt(5)-sqrt(2))/3 sage: e1.canonicalize_radical() 1/(sqrt(5) + sqrt(2)) sage: e2.canonicalize_radical() 1/3*sqrt(5) - 1/3*sqrt(2) sage: (e1-e2).canonicalize_radical() 0
The issue reported in github issue #3520 is a case where
canonicalize_radical()causes a numerical integral to be calculated incorrectly:sage: f1 = sqrt(25 - x) * sqrt( 1 + 1/(4*(25-x)) ) sage: f2 = f1.canonicalize_radical() sage: numerical_integral(f1.real(), 0, 1)[0] # abs tol 1e-10 4.974852579915647 sage: numerical_integral(f2.real(), 0, 1)[0] # abs tol 1e-10 -4.974852579915647
- coefficient(s, n=1)#
Return the coefficient of \(s^n\) in this symbolic expression.
INPUT:
s- expressionn- expression, default 1
OUTPUT:
A symbolic expression. The coefficient of \(s^n\).
Sometimes it may be necessary to expand or factor first, since this is not done automatically.
EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100 sage: f.collect(x) x^3*sin(x*y) + (a + y + 1/y)*x + 2*sin(x*y)/x + 100 sage: f.coefficient(x,0) 100 sage: f.coefficient(x,-1) 2*sin(x*y) sage: f.coefficient(x,1) a + y + 1/y sage: f.coefficient(x,2) 0 sage: f.coefficient(x,3) sin(x*y) sage: f.coefficient(x^3) sin(x*y) sage: f.coefficient(sin(x*y)) x^3 + 2/x sage: f.collect(sin(x*y)) a*x + x*y + (x^3 + 2/x)*sin(x*y) + x/y + 100 sage: var('a, x, y, z') (a, x, y, z) sage: f = (a*sqrt(2))*x^2 + sin(y)*x^(1/2) + z^z sage: f.coefficient(sin(y)) sqrt(x) sage: f.coefficient(x^2) sqrt(2)*a sage: f.coefficient(x^(1/2)) sin(y) sage: f.coefficient(1) 0 sage: f.coefficient(x, 0) z^z
Any coefficient can be queried:
sage: (x^2 + 3*x^pi).coefficient(x, pi) 3 sage: (2^x + 5*x^x).coefficient(x, x) 5
- coefficients(x=None, sparse=True)#
Return the coefficients of this symbolic expression as a polynomial in x.
INPUT:
x– optional variable.
OUTPUT:
Depending on the value of
sparse,A list of pairs
(expr, n), whereexpris a symbolic expression andnis a power (sparse=True, default)A list of expressions where the
n-th element is the coefficient ofx^nwhen self is seen as polynomial inx(sparse=False).
EXAMPLES:
sage: var('x, y, a') (x, y, a) sage: p = x^3 - (x-3)*(x^2+x) + 1 sage: p.coefficients() [[1, 0], [3, 1], [2, 2]] sage: p.coefficients(sparse=False) [1, 3, 2] sage: p = x - x^3 + 5/7*x^5 sage: p.coefficients() [[1, 1], [-1, 3], [5/7, 5]] sage: p.coefficients(sparse=False) [0, 1, 0, -1, 0, 5/7] sage: p = expand((x-a*sqrt(2))^2 + x + 1); p -2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1 sage: p.coefficients(a) [[x^2 + x + 1, 0], [-2*sqrt(2)*x, 1], [2, 2]] sage: p.coefficients(a, sparse=False) [x^2 + x + 1, -2*sqrt(2)*x, 2] sage: p.coefficients(x) [[2*a^2 + 1, 0], [-2*sqrt(2)*a + 1, 1], [1, 2]] sage: p.coefficients(x, sparse=False) [2*a^2 + 1, -2*sqrt(2)*a + 1, 1]
- collect(s)#
Collect the coefficients of
sinto a group.INPUT:
s– the symbol whose coefficients will be collected.
OUTPUT:
A new expression, equivalent to the original one, with the coefficients of
sgrouped.Note
The expression is not expanded or factored before the grouping takes place. For best results, call
expand()on the expression beforecollect().EXAMPLES:
In the first term of \(f\), \(x\) has a coefficient of \(4y\). In the second term, \(x\) has a coefficient of \(z\). Therefore, if we collect those coefficients, \(x\) will have a coefficient of \(4y+z\):
sage: x,y,z = var('x,y,z') sage: f = 4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2 + x^2*y^2*z^2 sage: f.collect(x) x^2*y^2*z^2 + x*(4*y + z) + 20*y^2 + 21*y*z + 4*z^2
Here we do the same thing for \(y\) and \(z\); however, note that we do not factor the \(y^{2}\) and \(z^{2}\) terms before collecting coefficients:
sage: f.collect(y) (x^2*z^2 + 20)*y^2 + (4*x + 21*z)*y + x*z + 4*z^2 sage: f.collect(z) (x^2*y^2 + 4)*z^2 + 4*x*y + 20*y^2 + (x + 21*y)*z
The terms are collected, whether the expression is expanded or not:
sage: f = (x + y)*(x - z) sage: f.collect(x) x^2 + x*(y - z) - y*z sage: f.expand().collect(x) x^2 + x*(y - z) - y*z
- collect_common_factors()#
This function does not perform a full factorization but only looks for factors which are already explicitly present.
Polynomials can often be brought into a more compact form by collecting common factors from the terms of sums. This is accomplished by this function.
EXAMPLES:
sage: var('x') x sage: (x/(x^2 + x)).collect_common_factors() 1/(x + 1) sage: var('a,b,c,x,y') (a, b, c, x, y) sage: (a*x+a*y).collect_common_factors() a*(x + y) sage: (a*x^2+2*a*x*y+a*y^2).collect_common_factors() (x^2 + 2*x*y + y^2)*a sage: (a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y)).collect_common_factors() ((x + y)*y + x)*(a + c)*a*b
- combine(deep=False)#
Return a simplified version of this symbolic expression by combining all toplevel terms with the same denominator into a single term.
Please use the keyword
deep=Trueto apply the process recursively.EXAMPLES:
sage: var('x, y, a, b, c') (x, y, a, b, c) sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a; f (x - 1)*x/(x^2 - 7) + y^2/(x^2 - 7) + b/a + c/a + 1/(x + 1) sage: f.combine() ((x - 1)*x + y^2)/(x^2 - 7) + (b + c)/a + 1/(x + 1) sage: (1/x + 1/x^2 + (x+1)/x).combine() (x + 2)/x + 1/x^2 sage: ex = 1/x + ((x + 1)/x - 1/x)/x^2 + (x+1)/x; ex (x + 1)/x + 1/x + ((x + 1)/x - 1/x)/x^2 sage: ex.combine() (x + 2)/x + ((x + 1)/x - 1/x)/x^2 sage: ex.combine(deep=True) (x + 2)/x + 1/x^2 sage: (1+sin((x + 1)/x - 1/x)).combine(deep=True) sin(1) + 1
- conjugate(hold=False)#
Return the complex conjugate of this symbolic expression.
EXAMPLES:
sage: a = 1 + 2*I sage: a.conjugate() -2*I + 1 sage: a = sqrt(2) + 3^(1/3)*I; a sqrt(2) + I*3^(1/3) sage: a.conjugate() sqrt(2) - I*3^(1/3) sage: SR(CDF.0).conjugate() -1.0*I sage: x.conjugate() conjugate(x) sage: SR(RDF(1.5)).conjugate() 1.5 sage: SR(float(1.5)).conjugate() 1.5 sage: SR(I).conjugate() -I sage: ( 1+I + (2-3*I)*x).conjugate() (3*I + 2)*conjugate(x) - I + 1
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(I).conjugate(hold=True) conjugate(I)
This also works in functional notation:
sage: conjugate(I) -I sage: conjugate(I,hold=True) conjugate(I)
To then evaluate again, we use
unhold():sage: a = SR(I).conjugate(hold=True); a.unhold() -I
- content(s)#
Return the content of this expression when considered as a polynomial in
s.See also
unit(),primitive_part(), andunit_content_primitive().INPUT:
s– a symbolic expression.
OUTPUT:
The content part of a polynomial as a symbolic expression. It is defined as the gcd of the coefficients.
Warning
The expression is considered to be a univariate polynomial in
s. The output is different from thecontent()method provided by multivariate polynomial rings in Sage.EXAMPLES:
sage: (2*x+4).content(x) 2 sage: (2*x+1).content(x) 1 sage: (2*x+1/2).content(x) 1/2 sage: var('y') y sage: (2*x + 4*sin(y)).content(sin(y)) 2
- contradicts(soln)#
Return
Trueif this relation is violated by the given variable assignment(s).EXAMPLES:
sage: (x<3).contradicts(x==0) False sage: (x<3).contradicts(x==3) True sage: (x<=3).contradicts(x==3) False sage: y = var('y') sage: (x<y).contradicts(x==30) False sage: (x<y).contradicts({x: 30, y: 20}) True
- convert(target=None)#
Call the convert function in the units package. For symbolic variables that are not units, this function just returns the variable.
INPUT:
self– the symbolic expression converting fromtarget– (default None) the symbolic expression converting to
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: units.length.foot.convert() 381/1250*meter sage: units.mass.kilogram.convert(units.mass.pound) 100000000/45359237*pound
We do not get anything new by converting an ordinary symbolic variable:
sage: a = var('a') sage: a - a.convert() 0
Raises ValueError if self and target are not convertible:
sage: units.mass.kilogram.convert(units.length.foot) Traceback (most recent call last): ... ValueError: Incompatible units sage: (units.length.meter^2).convert(units.length.foot) Traceback (most recent call last): ... ValueError: Incompatible units
Recognizes derived unit relationships to base units and other derived units:
sage: (units.length.foot/units.time.second^2).convert(units.acceleration.galileo) 762/25*galileo sage: (units.mass.kilogram*units.length.meter/units.time.second^2).convert(units.force.newton) newton sage: (units.length.foot^3).convert(units.area.acre*units.length.inch) 1/3630*(acre*inch) sage: (units.charge.coulomb).convert(units.current.ampere*units.time.second) (ampere*second) sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch) 1290320000000/8896443230521*pounds_per_square_inch
For decimal answers multiply by 1.0:
sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch)*1.0 0.145037737730209*pounds_per_square_inch
Converting temperatures works as well:
sage: s = 68*units.temperature.fahrenheit sage: s.convert(units.temperature.celsius) 20*celsius sage: s.convert() 293.150000000000*kelvin
Trying to multiply temperatures by another unit then converting raises a ValueError:
sage: wrong = 50*units.temperature.celsius*units.length.foot sage: wrong.convert() Traceback (most recent call last): ... ValueError: cannot convert
- cos(hold=False)#
Return the cosine of self.
EXAMPLES:
sage: var('x, y') (x, y) sage: cos(x^2 + y^2) cos(x^2 + y^2) sage: cos(sage.symbolic.constants.pi) -1 sage: cos(SR(1)) cos(1) sage: cos(SR(RealField(150)(1))) 0.54030230586813971740093660744297660373231042
In order to get a numeric approximation use .n():
sage: SR(RR(1)).cos().n() 0.540302305868140 sage: SR(float(1)).cos().n() 0.540302305868140
To prevent automatic evaluation use the
holdargument:sage: pi.cos() -1 sage: pi.cos(hold=True) cos(pi)
This also works using functional notation:
sage: cos(pi,hold=True) cos(pi) sage: cos(pi) -1
To then evaluate again, we use
unhold():sage: a = pi.cos(hold=True); a.unhold() -1
- cosh(hold=False)#
Return cosh of self.
We have \(\cosh(x) = (e^{x} + e^{-x})/2\).
EXAMPLES:
sage: x.cosh() cosh(x) sage: SR(1).cosh() cosh(1) sage: SR(0).cosh() 1 sage: SR(1.0).cosh() 1.54308063481524 sage: maxima('cosh(1.0)') 1.54308063481524... sage: SR(1.00000000000000000000000000).cosh() 1.5430806348152437784779056 sage: SR(RIF(1)).cosh() 1.543080634815244?
To prevent automatic evaluation use the
holdargument:sage: arcsinh(x).cosh() sqrt(x^2 + 1) sage: arcsinh(x).cosh(hold=True) cosh(arcsinh(x))
This also works using functional notation:
sage: cosh(arcsinh(x),hold=True) cosh(arcsinh(x)) sage: cosh(arcsinh(x)) sqrt(x^2 + 1)
To then evaluate again, we use
unhold():sage: a = arcsinh(x).cosh(hold=True); a.unhold() sqrt(x^2 + 1)
- csgn(hold=False)#
Return the sign of self, which is -1 if self < 0, 0 if self == 0, and 1 if self > 0, or unevaluated when self is a nonconstant symbolic expression.
If self is not real, return the complex half-plane (left or right) in which the number lies. If self is pure imaginary, return the sign of the imaginary part of self.
EXAMPLES:
sage: x = var('x') sage: SR(-2).csgn() -1 sage: SR(0.0).csgn() 0 sage: SR(10).csgn() 1 sage: x.csgn() csgn(x) sage: SR(CDF.0).csgn() 1 sage: SR(I).csgn() 1 sage: SR(-I).csgn() -1 sage: SR(1+I).csgn() 1 sage: SR(1-I).csgn() 1 sage: SR(-1+I).csgn() -1 sage: SR(-1-I).csgn() -1
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(I).csgn(hold=True) csgn(I)
- decl_assume(decl)#
- decl_forget(decl)#
- default_variable()#
Return the default variable, which is by definition the first variable in self, or \(x\) is there are no variables in self. The result is cached.
EXAMPLES:
sage: sqrt(2).default_variable() x sage: x, theta, a = var('x, theta, a') sage: f = x^2 + theta^3 - a^x sage: f.default_variable() a
Note that this is the first variable, not the first argument:
sage: f(theta, a, x) = a + theta^3 sage: f.default_variable() a sage: f.variables() (a, theta) sage: f.arguments() (theta, a, x)
- degree(s)#
Return the exponent of the highest power of
sinself.OUTPUT:
An integer
EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100 sage: f.degree(x) 3 sage: f.degree(y) 1 sage: f.degree(sin(x*y)) 1 sage: (x^-3+y).degree(x) 0 sage: (1/x+1/x**2).degree(x) -1
- demoivre(force=False)#
Return this symbolic expression with complex exponentials (optionally all exponentials) replaced by (at least partially) trigonometric/hyperbolic expressions.
EXAMPLES:
sage: x, a, b = SR.var("x, a, b") sage: exp(a + I*b).demoivre() (cos(b) + I*sin(b))*e^a sage: exp(I*x).demoivre() cos(x) + I*sin(x) sage: exp(x).demoivre() e^x sage: exp(x).demoivre(force=True) cosh(x) + sinh(x)
- denominator(normalize=True)#
Return the denominator of this symbolic expression
INPUT:
normalize– (default:True) a boolean.
If
normalizeisTrue, the expression is first normalized to have it as a fraction before getting the denominator.If
normalizeisFalse, the expression is kept and if it is not a quotient, then this will just return 1.See also
normalize(),numerator(),numerator_denominator(),combine()EXAMPLES:
sage: x, y, z, theta = var('x, y, z, theta') sage: f = (sqrt(x) + sqrt(y) + sqrt(z))/(x^10 - y^10 - sqrt(theta)) sage: f.numerator() sqrt(x) + sqrt(y) + sqrt(z) sage: f.denominator() x^10 - y^10 - sqrt(theta) sage: f.numerator(normalize=False) (sqrt(x) + sqrt(y) + sqrt(z)) sage: f.denominator(normalize=False) x^10 - y^10 - sqrt(theta) sage: y = var('y') sage: g = x + y/(x + 2); g x + y/(x + 2) sage: g.numerator(normalize=False) x + y/(x + 2) sage: g.denominator(normalize=False) 1
- derivative(*args)#
Return the derivative of this expressions with respect to the variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global
derivative()function for more details.See also
This is implemented in the
_derivativemethod (see the source code).EXAMPLES:
sage: var("x y") (x, y) sage: t = (x^2+y)^2 sage: t.derivative(x) 4*(x^2 + y)*x sage: t.derivative(x, 2) 12*x^2 + 4*y sage: t.derivative(x, 2, y) 4 sage: t.derivative(y) 2*x^2 + 2*y
If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:
sage: f(x) = x^3 + sin(x) sage: f.derivative() x |--> 3*x^2 + cos(x) sage: f.derivative(2) x |--> 6*x - sin(x)
Some expressions can’t be cleanly differentiated by the chain rule:
sage: _ = var('x', domain='real') sage: _ = var('w z') sage: (x^z).conjugate().diff(x) conjugate(x^(z - 1))*conjugate(z) sage: (w^z).conjugate().diff(w) w^(z - 1)*z*D[0](conjugate)(w^z) sage: atanh(x).real_part().diff(x) -1/(x^2 - 1) sage: atanh(x).imag_part().diff(x) 0 sage: atanh(w).real_part().diff(w) -D[0](real_part)(arctanh(w))/(w^2 - 1) sage: atanh(w).imag_part().diff(w) -D[0](imag_part)(arctanh(w))/(w^2 - 1) sage: abs(log(x)).diff(x) 1/2*(conjugate(log(x))/x + log(x)/x)/abs(log(x)) sage: abs(log(z)).diff(z) 1/2*(conjugate(log(z))/z + log(z)/conjugate(z))/abs(log(z)) sage: forget() sage: t = sin(x+y^2)*tan(x*y) sage: t.derivative(x) (tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y) sage: t.derivative(y) (tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)
sage: h = sin(x)/cos(x) sage: derivative(h,x,x,x) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2 sage: derivative(h,x,3) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: var('x, y') (x, y) sage: u = (sin(x) + cos(y))*(cos(x) - sin(y)) sage: derivative(u,x,y) -cos(x)*cos(y) + sin(x)*sin(y) sage: f = ((x^2+1)/(x^2-1))^(1/4) sage: g = derivative(f, x); g # this is a complex expression -1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4) sage: g.factor() -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
sage: y = var('y') sage: f = y^(sin(x)) sage: derivative(f, x) y^sin(x)*cos(x)*log(y)
sage: g(x) = sqrt(5-2*x) sage: g_3 = derivative(g, x, 3); g_3(2) -3
sage: f = x*e^(-x) sage: derivative(f, 100) x*e^(-x) - 100*e^(-x)
sage: g = 1/(sqrt((x^2-1)*(x+5)^6)) sage: derivative(g, x) -((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)
- diff(*args)#
Return the derivative of this expressions with respect to the variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global
derivative()function for more details.See also
This is implemented in the
_derivativemethod (see the source code).EXAMPLES:
sage: var("x y") (x, y) sage: t = (x^2+y)^2 sage: t.derivative(x) 4*(x^2 + y)*x sage: t.derivative(x, 2) 12*x^2 + 4*y sage: t.derivative(x, 2, y) 4 sage: t.derivative(y) 2*x^2 + 2*y
If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:
sage: f(x) = x^3 + sin(x) sage: f.derivative() x |--> 3*x^2 + cos(x) sage: f.derivative(2) x |--> 6*x - sin(x)
Some expressions can’t be cleanly differentiated by the chain rule:
sage: _ = var('x', domain='real') sage: _ = var('w z') sage: (x^z).conjugate().diff(x) conjugate(x^(z - 1))*conjugate(z) sage: (w^z).conjugate().diff(w) w^(z - 1)*z*D[0](conjugate)(w^z) sage: atanh(x).real_part().diff(x) -1/(x^2 - 1) sage: atanh(x).imag_part().diff(x) 0 sage: atanh(w).real_part().diff(w) -D[0](real_part)(arctanh(w))/(w^2 - 1) sage: atanh(w).imag_part().diff(w) -D[0](imag_part)(arctanh(w))/(w^2 - 1) sage: abs(log(x)).diff(x) 1/2*(conjugate(log(x))/x + log(x)/x)/abs(log(x)) sage: abs(log(z)).diff(z) 1/2*(conjugate(log(z))/z + log(z)/conjugate(z))/abs(log(z)) sage: forget() sage: t = sin(x+y^2)*tan(x*y) sage: t.derivative(x) (tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y) sage: t.derivative(y) (tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)
sage: h = sin(x)/cos(x) sage: derivative(h,x,x,x) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2 sage: derivative(h,x,3) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: var('x, y') (x, y) sage: u = (sin(x) + cos(y))*(cos(x) - sin(y)) sage: derivative(u,x,y) -cos(x)*cos(y) + sin(x)*sin(y) sage: f = ((x^2+1)/(x^2-1))^(1/4) sage: g = derivative(f, x); g # this is a complex expression -1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4) sage: g.factor() -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
sage: y = var('y') sage: f = y^(sin(x)) sage: derivative(f, x) y^sin(x)*cos(x)*log(y)
sage: g(x) = sqrt(5-2*x) sage: g_3 = derivative(g, x, 3); g_3(2) -3
sage: f = x*e^(-x) sage: derivative(f, 100) x*e^(-x) - 100*e^(-x)
sage: g = 1/(sqrt((x^2-1)*(x+5)^6)) sage: derivative(g, x) -((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)
- differentiate(*args)#
Return the derivative of this expressions with respect to the variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global
derivative()function for more details.See also
This is implemented in the
_derivativemethod (see the source code).EXAMPLES:
sage: var("x y") (x, y) sage: t = (x^2+y)^2 sage: t.derivative(x) 4*(x^2 + y)*x sage: t.derivative(x, 2) 12*x^2 + 4*y sage: t.derivative(x, 2, y) 4 sage: t.derivative(y) 2*x^2 + 2*y
If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:
sage: f(x) = x^3 + sin(x) sage: f.derivative() x |--> 3*x^2 + cos(x) sage: f.derivative(2) x |--> 6*x - sin(x)
Some expressions can’t be cleanly differentiated by the chain rule:
sage: _ = var('x', domain='real') sage: _ = var('w z') sage: (x^z).conjugate().diff(x) conjugate(x^(z - 1))*conjugate(z) sage: (w^z).conjugate().diff(w) w^(z - 1)*z*D[0](conjugate)(w^z) sage: atanh(x).real_part().diff(x) -1/(x^2 - 1) sage: atanh(x).imag_part().diff(x) 0 sage: atanh(w).real_part().diff(w) -D[0](real_part)(arctanh(w))/(w^2 - 1) sage: atanh(w).imag_part().diff(w) -D[0](imag_part)(arctanh(w))/(w^2 - 1) sage: abs(log(x)).diff(x) 1/2*(conjugate(log(x))/x + log(x)/x)/abs(log(x)) sage: abs(log(z)).diff(z) 1/2*(conjugate(log(z))/z + log(z)/conjugate(z))/abs(log(z)) sage: forget() sage: t = sin(x+y^2)*tan(x*y) sage: t.derivative(x) (tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y) sage: t.derivative(y) (tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)
sage: h = sin(x)/cos(x) sage: derivative(h,x,x,x) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2 sage: derivative(h,x,3) 8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: var('x, y') (x, y) sage: u = (sin(x) + cos(y))*(cos(x) - sin(y)) sage: derivative(u,x,y) -cos(x)*cos(y) + sin(x)*sin(y) sage: f = ((x^2+1)/(x^2-1))^(1/4) sage: g = derivative(f, x); g # this is a complex expression -1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4) sage: g.factor() -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
sage: y = var('y') sage: f = y^(sin(x)) sage: derivative(f, x) y^sin(x)*cos(x)*log(y)
sage: g(x) = sqrt(5-2*x) sage: g_3 = derivative(g, x, 3); g_3(2) -3
sage: f = x*e^(-x) sage: derivative(f, 100) x*e^(-x) - 100*e^(-x)
sage: g = 1/(sqrt((x^2-1)*(x+5)^6)) sage: derivative(g, x) -((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)
- distribute(recursive=True)#
Distribute some indexed operators over similar operators in order to allow further groupings or simplifications.
Implemented cases (so far):
Symbolic sum of a sum ==> sum of symbolic sums
Integral (definite or not) of a sum ==> sum of integrals.
Symbolic product of a product ==> product of symbolic products.
INPUT:
recursive– (default : True) the distribution proceeds along the subtrees of the expression.
AUTHORS:
Emmanuel Charpentier, Ralf Stephan (05-2017)
- divide_both_sides(x, checksign=None)#
Return a relation obtained by dividing both sides of this relation by
x.Note
The
checksignkeyword argument is currently ignored and is included for backward compatibility reasons only.EXAMPLES:
sage: theta = var('theta') sage: eqn = (x^3 + theta < sin(x*theta)) sage: eqn.divide_both_sides(theta, checksign=False) (x^3 + theta)/theta < sin(theta*x)/theta sage: eqn.divide_both_sides(theta) (x^3 + theta)/theta < sin(theta*x)/theta sage: eqn/theta (x^3 + theta)/theta < sin(theta*x)/theta
- exp(hold=False)#
Return exponential function of self, i.e., e to the power of self.
EXAMPLES:
sage: x.exp() e^x sage: SR(0).exp() 1 sage: SR(1/2).exp() e^(1/2) sage: SR(0.5).exp() 1.64872127070013 sage: math.exp(0.5) 1.6487212707001282 sage: SR(0.5).exp().log() 0.500000000000000 sage: (pi*I).exp() -1
To prevent automatic evaluation use the
holdargument:sage: (pi*I).exp(hold=True) e^(I*pi)
This also works using functional notation:
sage: exp(I*pi,hold=True) e^(I*pi) sage: exp(I*pi) -1
To then evaluate again, we use
unhold():sage: a = (pi*I).exp(hold=True); a.unhold() -1
- expand(side=None)#
Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.
EXAMPLES:
We expand the expression \((x-y)^5\) using both method and functional notation.
sage: x,y = var('x,y') sage: a = (x-y)^5 sage: a.expand() x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5 sage: expand(a) x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
We expand some other expressions:
sage: expand((x-1)^3/(y-1)) x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1) sage: expand((x+sin((x+y)^2))^2) x^2 + 2*x*sin(x^2 + 2*x*y + y^2) + sin(x^2 + 2*x*y + y^2)^2
Observe that
expand()also expands function arguments:sage: f(x) = function('f')(x) sage: fx = f(x*(x+1)); fx f((x + 1)*x) sage: fx.expand() f(x^2 + x)
We can expand individual sides of a relation:
sage: a = (16*x-13)^2 == (3*x+5)^2/2 sage: a.expand() 256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2 sage: a.expand('left') 256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2 sage: a.expand('right') (16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2
- expand_log(algorithm='products')#
Simplify symbolic expression, which can contain logs.
Expands logarithms of powers, logarithms of products and logarithms of quotients. The option
algorithmspecifies which expression types should be expanded.INPUT:
self- expression to be simplifiedalgorithm- (default: ‘products’) optional, governs which expression is expanded. Possible values are‘nothing’ (no expansion),
‘powers’ (log(a^r) is expanded),
‘products’ (like ‘powers’ and also log(a*b) are expanded),
‘all’ (all possible expansion).
See also examples below.
DETAILS: This uses the Maxima simplifier and sets
logexpandoption for this simplifier. From the Maxima documentation: “Logexpand:true causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off. “ALIAS:
log_expand()andexpand_log()are the sameEXAMPLES:
By default powers and products (and quotients) are expanded, but not quotients of integers:
sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4)
To expand also log(3/4) use
algorithm='all':sage: (log(3/4*x^pi)).log_expand('all') pi*log(x) + log(3) - 2*log(2)
To expand only the power use
algorithm='powers'.:sage: (log(x^6)).log_expand('powers') 6*log(x)
The expression
log((3*x)^6)is not expanded withalgorithm='powers', since it is converted into product first:sage: (log((3*x)^6)).log_expand('powers') log(729*x^6)
This shows that the option
algorithmfrom the previous call has no influence to future calls (we changed some default Maxima flag, and have to ensure that this flag has been restored):sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4) sage: (log(3/4*x^pi)).log_expand('all') pi*log(x) + log(3) - 2*log(2) sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4)
AUTHORS:
Robert Marik (11-2009)
- expand_rational(side=None)#
Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.
EXAMPLES:
We expand the expression \((x-y)^5\) using both method and functional notation.
sage: x,y = var('x,y') sage: a = (x-y)^5 sage: a.expand() x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5 sage: expand(a) x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
We expand some other expressions:
sage: expand((x-1)^3/(y-1)) x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1) sage: expand((x+sin((x+y)^2))^2) x^2 + 2*x*sin(x^2 + 2*x*y + y^2) + sin(x^2 + 2*x*y + y^2)^2
Observe that
expand()also expands function arguments:sage: f(x) = function('f')(x) sage: fx = f(x*(x+1)); fx f((x + 1)*x) sage: fx.expand() f(x^2 + x)
We can expand individual sides of a relation:
sage: a = (16*x-13)^2 == (3*x+5)^2/2 sage: a.expand() 256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2 sage: a.expand('left') 256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2 sage: a.expand('right') (16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2
- expand_sum()#
For every symbolic sum in the given expression, try to expand it, symbolically or numerically.
While symbolic sum expressions with constant limits are evaluated immediately on the command line, unevaluated sums of this kind can result from, e.g., substitution of limit variables.
INPUT:
self- symbolic expression
EXAMPLES:
sage: (k,n) = var('k,n') sage: ex = sum(abs(-k*k+n),k,1,n)(n=8); ex sum(abs(-k^2 + 8), k, 1, 8) sage: ex.expand_sum() 162 sage: f(x,k) = sum((2/n)*(sin(n*x)*(-1)^(n+1)), n, 1, k) sage: f(x,2) -2*sum((-1)^n*sin(n*x)/n, n, 1, 2) sage: f(x,2).expand_sum() -sin(2*x) + 2*sin(x)
We can use this to do floating-point approximation as well:
sage: (k,n) = var('k,n') sage: f(n)=sum(sqrt(abs(-k*k+n)),k,1,n) sage: f(n=8) sum(sqrt(abs(-k^2 + 8)), k, 1, 8) sage: f(8).expand_sum() sqrt(41) + sqrt(17) + 2*sqrt(14) + 3*sqrt(7) + 2*sqrt(2) + 3 sage: f(8).expand_sum().n() 31.7752256945384
See github issue #9424 for making the following no longer raise an error:
sage: f(8).n() 31.7752256945384
- expand_trig(full=False, half_angles=False, plus=True, times=True)#
Expand trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in
self.For best results,
selfshould already be expanded.INPUT:
full– (default:False) To enhance user control of simplification, this function expands only one level at a time by default, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the optional parameter full toTrue.half_angles- (default:False) IfTrue, causes half-angles to be simplified away.plus– (default:True) Controls the sum rule; expansion of sums (e.g. \(\sin(x + y)\)) will take place only ifplusisTrue.times– (default:True) Controls the product rule, expansion of products (e.g. \(\sin(2 x)\)) will take place only iftimesisTrue.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: sin(5*x).expand_trig() 5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5 sage: cos(2*x + var('y')).expand_trig() cos(2*x)*cos(y) - sin(2*x)*sin(y)
We illustrate various options to this function:
sage: f = sin(sin(3*cos(2*x))*x) sage: f.expand_trig() sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x) sage: f.expand_trig(full=True) sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x) sage: sin(2*x).expand_trig(times=False) sin(2*x) sage: sin(2*x).expand_trig(times=True) 2*cos(x)*sin(x) sage: sin(2 + x).expand_trig(plus=False) sin(x + 2) sage: sin(2 + x).expand_trig(plus=True) cos(x)*sin(2) + cos(2)*sin(x) sage: sin(x/2).expand_trig(half_angles=False) sin(1/2*x) sage: sin(x/2).expand_trig(half_angles=True) (-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)
If the expression contains terms which are factored, we expand first:
sage: (x, k1, k2) = var('x, k1, k2') sage: cos((k1-k2)*x).expand().expand_trig() cos(k1*x)*cos(k2*x) + sin(k1*x)*sin(k2*x)
ALIAS:
trig_expand()andexpand_trig()are the same
- exponentialize()#
Return this symbolic expression with all circular and hyperbolic functions replaced by their respective exponential expressions.
EXAMPLES:
sage: x = SR.var("x") sage: sin(x).exponentialize() -1/2*I*e^(I*x) + 1/2*I*e^(-I*x) sage: sec(x).exponentialize() 2/(e^(I*x) + e^(-I*x)) sage: tan(x).exponentialize() (-I*e^(I*x) + I*e^(-I*x))/(e^(I*x) + e^(-I*x)) sage: sinh(x).exponentialize() -1/2*e^(-x) + 1/2*e^x sage: sech(x).exponentialize() 2/(e^(-x) + e^x) sage: tanh(x).exponentialize() -(e^(-x) - e^x)/(e^(-x) + e^x)
- factor(dontfactor=None)#
Factor the expression, containing any number of variables or functions, into factors irreducible over the integers.
INPUT:
self- a symbolic expressiondontfactor- list (default: []), a list of variables with respect to which factoring is not to occur. Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the ‘dontfactor’ list.
EXAMPLES:
sage: x,y,z = var('x, y, z') sage: (x^3-y^3).factor() (x^2 + x*y + y^2)*(x - y) sage: factor(-8*y - 4*x + z^2*(2*y + x)) (x + 2*y)*(z + 2)*(z - 2) sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2 sage: F = factor(f/(36*(1 + 2*y + y^2)), dontfactor=[x]); F 1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
If you are factoring a polynomial with rational coefficients (and dontfactor is empty) the factorization is done using Singular instead of Maxima, so the following is very fast instead of dreadfully slow:
sage: var('x,y') (x, y) sage: (x^99 + y^99).factor() (x^60 + x^57*y^3 - x^51*y^9 - x^48*y^12 + x^42*y^18 + x^39*y^21 - x^33*y^27 - x^30*y^30 - x^27*y^33 + x^21*y^39 + x^18*y^42 - x^12*y^48 - x^9*y^51 + x^3*y^57 + y^60)*(x^20 + x^19*y - x^17*y^3 - x^16*y^4 + x^14*y^6 + x^13*y^7 - x^11*y^9 - x^10*y^10 - x^9*y^11 + x^7*y^13 + x^6*y^14 - x^4*y^16 - x^3*y^17 + x*y^19 + y^20)*(x^10 - x^9*y + x^8*y^2 - x^7*y^3 + x^6*y^4 - x^5*y^5 + x^4*y^6 - x^3*y^7 + x^2*y^8 - x*y^9 + y^10)*(x^6 - x^3*y^3 + y^6)*(x^2 - x*y + y^2)*(x + y)
- factor_list(dontfactor=None)#
Return a list of the factors of self, as computed by the factor command.
INPUT:
self– a symbolic expressiondontfactor– see docs forfactor()
Note
If you already have a factored expression and just want to get at the individual factors, use the
_factor_listmethod instead.EXAMPLES:
sage: var('x, y, z') (x, y, z) sage: f = x^3-y^3 sage: f.factor() (x^2 + x*y + y^2)*(x - y)
Notice that the -1 factor is separated out:
sage: f.factor_list() [(x^2 + x*y + y^2, 1), (x - y, 1)]
We factor a fairly straightforward expression:
sage: factor(-8*y - 4*x + z^2*(2*y + x)).factor_list() [(x + 2*y, 1), (z + 2, 1), (z - 2, 1)]
A more complicated example:
sage: var('x, u, v') (x, u, v) sage: f = expand((2*u*v^2-v^2-4*u^3)^2 * (-u)^3 * (x-sin(x))^3) sage: f.factor() -(4*u^3 - 2*u*v^2 + v^2)^2*u^3*(x - sin(x))^3 sage: g = f.factor_list(); g [(4*u^3 - 2*u*v^2 + v^2, 2), (u, 3), (x - sin(x), 3), (-1, 1)]
This function also works for quotients:
sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2 sage: g = f/(36*(1 + 2*y + y^2)); g 1/36*(x^2*y^2 + 2*x*y^2 - x^2 + y^2 - 2*x - 1)/(y^2 + 2*y + 1) sage: g.factor(dontfactor=[x]) 1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1) sage: g.factor_list(dontfactor=[x]) [(x^2 + 2*x + 1, 1), (y + 1, -1), (y - 1, 1), (1/36, 1)]
This example also illustrates that the exponents do not have to be integers:
sage: f = x^(2*sin(x)) * (x-1)^(sqrt(2)*x); f (x - 1)^(sqrt(2)*x)*x^(2*sin(x)) sage: f.factor_list() [(x - 1, sqrt(2)*x), (x, 2*sin(x))]
- factorial(hold=False)#
Return the factorial of self.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: var('x, y') (x, y) sage: SR(5).factorial() 120 sage: x.factorial() factorial(x) sage: (x^2+y^3).factorial() factorial(y^3 + x^2)
To prevent automatic evaluation use the
holdargument:sage: SR(5).factorial(hold=True) factorial(5)
This also works using functional notation:
sage: factorial(5,hold=True) factorial(5) sage: factorial(5) 120
To then evaluate again, we use
unhold():sage: a = SR(5).factorial(hold=True); a.unhold() 120
- factorial_simplify()#
Simplify by combining expressions with factorials, and by expanding binomials into factorials.
ALIAS: factorial_simplify and simplify_factorial are the same
EXAMPLES:
Some examples are relatively clear:
sage: var('n,k') (n, k) sage: f = factorial(n+1)/factorial(n); f factorial(n + 1)/factorial(n) sage: f.simplify_factorial() n + 1
sage: f = factorial(n)*(n+1); f (n + 1)*factorial(n) sage: simplify(f) (n + 1)*factorial(n) sage: f.simplify_factorial() factorial(n + 1)
sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f binomial(n, k)*factorial(k)*factorial(-k + n) sage: f.simplify_factorial() factorial(n)
A more complicated example, which needs further processing:
sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f 1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2) sage: g = f.simplify_factorial(); g 1/2*(x - 1)*x + 1/2*x + 1/2 sage: g.simplify_rational() 1/2*x^2 + 1/2
- find(pattern)#
Find all occurrences of the given pattern in this expression.
Note that once a subexpression matches the pattern, the search does not extend to subexpressions of it.
EXAMPLES:
sage: var('x,y,z,a,b') (x, y, z, a, b) sage: w0 = SR.wild(0); w1 = SR.wild(1) sage: (sin(x)*sin(y)).find(sin(w0)) [sin(y), sin(x)] sage: ((sin(x)+sin(y))*(a+b)).expand().find(sin(w0)) [sin(y), sin(x)] sage: (1+x+x^2+x^3).find(x) [x] sage: (1+x+x^2+x^3).find(x^w0) [x^2, x^3] sage: (1+x+x^2+x^3).find(y) [] # subexpressions of a match are not listed sage: ((x^y)^z).find(w0^w1) [(x^y)^z]
- find_local_maximum(a, b, var=None, tol=1.48e-08, maxfun=500, imaginary_tolerance=1e-08)#
Numerically find a local maximum of the expression
selfon the interval [a,b] (or [b,a]) along with the point at which the maximum is attained.See the documentation for
find_local_minimum()for more details.EXAMPLES:
sage: f = x*cos(x) sage: f.find_local_maximum(0,5) # needs scipy (0.5610963381910451, 0.8603335890...) sage: f.find_local_maximum(0,5, tol=0.1, maxfun=10) # needs scipy (0.561090323458081..., 0.857926501456...)
- find_local_minimum(a, b, var=None, tol=1.48e-08, maxfun=500, imaginary_tolerance=1e-08)#
Numerically find a local minimum of the expression
selfon the interval [a,b] (or [b,a]) and the point at which it attains that minimum. Note thatselfmust be a function of (at most) one variable.INPUT:
a- real number; left endpoint of interval on which to minimizeb- real number; right endpoint of interval on which to minimizevar- variable (default: first variable in self); the variable in self to maximize overtol- positive real (default: 1.48e-08); the convergence tolerancemaxfun- natural number (default: 500); maximum function evaluationsimaginary_tolerance– (default:1e-8); if an imaginary number arises (due, for example, to numerical issues), this tolerance specifies how large it has to be in magnitude before we raise an error. In other words, imaginary parts smaller than this are ignored when we are expecting a real answer.
OUTPUT:
A tuple
(minval, x), whereminval– float. The minimum value that self takes on in the interval[a,b].x– float. The point at which self takes on the minimum value.
EXAMPLES:
sage: # needs scipy sage: f = x*cos(x) sage: f.find_local_minimum(1, 5) (-3.288371395590..., 3.4256184695...) sage: f.find_local_minimum(1, 5, tol=1e-3) (-3.288371361890..., 3.4257507903...) sage: f.find_local_minimum(1, 5, tol=1e-2, maxfun=10) (-3.288370845983..., 3.4250840220...) sage: show(f.plot(0, 20)) # needs sage.plot sage: f.find_local_minimum(1, 15) (-9.477294259479..., 9.5293344109...)
ALGORITHM:
Uses
sage.numerical.optimize.find_local_minimum().AUTHORS:
William Stein (2007-12-07)
- find_root(a, b, var=None, xtol=1e-12, rtol=8.881784197001252e-16, maxiter=100, full_output=False, imaginary_tolerance=1e-08)#
Numerically find a root of self on the closed interval [a,b] (or [b,a]) if possible, where self is a function in the one variable. Note: this function only works in fixed (machine) precision, it is not possible to get arbitrary precision approximations with it.
INPUT:
a, b- endpoints of the intervalvar- optional variablextol, rtol- the routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles.maxiter- integer; if convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.full_output- bool (default: False), if True, also return object that contains information about convergence.imaginary_tolerance– (default:1e-8); if an imaginary number arises (due, for example, to numerical issues), this tolerance specifies how large it has to be in magnitude before we raise an error. In other words, imaginary parts smaller than this are ignored when we are expecting a real answer.
EXAMPLES:
Note that in this example both f(-2) and f(3) are positive, yet we still find a root in that interval:
sage: # needs scipy sage: f = x^2 - 1 sage: f.find_root(-2, 3) 1.0 sage: f.find_root(-2, 3, x) 1.0 sage: z, result = f.find_root(-2, 3, full_output=True) sage: result.converged True sage: result.flag 'converged' sage: result.function_calls 11 sage: result.iterations 10 sage: result.root 1.0
More examples:
sage: (sin(x) + exp(x)).find_root(-10, 10) # needs scipy -0.588532743981862... sage: sin(x).find_root(-1,1) # needs scipy 0.0
This example was fixed along with github issue #4942 - there was an error in the example pi is a root for tan(x), but an asymptote to 1/tan(x) added an example to show handling of both cases:
sage: (tan(x)).find_root(3,3.5) # needs scipy 3.1415926535... sage: (1/tan(x)).find_root(3, 3.5) # needs scipy Traceback (most recent call last): ... NotImplementedError: Brent's method failed to find a zero for f on the interval
An example with a square root:
sage: f = 1 + x + sqrt(x+2); f.find_root(-2,10) # needs scipy -1.618033988749895
Some examples that Ted Kosan came up with:
sage: t = var('t') sage: v = 0.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t))) sage: v.find_root(0, 0.002) # needs scipy 0.001540327067911417...
With this expression, we can see there is a zero very close to the origin:
sage: a = .004*(8*e^(-(300*t)) - 8*e^(-(1200*t)))*(720000*e^(-(300*t)) - 11520000*e^(-(1200*t))) +.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t)))^2 sage: show(plot(a, 0, .002), xmin=0, xmax=.002) # needs sage.plot
It is easy to approximate with
find_root:sage: a.find_root(0,0.002) # needs scipy 0.0004110514049349...
Using solve takes more effort, and even then gives only a solution with free (integer) variables:
sage: a.solve(t) [] sage: b = a.canonicalize_radical(); b (46080.0*e^(1800*t) - 576000.0*e^(900*t) + 737280.0)*e^(-2400*t) sage: b.solve(t) [] sage: b.solve(t, to_poly_solve=True) [t == 1/450*I*pi*z... + 1/900*log(-3/4*sqrt(41) + 25/4), t == 1/450*I*pi*z... + 1/900*log(3/4*sqrt(41) + 25/4)] sage: n(1/900*log(-3/4*sqrt(41) + 25/4)) 0.000411051404934985
We illustrate that root finding is only implemented in one dimension:
sage: x, y = var('x,y') sage: (x-y).find_root(-2,2) Traceback (most recent call last): ... NotImplementedError: root finding currently only implemented in 1 dimension.
- forget()#
Forget the given constraint.
EXAMPLES:
sage: var('x,y') (x, y) sage: forget() sage: assume(x>0, y < 2) sage: assumptions() [x > 0, y < 2] sage: forget(y < 2) sage: assumptions() [x > 0]
- fraction(base_ring)#
Return this expression as element of the algebraic fraction field over the base ring given.
EXAMPLES:
sage: fr = (1/x).fraction(ZZ); fr 1/x sage: parent(fr) Fraction Field of Univariate Polynomial Ring in x over Integer Ring sage: parent(((pi+sqrt(2)/x).fraction(SR))) Fraction Field of Univariate Polynomial Ring in x over Symbolic Ring sage: parent(((pi+sqrt(2))/x).fraction(SR)) Fraction Field of Univariate Polynomial Ring in x over Symbolic Ring sage: y = var('y') sage: fr = ((3*x^5 - 5*y^5)^7/(x*y)).fraction(GF(7)); fr (3*x^35 + 2*y^35)/(x*y) sage: parent(fr) Fraction Field of Multivariate Polynomial Ring in x, y over Finite Field of size 7
- free_variables()#
Return sorted tuple of unbound variables that occur in this expression.
EXAMPLES:
sage: (x,y,z) = var('x,y,z') sage: (x+y).free_variables() (x, y) sage: (2*x).free_variables() (x,) sage: (x^y).free_variables() (x, y) sage: sin(x+y^z).free_variables() (x, y, z) sage: _ = function('f') sage: e = limit( f(x,y), x=0 ); e limit(f(x, y), x, 0) sage: e.free_variables() (y,)
- full_simplify()#
Apply
simplify_factorial(),simplify_rectform(),simplify_trig(),simplify_rational(), and thenexpand_sum()to self (in that order).ALIAS:
simplify_fullandfull_simplifyare the same.EXAMPLES:
sage: f = sin(x)^2 + cos(x)^2 sage: f.simplify_full() 1
sage: f = sin(x/(x^2 + x)) sage: f.simplify_full() sin(1/(x + 1))
sage: var('n,k') (n, k) sage: f = binomial(n,k)*factorial(k)*factorial(n-k) sage: f.simplify_full() factorial(n)
- function(*args)#
Return a callable symbolic expression with the given variables.
EXAMPLES:
We will use several symbolic variables in the examples below:
sage: var('x, y, z, t, a, w, n') (x, y, z, t, a, w, n)
sage: u = sin(x) + x*cos(y) sage: g = u.function(x,y) sage: g(x,y) x*cos(y) + sin(x) sage: g(t,z) t*cos(z) + sin(t) sage: g(x^2, x^y) x^2*cos(x^y) + sin(x^2)
sage: f = (x^2 + sin(a*w)).function(a,x,w); f (a, x, w) |--> x^2 + sin(a*w) sage: f(1,2,3) sin(3) + 4
Using the
function()method we can obtain the above function \(f\), but viewed as a function of different variables:sage: h = f.function(w,a); h (w, a) |--> x^2 + sin(a*w)
This notation also works:
sage: h(w,a) = f sage: h (w, a) |--> x^2 + sin(a*w)
You can even make a symbolic expression \(f\) into a function by writing
f(x,y) = f:sage: f = x^n + y^n; f x^n + y^n sage: f(x,y) = f sage: f (x, y) |--> x^n + y^n sage: f(2,3) 3^n + 2^n
- gamma(hold=False)#
Return the Gamma function evaluated at self.
EXAMPLES:
sage: x = var('x') sage: x.gamma() gamma(x) sage: SR(2).gamma() 1 sage: SR(10).gamma() 362880 sage: SR(10.0r).gamma() # For ARM: rel tol 2e-15 362880.0 sage: SR(CDF(1,1)).gamma() 0.49801566811835607 - 0.15494982830181067*I
sage: gp('gamma(1+I)') 0.4980156681183560427136911175 - 0.1549498283018106851249551305*I # 32-bit 0.49801566811835604271369111746219809195 - 0.15494982830181068512495513048388660520*I # 64-bit
We plot the familiar plot of this log-convex function:
sage: plot(gamma(x), -6, 4).show(ymin=-3, ymax=3) # needs sage.plot
To prevent automatic evaluation use the
holdargument:sage: SR(1/2).gamma() sqrt(pi) sage: SR(1/2).gamma(hold=True) gamma(1/2)
This also works using functional notation:
sage: gamma(1/2,hold=True) gamma(1/2) sage: gamma(1/2) sqrt(pi)
To then evaluate again, we use
unhold():sage: a = SR(1/2).gamma(hold=True); a.unhold() sqrt(pi)
- gamma_normalize()#
Return the expression with any gamma functions that have a common base converted to that base.
Additionally the expression is normalized so any fractions can be simplified through cancellation.
EXAMPLES:
sage: m,n = var('m n', domain='integer') sage: (gamma(n+2)/gamma(n)).gamma_normalize() (n + 1)*n sage: (gamma(n+2)*gamma(n)).gamma_normalize() (n + 1)*n*gamma(n)^2 sage: (gamma(n+2)*gamma(m-1)/gamma(n)/gamma(m+1)).gamma_normalize() (n + 1)*n/((m - 1)*m)
Check that github issue #22826 is fixed:
sage: _ = var('n') sage: (n-1).gcd(n+1) 1 sage: ex = (n-1)^2*gamma(2*n+5)/gamma(n+3) + gamma(2*n+3)/gamma(n+1) sage: ex.gamma_normalize() (4*n^3 - 2*n^2 - 7*n + 7)*gamma(2*n + 3)/((n + 1)*gamma(n + 1))
- gcd(b)#
Return the symbolic gcd of
selfandb.Note that the polynomial GCD is unique up to the multiplication by an invertible constant. The following examples make sure all results are caught.
EXAMPLES:
sage: var('x,y') (x, y) sage: SR(10).gcd(SR(15)) 5 sage: (x^3 - 1).gcd(x-1) / (x-1) in QQ True sage: (x^3 - 1).gcd(x^2+x+1) / (x^2+x+1) in QQ True sage: (x^3 - x^2*pi + x^2 - pi^2).gcd(x-pi) / (x-pi) in QQ True sage: gcd(sin(x)^2 + sin(x), sin(x)^2 - 1) / (sin(x) + 1) in QQ True sage: gcd(x^3 - y^3, x-y) / (x-y) in QQ True sage: gcd(x^100-y^100, x^10-y^10) / (x^10-y^10) in QQ True sage: r = gcd(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) ) sage: r / (x^5 - 17*y + 2/3) in QQ True
Embedded Sage objects of all kinds get basic support. Note that full algebraic GCD is not implemented yet:
sage: gcd(I - I*x, x^2 - 1) x - 1 sage: gcd(I + I*x, x^2 - 1) x + 1 sage: alg = SR(QQbar(sqrt(2) + I*sqrt(3))) sage: gcd(alg + alg*x, x^2 - 1) # known bug (trac #28489) x + 1 sage: gcd(alg - alg*x, x^2 - 1) # known bug (trac #28489) x - 1 sage: sqrt2 = SR(QQbar(sqrt(2))) sage: gcd(sqrt2 + x, x^2 - 2) # known bug 1
- gosper_sum(*args)#
Return the summation of this hypergeometric expression using Gosper’s algorithm.
INPUT:
a symbolic expression that may contain rational functions, powers, factorials, gamma function terms, binomial coefficients, and Pochhammer symbols that are rational-linear in their arguments
the main variable and, optionally, summation limits
EXAMPLES:
sage: a,b,k,m,n = var('a b k m n') sage: SR(1).gosper_sum(n) n sage: SR(1).gosper_sum(n,5,8) 4 sage: n.gosper_sum(n) 1/2*(n - 1)*n sage: n.gosper_sum(n,0,5) 15 sage: n.gosper_sum(n,0,m) 1/2*(m + 1)*m sage: n.gosper_sum(n,a,b) -1/2*(a + b)*(a - b - 1)
sage: (factorial(m + n)/factorial(n)).gosper_sum(n) n*factorial(m + n)/((m + 1)*factorial(n)) sage: (binomial(m + n, n)).gosper_sum(n) n*binomial(m + n, n)/(m + 1) sage: (binomial(m + n, n)).gosper_sum(n, 0, a) (a + m + 1)*binomial(a + m, a)/(m + 1) sage: (binomial(m + n, n)).gosper_sum(n, 0, 5) 1/120*(m + 6)*(m + 5)*(m + 4)*(m + 3)*(m + 2) sage: (rising_factorial(a,n)/rising_factorial(b,n)).gosper_sum(n) (b + n - 1)*gamma(a + n)*gamma(b)/((a - b + 1)*gamma(a)*gamma(b + n)) sage: factorial(n).gosper_term(n) Traceback (most recent call last): ... ValueError: expression not Gosper-summable
- gosper_term(n)#
Return Gosper’s hypergeometric term for
self.Suppose
f``=``selfis a hypergeometric term such that:\[s_n = \sum_{k=0}^{n-1} f_k\]and \(f_k\) doesn’t depend on \(n\). Return a hypergeometric term \(g_n\) such that \(g_{n+1} - g_n = f_n\).
EXAMPLES:
sage: _ = var('n') sage: SR(1).gosper_term(n) n sage: n.gosper_term(n) 1/2*(n^2 - n)/n sage: (n*factorial(n)).gosper_term(n) 1/n sage: factorial(n).gosper_term(n) Traceback (most recent call last): ... ValueError: expression not Gosper-summable
- gradient(variables=None)#
Compute the gradient of a symbolic function.
This function returns a vector whose components are the derivatives of the original function with respect to the arguments of the original function. Alternatively, you can specify the variables as a list.
EXAMPLES:
sage: x,y = var('x y') sage: f = x^2+y^2 sage: f.gradient() (2*x, 2*y) sage: g(x,y) = x^2+y^2 sage: g.gradient() (x, y) |--> (2*x, 2*y) sage: n = var('n') sage: f(x,y) = x^n+y^n sage: f.gradient() (x, y) |--> (n*x^(n - 1), n*y^(n - 1)) sage: f.gradient([y,x]) (x, y) |--> (n*y^(n - 1), n*x^(n - 1))
See also
gradient()of scalar fields on Euclidean spaces (and more generally pseudo-Riemannian manifolds), in particular for computing the gradient in curvilinear coordinates.
- has(pattern)#
EXAMPLES:
sage: var('x,y,a'); w0 = SR.wild(); w1 = SR.wild() (x, y, a) sage: (x*sin(x + y + 2*a)).has(y) True
Here “x+y” is not a subexpression of “x+y+2*a” (which has the subexpressions “x”, “y” and “2*a”):
sage: (x*sin(x + y + 2*a)).has(x+y) False sage: (x*sin(x + y + 2*a)).has(x + y + w0) True
The following fails because “2*(x+y)” automatically gets converted to “2*x+2*y” of which “x+y” is not a subexpression:
sage: (x*sin(2*(x+y) + 2*a)).has(x+y) False
Although x^1==x and x^0==1, neither “x” nor “1” are actually of the form “x^something”:
sage: (x+1).has(x^w0) False
Here is another possible pitfall, where the first expression matches because the term “-x” has the form “(-1)*x” in GiNaC. To check whether a polynomial contains a linear term you should use the coeff() function instead.
sage: (4*x^2 - x + 3).has(w0*x) True sage: (4*x^2 + x + 3).has(w0*x) False sage: (4*x^2 + x + 3).has(x) True sage: (4*x^2 - x + 3).coefficient(x,1) -1 sage: (4*x^2 + x + 3).coefficient(x,1) 1
- has_wild()#
Return
Trueif this expression contains a wildcard.EXAMPLES:
sage: (1 + x^2).has_wild() False sage: (SR.wild(0) + x^2).has_wild() True sage: SR.wild(0).has_wild() True
- hessian()#
Compute the hessian of a function. This returns a matrix components are the 2nd partial derivatives of the original function.
EXAMPLES:
sage: x,y = var('x y') sage: f = x^2+y^2 sage: f.hessian() [2 0] [0 2] sage: g(x,y) = x^2+y^2 sage: g.hessian() [(x, y) |--> 2 (x, y) |--> 0] [(x, y) |--> 0 (x, y) |--> 2]
- horner(x)#
Rewrite this expression as a polynomial in Horner form in
x.EXAMPLES:
sage: add((i+1)*x^i for i in range(5)).horner(x) (((5*x + 4)*x + 3)*x + 2)*x + 1 sage: x, y, z = SR.var('x,y,z') sage: (x^5 + y*cos(x) + z^3 + (x + y)^2 + y^x).horner(x) z^3 + ((x^3 + 1)*x + 2*y)*x + y^2 + y*cos(x) + y^x sage: expr = sin(5*x).expand_trig(); expr 5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5 sage: expr.horner(sin(x)) (5*cos(x)^4 - (10*cos(x)^2 - sin(x)^2)*sin(x)^2)*sin(x) sage: expr.horner(cos(x)) sin(x)^5 + 5*(cos(x)^2*sin(x) - 2*sin(x)^3)*cos(x)^2
- hypergeometric_simplify(algorithm='maxima')#
Simplify an expression containing hypergeometric or confluent hypergeometric functions.
INPUT:
algorithm– (default:'maxima') the algorithm to use for for simplification. Implemented are'maxima', which uses Maxima’shgfredfunction, and'sage', which uses an algorithm implemented in the hypergeometric module
ALIAS:
hypergeometric_simplify()andsimplify_hypergeometric()are the sameEXAMPLES:
sage: hypergeometric((5, 4), (4, 1, 2, 3), ....: x).simplify_hypergeometric() 1/144*x^2*hypergeometric((), (3, 4), x) +... 1/3*x*hypergeometric((), (2, 3), x) + hypergeometric((), (1, 2), x) sage: (2*hypergeometric((), (), x)).simplify_hypergeometric() 2*e^x sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1) # not tested, unstable ....: .simplify_hypergeometric()) laguerre(-laguerre(-e^x, x), x) sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1) # not tested, unstable ....: .simplify_hypergeometric(algorithm='sage')) hypergeometric((hypergeometric((e^x,), (1,), x),), (1,), x) sage: hypergeometric_M(1, 3, x).simplify_hypergeometric() -2*((x + 1)*e^(-x) - 1)*e^x/x^2 sage: (2 * hypergeometric_U(1, 3, x)).simplify_hypergeometric() 2*(x + 1)/x^2
- imag(hold=False)#
Return the imaginary part of this symbolic expression.
EXAMPLES:
sage: sqrt(-2).imag_part() sqrt(2)
We simplify \(\ln(\exp(z))\) to \(z\). This should only be for \(-\pi<{\rm Im}(z)<=\pi\), but Maxima does not have a symbolic imaginary part function, so we cannot use
assumeto assume that first:sage: z = var('z') sage: f = log(exp(z)) sage: f log(e^z) sage: f.simplify() z sage: forget()
A more symbolic example:
sage: var('a, b') (a, b) sage: f = log(a + b*I) sage: f.imag_part() arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(I).imag_part() 1 sage: SR(I).imag_part(hold=True) imag_part(I)
This also works using functional notation:
sage: imag_part(I, hold=True) imag_part(I) sage: imag_part(SR(I)) 1
To then evaluate again, we use
unhold():sage: a = SR(I).imag_part(hold=True); a.unhold() 1
- imag_part(hold=False)#
Return the imaginary part of this symbolic expression.
EXAMPLES:
sage: sqrt(-2).imag_part() sqrt(2)
We simplify \(\ln(\exp(z))\) to \(z\). This should only be for \(-\pi<{\rm Im}(z)<=\pi\), but Maxima does not have a symbolic imaginary part function, so we cannot use
assumeto assume that first:sage: z = var('z') sage: f = log(exp(z)) sage: f log(e^z) sage: f.simplify() z sage: forget()
A more symbolic example:
sage: var('a, b') (a, b) sage: f = log(a + b*I) sage: f.imag_part() arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(I).imag_part() 1 sage: SR(I).imag_part(hold=True) imag_part(I)
This also works using functional notation:
sage: imag_part(I, hold=True) imag_part(I) sage: imag_part(SR(I)) 1
To then evaluate again, we use
unhold():sage: a = SR(I).imag_part(hold=True); a.unhold() 1
- implicit_derivative(Y, X, n=1)#
Return the \(n\)-th derivative of \(Y\) with respect to \(X\) given implicitly by this expression.
INPUT:
Y– The dependent variable of the implicit expression.X– The independent variable with respect to which the derivative is taken.n– (default: 1) the order of the derivative.
EXAMPLES:
sage: var('x, y') (x, y) sage: f = cos(x)*sin(y) sage: f.implicit_derivative(y, x) sin(x)*sin(y)/(cos(x)*cos(y)) sage: g = x*y^2 sage: g.implicit_derivative(y, x, 3) -1/4*(y + 2*y/x)/x^2 + 1/4*(2*y^2/x - y^2/x^2)/(x*y) - 3/4*y/x^3
It is an error to not include an independent variable term in the expression:
sage: (cos(x)*sin(x)).implicit_derivative(y, x) Traceback (most recent call last): ... ValueError: Expression cos(x)*sin(x) contains no y terms
- integral(*args, **kwds)#
Compute the integral of
self.Please see
sage.symbolic.integration.integral.integrate()for more details.EXAMPLES:
sage: sin(x).integral(x,0,3) -cos(3) + 1 sage: sin(x).integral(x) -cos(x)
- integrate(*args, **kwds)#
Compute the integral of
self.Please see
sage.symbolic.integration.integral.integrate()for more details.EXAMPLES:
sage: sin(x).integral(x,0,3) -cos(3) + 1 sage: sin(x).integral(x) -cos(x)
- inverse_laplace(t, s)#
Return inverse Laplace transform of
self.See
sage.calculus.calculus.inverse_laplaceEXAMPLES:
sage: var('w, m') (w, m) sage: f = (1/(w^2+10)).inverse_laplace(w, m); f 1/10*sqrt(10)*sin(sqrt(10)*m)
- is_algebraic()#
Return True if this expression is known to be algebraic.
EXAMPLES:
sage: sqrt(2).is_algebraic() True sage: (5*sqrt(2)).is_algebraic() True sage: (sqrt(2) + 2^(1/3) - 1).is_algebraic() True sage: (I*golden_ratio + sqrt(2)).is_algebraic() True sage: (sqrt(2) + pi).is_algebraic() False sage: SR(QQ(2/3)).is_algebraic() True sage: SR(1.2).is_algebraic() False sage: complex_root_of(x^3 - x^2 - x - 1, 0).is_algebraic() True
- is_callable()#
Return
Trueifselfis a callable symbolic expression.EXAMPLES:
sage: var('a x y z') (a, x, y, z) sage: f(x, y) = a + 2*x + 3*y + z sage: f.is_callable() True sage: (a+2*x).is_callable() False
- is_constant()#
Return whether this symbolic expression is a constant.
A symbolic expression is constant if it does not contain any variables.
EXAMPLES:
sage: pi.is_constant() True sage: SR(1).is_constant() True sage: SR(2).is_constant() True sage: log(2).is_constant() True sage: SR(I).is_constant() True sage: x.is_constant() False
- is_exact()#
Return
Trueif this expression only contains exact numerical coefficients.EXAMPLES:
sage: x, y = var('x, y') sage: (x+y-1).is_exact() True sage: (x+y-1.9).is_exact() False sage: x.is_exact() True sage: pi.is_exact() True sage: (sqrt(x-y) - 2*x + 1).is_exact() True sage: ((x-y)^0.5 - 2*x + 1).is_exact() False
- is_infinity()#
Return
Trueifselfis an infinite expression.EXAMPLES:
sage: SR(oo).is_infinity() True sage: x.is_infinity() False
- is_integer()#
Return True if this expression is known to be an integer.
EXAMPLES:
sage: SR(5).is_integer() True
- is_negative()#
Return True if this expression is known to be negative.
EXAMPLES:
sage: SR(-5).is_negative() True
Check if we can correctly deduce negativity of mul objects:
sage: t0 = SR.symbol("t0", domain='positive') sage: t0.is_negative() False sage: (-t0).is_negative() True sage: (-pi).is_negative() True
Assumptions on symbols are handled correctly:
sage: y = var('y') sage: assume(y < 0) sage: y.is_positive() False sage: y.is_negative() True sage: forget()
- is_negative_infinity()#
Return
Trueifselfis a negative infinite expression.EXAMPLES:
sage: SR(oo).is_negative_infinity() False sage: SR(-oo).is_negative_infinity() True sage: x.is_negative_infinity() False
- is_numeric()#
A Pynac numeric is an object you can do arithmetic with that is not a symbolic variable, function, or constant. Return True if this expression only consists of a numeric object.
EXAMPLES:
sage: SR(1).is_numeric() True sage: x.is_numeric() False sage: pi.is_numeric() False sage: sin(x).is_numeric() False
- is_polynomial(var)#
Return
Trueifselfis a polynomial in the given variable.EXAMPLES:
sage: var('x,y,z') (x, y, z) sage: t = x^2 + y; t x^2 + y sage: t.is_polynomial(x) True sage: t.is_polynomial(y) True sage: t.is_polynomial(z) True sage: t = sin(x) + y; t y + sin(x) sage: t.is_polynomial(x) False sage: t.is_polynomial(y) True sage: t.is_polynomial(sin(x)) True
- is_positive()#
Return True if this expression is known to be positive.
EXAMPLES:
sage: t0 = SR.symbol("t0", domain='positive') sage: t0.is_positive() True sage: t0.is_negative() False sage: t0.is_real() True sage: t1 = SR.symbol("t1", domain='positive') sage: (t0*t1).is_positive() True sage: (t0 + t1).is_positive() True sage: (t0*x).is_positive() False
sage: forget() sage: assume(x>0) sage: x.is_positive() True sage: cosh(x).is_positive() True sage: f = function('f')(x) sage: assume(f>0) sage: f.is_positive() True sage: forget()
- is_positive_infinity()#
Return
Trueifselfis a positive infinite expression.EXAMPLES:
sage: SR(oo).is_positive_infinity() True sage: SR(-oo).is_positive_infinity() False sage: x.is_infinity() False
- is_rational_expression()#
Return
Trueif this expression if a rational expression, i.e., a quotient of polynomials.EXAMPLES:
sage: var('x y z') (x, y, z) sage: ((x + y + z)/(1 + x^2)).is_rational_expression() True sage: ((1 + x + y)^10).is_rational_expression() True sage: ((1/x + z)^5 - 1).is_rational_expression() True sage: (1/(x + y)).is_rational_expression() True sage: (exp(x) + 1).is_rational_expression() False sage: (sin(x*y) + z^3).is_rational_expression() False sage: (exp(x) + exp(-x)).is_rational_expression() False
- is_real()#
Return True if this expression is known to be a real number.
EXAMPLES:
sage: t0 = SR.symbol("t0", domain='real') sage: t0.is_real() True sage: t0.is_positive() False sage: t1 = SR.symbol("t1", domain='positive') sage: (t0+t1).is_real() True sage: (t0+x).is_real() False sage: (t0*t1).is_real() True sage: t2 = SR.symbol("t2", domain='positive') sage: (t1**t2).is_real() True sage: (t0*x).is_real() False sage: (t0^t1).is_real() False sage: (t1^t2).is_real() True sage: gamma(pi).is_real() True sage: cosh(-3).is_real() True sage: cos(exp(-3) + log(2)).is_real() True sage: gamma(t1).is_real() True sage: (x^pi).is_real() False sage: (cos(exp(t0) + log(t1))^8).is_real() True sage: cos(I + 1).is_real() False sage: sin(2 - I).is_real() False sage: (2^t0).is_real() True
The following is real, but we cannot deduce that.:
sage: (x*x.conjugate()).is_real() False
Assumption of real has the same effect as setting the domain:
sage: forget() sage: assume(x, 'real') sage: x.is_real() True sage: cosh(x).is_real() True sage: forget()
The real domain is also set with the integer domain:
sage: SR.var('x', domain='integer').is_real() True
- is_relational()#
Return
Trueifselfis a relational expression.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 == x^2 - 2*x + 3 sage: eqn.is_relational() True sage: sin(x).is_relational() False
- is_square()#
Return
Trueifselfis the square of another symbolic expression.This is
Truefor all constant, non-relational expressions (containing no variables or comparison), and not implemented otherwise.EXAMPLES:
sage: SR(4).is_square() True sage: SR(5).is_square() True sage: pi.is_square() True sage: x.is_square() Traceback (most recent call last): ... NotImplementedError: is_square() not implemented for non-constant or relational elements of Symbolic Ring sage: r = SR(4) == SR(5) sage: r.is_square() Traceback (most recent call last): ... NotImplementedError: is_square() not implemented for non-constant or relational elements of Symbolic Ring
- is_symbol()#
Return True if this symbolic expression consists of only a symbol, i.e., a symbolic variable.
EXAMPLES:
sage: x.is_symbol() True sage: var('y') y sage: y.is_symbol() True sage: (x*y).is_symbol() False sage: pi.is_symbol() False
sage: ((x*y)/y).is_symbol() True sage: (x^y).is_symbol() False
- is_terminating_series()#
Return True if
selfis a series without order term.A series is terminating if it can be represented exactly, without requiring an order term. You can explicitly request terminating series by setting the order to positive infinity.
OUTPUT:
Boolean. Whether
selfwas constructed byseries()and has no order term.EXAMPLES:
sage: (x^5+x^2+1).series(x, +oo) 1 + 1*x^2 + 1*x^5 sage: (x^5+x^2+1).series(x,+oo).is_terminating_series() True sage: SR(5).is_terminating_series() False sage: var('x') x sage: x.is_terminating_series() False sage: exp(x).series(x,10).is_terminating_series() False
- is_trivial_zero()#
Check if this expression is trivially equal to zero without any simplification.
This method is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable.
EXAMPLES:
sage: SR(0).is_trivial_zero() True sage: SR(0.0).is_trivial_zero() True sage: SR(float(0.0)).is_trivial_zero() True sage: (SR(1)/2^1000).is_trivial_zero() False sage: SR(1./2^10000).is_trivial_zero() False
The
is_zero()method is more capable:sage: t = pi + (pi - 1)*pi - pi^2 sage: t.is_trivial_zero() False sage: t.is_zero() True sage: t = pi + x*pi + (pi - 1 - x)*pi - pi^2 sage: t.is_zero() True sage: u = sin(x)^2 + cos(x)^2 - 1 sage: u.is_trivial_zero() False sage: u.is_zero() True
- is_trivially_equal(other)#
Check if this expression is trivially equal to the argument expression, without any simplification.
Note that the expressions may still be subject to immediate evaluation.
This method is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable.
EXAMPLES:
sage: (x^2).is_trivially_equal(x^2) True sage: ((x+1)^2 - 2*x - 1).is_trivially_equal(x^2) False sage: (x*(x+1)).is_trivially_equal((x+1)*x) True sage: (x^2 + x).is_trivially_equal((x+1)*x) False sage: ((x+1)*(x+1)).is_trivially_equal((x+1)^2) True sage: (x^2 + 2*x + 1).is_trivially_equal((x+1)^2) False sage: (x^-1).is_trivially_equal(1/x) True sage: (x/x^2).is_trivially_equal(1/x) True sage: ((x^2+x) / (x+1)).is_trivially_equal(1/x) False
- is_unit()#
Return True if this expression is a unit of the symbolic ring.
Note that a proof may be attempted to get the result. To avoid this use
(ex-1).is_trivial_zero().EXAMPLES:
sage: SR(1).is_unit() True sage: SR(-1).is_unit() True sage: SR(0).is_unit() False
- iterator()#
Return an iterator over the operands of this expression.
EXAMPLES:
sage: x,y,z = var('x,y,z') sage: list((x+y+z).iterator()) [x, y, z] sage: list((x*y*z).iterator()) [x, y, z] sage: list((x^y*z*(x+y)).iterator()) [x + y, x^y, z]
Note that symbols, constants and numeric objects do not have operands, so the iterator function raises an error in these cases:
sage: x.iterator() Traceback (most recent call last): ... ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands sage: pi.iterator() Traceback (most recent call last): ... ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands sage: SR(5).iterator() Traceback (most recent call last): ... ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
- laplace(t, s)#
Return Laplace transform of
self.See
sage.calculus.calculus.laplaceEXAMPLES:
sage: var('x,s,z') (x, s, z) sage: (z + exp(x)).laplace(x, s) z/s + 1/(s - 1)
- laurent_polynomial(base_ring=None, ring=None)#
Return this symbolic expression as a Laurent polynomial over the given base ring, if possible.
INPUT:
base_ring- (optional) the base ring for the polynomialring- (optional) the parent for the polynomial
You can specify either the base ring (
base_ring) you want the output Laurent polynomial to be over, or you can specify the full laurent polynomial ring (ring) you want the output laurent polynomial to be an element of.EXAMPLES:
sage: f = x^2 -2/3/x + 1 sage: f.laurent_polynomial(QQ) -2/3*x^-1 + 1 + x^2 sage: f.laurent_polynomial(GF(19)) 12*x^-1 + 1 + x^2
- lcm(b)#
Return the lcm of
selfandb.The lcm is computed from the gcd of
selfandbimplicitly from the relation self * b = gcd(self, b) * lcm(self, b).Note
In agreement with the convention in use for integers, if self * b == 0, then gcd(self, b) == max(self, b) and lcm(self, b) == 0.
Note
Since the polynomial lcm is computed from the gcd, and the polynomial gcd is unique up to a constant factor (which can be negative), the polynomial lcm is unique up to a factor of -1.
EXAMPLES:
sage: var('x,y') (x, y) sage: SR(10).lcm(SR(15)) 30 sage: (x^3 - 1).lcm(x-1) x^3 - 1 sage: (x^3 - 1).lcm(x^2+x+1) x^3 - 1 sage: (x^3 - sage.symbolic.constants.pi).lcm(x-sage.symbolic.constants.pi) (pi - x^3)*(pi - x) sage: lcm(x^3 - y^3, x-y) / (x^3 - y^3) in [1,-1] True sage: lcm(x^100-y^100, x^10-y^10) / (x^100 - y^100) in [1,-1] True sage: a = expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ) sage: b = expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3) ) sage: gcd(a,b) * lcm(a,b) / (a * b) in [1,-1] True
The result is not automatically simplified:
sage: ex = lcm(sin(x)^2 - 1, sin(x)^2 + sin(x)); ex (sin(x)^2 + sin(x))*(sin(x)^2 - 1)/(sin(x) + 1) sage: ex.simplify_full() sin(x)^3 - sin(x)
- leading_coeff(s)#
Return the leading coefficient of
sinself.EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100 sage: f.leading_coefficient(x) sin(x*y) sage: f.leading_coefficient(y) x sage: f.leading_coefficient(sin(x*y)) x^3 + 2/x
- leading_coefficient(s)#
Return the leading coefficient of
sinself.EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100 sage: f.leading_coefficient(x) sin(x*y) sage: f.leading_coefficient(y) x sage: f.leading_coefficient(sin(x*y)) x^3 + 2/x
- left()#
If
selfis a relational expression, return the left hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 == x^2 - 2*x + 3 sage: eqn.left_hand_side() (x - 1)^2 sage: eqn.lhs() (x - 1)^2 sage: eqn.left() (x - 1)^2
- left_hand_side()#
If
selfis a relational expression, return the left hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 == x^2 - 2*x + 3 sage: eqn.left_hand_side() (x - 1)^2 sage: eqn.lhs() (x - 1)^2 sage: eqn.left() (x - 1)^2
- lhs()#
If
selfis a relational expression, return the left hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 == x^2 - 2*x + 3 sage: eqn.left_hand_side() (x - 1)^2 sage: eqn.lhs() (x - 1)^2 sage: eqn.left() (x - 1)^2
- limit(*args, **kwds)#
Return a symbolic limit.
See
sage.calculus.calculus.limitEXAMPLES:
sage: (sin(x)/x).limit(x=0) 1
- list(x=None)#
Return the coefficients of this symbolic expression as a polynomial in x.
INPUT:
x– optional variable.
OUTPUT:
A list of expressions where the
n-th element is the coefficient ofx^nwhen self is seen as polynomial inx.EXAMPLES:
sage: var('x, y, a') (x, y, a) sage: (x^5).list() [0, 0, 0, 0, 0, 1] sage: p = x - x^3 + 5/7*x^5 sage: p.list() [0, 1, 0, -1, 0, 5/7] sage: p = expand((x-a*sqrt(2))^2 + x + 1); p -2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1 sage: p.list(a) [x^2 + x + 1, -2*sqrt(2)*x, 2] sage: s = (1/(1-x)).series(x,6); s 1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 + Order(x^6) sage: s.list() [1, 1, 1, 1, 1, 1]
- log(b=None, hold=False)#
Return the logarithm of self.
EXAMPLES:
sage: x, y = var('x, y') sage: x.log() log(x) sage: (x^y + y^x).log() log(x^y + y^x) sage: SR(0).log() -Infinity sage: SR(-1).log() I*pi sage: SR(1).log() 0 sage: SR(1/2).log() log(1/2) sage: SR(0.5).log() -0.693147180559945 sage: SR(0.5).log().exp() 0.500000000000000 sage: math.log(0.5) -0.6931471805599453 sage: plot(lambda x: SR(x).log(), 0.1,10) # needs sage.plot Graphics object consisting of 1 graphics primitive
To prevent automatic evaluation use the
holdargument:sage: I.log() 1/2*I*pi sage: I.log(hold=True) log(I)
To then evaluate again, we use
unhold():sage: a = I.log(hold=True); a.unhold() 1/2*I*pi
The
holdparameter also works in functional notation:sage: log(-1,hold=True) log(-1) sage: log(-1) I*pi
- log_expand(algorithm='products')#
Simplify symbolic expression, which can contain logs.
Expands logarithms of powers, logarithms of products and logarithms of quotients. The option
algorithmspecifies which expression types should be expanded.INPUT:
self- expression to be simplifiedalgorithm- (default: ‘products’) optional, governs which expression is expanded. Possible values are‘nothing’ (no expansion),
‘powers’ (log(a^r) is expanded),
‘products’ (like ‘powers’ and also log(a*b) are expanded),
‘all’ (all possible expansion).
See also examples below.
DETAILS: This uses the Maxima simplifier and sets
logexpandoption for this simplifier. From the Maxima documentation: “Logexpand:true causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off. “ALIAS:
log_expand()andexpand_log()are the sameEXAMPLES:
By default powers and products (and quotients) are expanded, but not quotients of integers:
sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4)
To expand also log(3/4) use
algorithm='all':sage: (log(3/4*x^pi)).log_expand('all') pi*log(x) + log(3) - 2*log(2)
To expand only the power use
algorithm='powers'.:sage: (log(x^6)).log_expand('powers') 6*log(x)
The expression
log((3*x)^6)is not expanded withalgorithm='powers', since it is converted into product first:sage: (log((3*x)^6)).log_expand('powers') log(729*x^6)
This shows that the option
algorithmfrom the previous call has no influence to future calls (we changed some default Maxima flag, and have to ensure that this flag has been restored):sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4) sage: (log(3/4*x^pi)).log_expand('all') pi*log(x) + log(3) - 2*log(2) sage: (log(3/4*x^pi)).log_expand() pi*log(x) + log(3/4)
AUTHORS:
Robert Marik (11-2009)
- log_gamma(hold=False)#
Return the log gamma function evaluated at self. This is the logarithm of gamma of self, where gamma is a complex function such that \(gamma(n)\) equals \(factorial(n-1)\).
EXAMPLES:
sage: x = var('x') sage: x.log_gamma() log_gamma(x) sage: SR(2).log_gamma() 0 sage: SR(5).log_gamma() log(24) sage: a = SR(5).log_gamma(); a.n() 3.17805383034795 sage: SR(5-1).factorial().log() log(24) sage: from sage.misc.verbose import set_verbose sage: set_verbose(-1) sage: plot(lambda x: SR(x).log_gamma(), -7,8, plot_points=1000).show() # needs sage.plot sage: math.exp(0.5) 1.6487212707001282 sage: plot(lambda x: (SR(x).exp() - SR(-x).exp())/2 - SR(x).sinh(), -1, 1) # needs sage.plot Graphics object consisting of 1 graphics primitive
To prevent automatic evaluation use the
holdargument:sage: SR(5).log_gamma(hold=True) log_gamma(5)
To evaluate again, currently we must use numerical evaluation via
n():sage: a = SR(5).log_gamma(hold=True); a.n() 3.17805383034795
- log_simplify(algorithm=None)#
Simplify a (real) symbolic expression that contains logarithms.
The given expression is scanned recursively, transforming subexpressions of the form \(a \log(b) + c \log(d)\) into \(\log(b^{a} d^{c})\) before simplifying within the
log().The user can specify conditions that \(a\) and \(c\) must satisfy before this transformation will be performed using the optional parameter
algorithm.Warning
This is only safe to call if every variable in the given expression is assumed to be real. The simplification it performs is in general not valid over the complex numbers. For example:
sage: x,y = SR.var('x,y') sage: f = log(x*y) - (log(x) + log(y)) sage: f(x=-1, y=i) -2*I*pi sage: f.simplify_log() 0
INPUT:
self- expression to be simplifiedalgorithm- (default: None) optional, governs the condition on \(a\) and \(c\) which must be satisfied to contract expression \(a \log(b) + c \log(d)\). Values areNone(use Maxima default, integers),'one'(1 and -1),'ratios'(rational numbers),'constants'(constants),'all'(all expressions).
ALGORITHM:
This uses the Maxima
logcontract()command.ALIAS:
log_simplify()andsimplify_log()are the same.EXAMPLES:
sage: x,y,t = var('x y t')
Only two first terms are contracted in the following example; the logarithm with coefficient \(\frac{1}{2}\) is not contracted:
sage: f = log(x)+2*log(y)+1/2*log(t) sage: f.simplify_log() log(x*y^2) + 1/2*log(t)
To contract all terms in the previous example, we use the
'ratios'algorithm:sage: f.simplify_log(algorithm='ratios') log(sqrt(t)*x*y^2)
To contract terms with no coefficient (more precisely, with coefficients \(1\) and \(-1\)), we use the
'one'algorithm:sage: f = log(x)+2*log(y)-log(t) sage: f.simplify_log('one') 2*log(y) + log(x/t)
sage: f = log(x)+log(y)-1/3*log((x+1)) sage: f.simplify_log() log(x*y) - 1/3*log(x + 1) sage: f.simplify_log('ratios') log(x*y/(x + 1)^(1/3))
\(\pi\) is an irrational number; to contract logarithms in the following example we have to set
algorithmto'constants'or'all':sage: f = log(x)+log(y)-pi*log((x+1)) sage: f.simplify_log('constants') log(x*y/(x + 1)^pi)
x*log(9)is contracted only ifalgorithmis'all':sage: (x*log(9)).simplify_log() 2*x*log(3) sage: (x*log(9)).simplify_log('all') log(3^(2*x))
AUTHORS:
Robert Marik (11-2009)
- low_degree(s)#
Return the exponent of the lowest power of
sinself.OUTPUT:
An integer
EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100 sage: f.low_degree(x) -1 sage: f.low_degree(y) -10 sage: f.low_degree(sin(x*y)) 0 sage: (x^3+y).low_degree(x) 0 sage: (x+x**2).low_degree(x) 1
- match(pattern)#
Check if
selfmatches the given pattern.INPUT:
pattern– a symbolic expression, possibly containing wildcards to match for
OUTPUT:
One of
Noneif there is no match, or a dictionary mapping the wildcards to the matching values if a match was found. Note that the dictionary is empty if there were no wildcards in the given pattern.See also http://www.ginac.de/tutorial/Pattern-matching-and-advanced-substitutions.html
EXAMPLES:
sage: var('x,y,z,a,b,c,d,f,g') (x, y, z, a, b, c, d, f, g) sage: w0 = SR.wild(0); w1 = SR.wild(1); w2 = SR.wild(2) sage: ((x+y)^a).match((x+y)^a) # no wildcards, so empty dict {} sage: print(((x+y)^a).match((x+y)^b)) None sage: t = ((x+y)^a).match(w0^w1) sage: t[w0], t[w1] (x + y, a) sage: print(((x+y)^a).match(w0^w0)) None sage: ((x+y)^(x+y)).match(w0^w0) {$0: x + y} sage: t = ((a+b)*(a+c)).match((a+w0)*(a+w1)) sage: set([t[w0], t[w1]]) == set([b, c]) True sage: ((a+b)*(a+c)).match((w0+b)*(w0+c)) {$0: a} sage: t = ((a+b)*(a+c)).match((w0+w1)*(w0+w2)) sage: t[w0] a sage: set([t[w1], t[w2]]) == set([b, c]) True sage: t = ((a+b)*(a+c)).match((w0+w1)*(w1+w2)) sage: t[w1] a sage: set([t[w0], t[w2]]) == set([b, c]) True sage: t = (a*(x+y)+a*z+b).match(a*w0+w1) sage: s = set([t[w0], t[w1]]) sage: s == set([x+y, a*z+b]) or s == set([z, a*(x+y)+b]) True sage: print((a+b+c+d+f+g).match(c)) None sage: (a+b+c+d+f+g).has(c) True sage: (a+b+c+d+f+g).match(c+w0) {$0: a + b + d + f + g} sage: (a+b+c+d+f+g).match(c+g+w0) {$0: a + b + d + f} sage: (a+b).match(a+b+w0) # known bug {$0: 0} sage: print((a*b^2).match(a^w0*b^w1)) None sage: (a*b^2).match(a*b^w1) {$1: 2} sage: (x*x.arctan2(x^2)).match(w0*w0.arctan2(w0^2)) {$0: x}
Beware that behind-the-scenes simplification can lead to surprising results in matching:
sage: print((x+x).match(w0+w1)) None sage: t = x+x; t 2*x sage: t.operator() <function mul_vararg ...>
Since asking to match w0+w1 looks for an addition operator, there is no match.
- maxima_methods()#
Provide easy access to maxima methods, converting the result to a Sage expression automatically.
EXAMPLES:
sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t log(sqrt(2) + 1) + log(sqrt(2) - 1) sage: res = t.maxima_methods().logcontract(); res log((sqrt(2) + 1)*(sqrt(2) - 1)) sage: type(res) <class 'sage.symbolic.expression.Expression'>
- minpoly(*args, **kwds)#
Return the minimal polynomial of this symbolic expression.
EXAMPLES:
sage: golden_ratio.minpoly() x^2 - x - 1
- mul(hold=False, *args)#
Return the product of the current expression and the given arguments.
To prevent automatic evaluation use the
holdargument.EXAMPLES:
sage: x.mul(x) x^2 sage: x.mul(x, hold=True) x*x sage: x.mul(x, (2+x), hold=True) (x + 2)*x*x sage: x.mul(x, (2+x), x, hold=True) (x + 2)*x*x*x sage: x.mul(x, (2+x), x, 2*x, hold=True) (2*x)*(x + 2)*x*x*x
To then evaluate again, we use
unhold():sage: a = x.mul(x, hold=True); a.unhold() x^2
- multiply_both_sides(x, checksign=None)#
Return a relation obtained by multiplying both sides of this relation by
x.Note
The checksign keyword argument is currently ignored and is included for backward compatibility reasons only.
EXAMPLES:
sage: var('x,y'); f = x + 3 < y - 2 (x, y) sage: f.multiply_both_sides(7) 7*x + 21 < 7*y - 14 sage: f.multiply_both_sides(-1/2) -1/2*x - 3/2 < -1/2*y + 1 sage: f*(-2/3) -2/3*x - 2 < -2/3*y + 4/3 sage: f*(-pi) -pi*(x + 3) < -pi*(y - 2)
Since the direction of the inequality never changes when doing arithmetic with equations, you can multiply or divide the equation by a quantity with unknown sign:
sage: f*(1+I) (I + 1)*x + 3*I + 3 < (I + 1)*y - 2*I - 2 sage: f = sqrt(2) + x == y^3 sage: f.multiply_both_sides(I) I*x + I*sqrt(2) == I*y^3 sage: f.multiply_both_sides(-1) -x - sqrt(2) == -y^3
Note that the direction of the following inequalities is not reversed:
sage: (x^3 + 1 > 2*sqrt(3)) * (-1) -x^3 - 1 > -2*sqrt(3) sage: (x^3 + 1 >= 2*sqrt(3)) * (-1) -x^3 - 1 >= -2*sqrt(3) sage: (x^3 + 1 <= 2*sqrt(3)) * (-1) -x^3 - 1 <= -2*sqrt(3)
- negation()#
Return the negated version of
self.This is the relation that is
Falseiff self isTrue.EXAMPLES:
sage: (x < 5).negation() x >= 5 sage: (x == sin(3)).negation() x != sin(3) sage: (2*x >= sqrt(2)).negation() 2*x < sqrt(2)
- nintegral(*args, **kwds)#
Compute the numerical integral of
self.Please see
sage.calculus.calculus.nintegralfor more details.EXAMPLES:
sage: sin(x).nintegral(x,0,3) (1.989992496600..., 2.209335488557...e-14, 21, 0)
- nintegrate(*args, **kwds)#
Compute the numerical integral of
self.Please see
sage.calculus.calculus.nintegralfor more details.EXAMPLES:
sage: sin(x).nintegral(x,0,3) (1.989992496600..., 2.209335488557...e-14, 21, 0)
- nops()#
Return the number of operands of this expression.
EXAMPLES:
sage: var('a,b,c,x,y') (a, b, c, x, y) sage: a.number_of_operands() 0 sage: (a^2 + b^2 + (x+y)^2).number_of_operands() 3 sage: (a^2).number_of_operands() 2 sage: (a*b^2*c).number_of_operands() 3
- norm()#
Return the complex norm of this symbolic expression, i.e., the expression times its complex conjugate. If \(c = a + bi\) is a complex number, then the norm of \(c\) is defined as the product of \(c\) and its complex conjugate
\[\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.\]The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain \(\ZZ[i]\) of Gaussian integers, where the norm of each Gaussian integer \(c = a + bi\) is defined as its complex norm.
See also
EXAMPLES:
sage: a = 1 + 2*I sage: a.norm() 5 sage: a = sqrt(2) + 3^(1/3)*I; a sqrt(2) + I*3^(1/3) sage: a.norm() 3^(2/3) + 2 sage: CDF(a).norm() 4.080083823051... sage: CDF(a.norm()) 4.080083823051904
- normalize()#
Return this expression normalized as a fraction
See also
numerator(),denominator(),numerator_denominator(),combine()EXAMPLES:
sage: var('x, y, a, b, c') (x, y, a, b, c) sage: g = x + y/(x + 2) sage: g.normalize() (x^2 + 2*x + y)/(x + 2) sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a sage: f.normalize() (a*x^3 + b*x^3 + c*x^3 + a*x*y^2 + a*x^2 + b*x^2 + c*x^2 + a*y^2 - a*x - 7*b*x - 7*c*x - 7*a - 7*b - 7*c)/((x^2 - 7)*a*(x + 1))
ALGORITHM: Uses GiNaC.
- number_of_arguments()#
EXAMPLES:
sage: x,y = var('x,y') sage: f = x + y sage: f.number_of_arguments() 2 sage: g = f.function(x) sage: g.number_of_arguments() 1
sage: x,y,z = var('x,y,z') sage: (x+y).number_of_arguments() 2 sage: (x+1).number_of_arguments() 1 sage: (sin(x)+1).number_of_arguments() 1 sage: (sin(z)+x+y).number_of_arguments() 3 sage: (sin(x+y)).number_of_arguments() 2
- number_of_operands()#
Return the number of operands of this expression.
EXAMPLES:
sage: var('a,b,c,x,y') (a, b, c, x, y) sage: a.number_of_operands() 0 sage: (a^2 + b^2 + (x+y)^2).number_of_operands() 3 sage: (a^2).number_of_operands() 2 sage: (a*b^2*c).number_of_operands() 3
- numerator(normalize=True)#
Return the numerator of this symbolic expression
INPUT:
normalize– (default:True) a boolean.
If
normalizeisTrue, the expression is first normalized to have it as a fraction before getting the numerator.If
normalizeisFalse, the expression is kept and if it is not a quotient, then this will return the expression itself.See also
normalize(),denominator(),numerator_denominator(),combine()EXAMPLES:
sage: a, x, y = var('a,x,y') sage: f = x*(x-a)/((x^2 - y)*(x-a)); f x/(x^2 - y) sage: f.numerator() x sage: f.denominator() x^2 - y sage: f.numerator(normalize=False) x sage: f.denominator(normalize=False) x^2 - y sage: y = var('y') sage: g = x + y/(x + 2); g x + y/(x + 2) sage: g.numerator() x^2 + 2*x + y sage: g.denominator() x + 2 sage: g.numerator(normalize=False) x + y/(x + 2) sage: g.denominator(normalize=False) 1
- numerator_denominator(normalize=True)#
Return the numerator and the denominator of this symbolic expression
INPUT:
normalize– (default:True) a boolean.
If
normalizeisTrue, the expression is first normalized to have it as a fraction before getting the numerator and denominator.If
normalizeisFalse, the expression is kept and if it is not a quotient, then this will return the expression itself together with 1.See also
EXAMPLES:
sage: x, y, a = var("x y a") sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator() ((x + y)^2*x^3, (x - y)^3) sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator(False) ((x + y)^2*x^3, (x - y)^3) sage: g = x + y/(x + 2) sage: g.numerator_denominator() (x^2 + 2*x + y, x + 2) sage: g.numerator_denominator(normalize=False) (x + y/(x + 2), 1) sage: g = x^2*(x + 2) sage: g.numerator_denominator() ((x + 2)*x^2, 1) sage: g.numerator_denominator(normalize=False) ((x + 2)*x^2, 1)
- numerical_approx(prec=None, digits=None, algorithm=None)#
Return a numerical approximation of
selfwithprecbits (or decimaldigits) of precision.No guarantee is made about the accuracy of the result.
INPUT:
prec– precision in bitsdigits– precision in decimal digits (only used ifprecis not given)algorithm– which algorithm to use to compute this approximation
If neither
precnordigitsis given, the default precision is 53 bits (roughly 16 digits).EXAMPLES:
sage: sin(x).subs(x=5).n() -0.958924274663138 sage: sin(x).subs(x=5).n(100) -0.95892427466313846889315440616 sage: sin(x).subs(x=5).n(digits=50) -0.95892427466313846889315440615599397335246154396460 sage: zeta(x).subs(x=2).numerical_approx(digits=50) 1.6449340668482264364724151666460251892189499012068 sage: cos(3).numerical_approx(200) -0.98999249660044545727157279473126130239367909661558832881409 sage: numerical_approx(cos(3),200) -0.98999249660044545727157279473126130239367909661558832881409 sage: numerical_approx(cos(3), digits=10) -0.9899924966 sage: (i + 1).numerical_approx(32) 1.00000000 + 1.00000000*I sage: (pi + e + sqrt(2)).numerical_approx(100) 7.2740880444219335226246195788
- op#
Provide access to the operands of an expression through a property.
EXAMPLES:
sage: t = 1+x+x^2 sage: t.op Operands of x^2 + x + 1 sage: x.op Traceback (most recent call last): ... TypeError: expressions containing only a numeric coefficient, constant or symbol have no operands sage: t.op[0] x^2
Indexing directly with
t[1]causes problems with numpy types.sage: t[1] Traceback (most recent call last): … TypeError: ‘sage.symbolic.expression.Expression’ object …
- operands()#
Return a list containing the operands of this expression.
EXAMPLES:
sage: var('a,b,c,x,y') (a, b, c, x, y) sage: (a^2 + b^2 + (x+y)^2).operands() [a^2, b^2, (x + y)^2] sage: (a^2).operands() [a, 2] sage: (a*b^2*c).operands() [a, b^2, c]
- operator()#
Return the topmost operator in this expression.
EXAMPLES:
sage: x,y,z = var('x,y,z') sage: (x+y).operator() <function add_vararg ...> sage: (x^y).operator() <built-in function pow> sage: (x^y * z).operator() <function mul_vararg ...> sage: (x < y).operator() <built-in function lt> sage: abs(x).operator() abs sage: r = gamma(x).operator(); type(r) <class 'sage.functions.gamma.Function_gamma'> sage: psi = function('psi', nargs=1) sage: psi(x).operator() psi sage: r = psi(x).operator() sage: r == psi True sage: f = function('f', nargs=1, conjugate_func=lambda self, x: 2*x) sage: nf = f(x).operator() sage: nf(x).conjugate() 2*x sage: f = function('f') sage: a = f(x).diff(x); a diff(f(x), x) sage: a.operator() D[0](f)
- partial_fraction(var=None)#
Return the partial fraction expansion of
selfwith respect to the given variable.INPUT:
var– variable name or string (default: first variable)
OUTPUT:
A symbolic expression
See also
EXAMPLES:
sage: f = x^2/(x+1)^3 sage: f.partial_fraction() 1/(x + 1) - 2/(x + 1)^2 + 1/(x + 1)^3
Notice that the first variable in the expression is used by default:
sage: y = var('y') sage: f = y^2/(y+1)^3 sage: f.partial_fraction() 1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3 sage: f = y^2/(y+1)^3 + x/(x-1)^3 sage: f.partial_fraction() y^2/(y^3 + 3*y^2 + 3*y + 1) + 1/(x - 1)^2 + 1/(x - 1)^3
You can explicitly specify which variable is used:
sage: f.partial_fraction(y) x/(x^3 - 3*x^2 + 3*x - 1) + 1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3
- partial_fraction_decomposition(var=None)#
Return the partial fraction decomposition of
selfwith respect to the given variable.INPUT:
var– variable name or string (default: first variable)
OUTPUT:
A list of symbolic expressions
See also
EXAMPLES:
sage: f = x^2/(x+1)^3 sage: f.partial_fraction_decomposition() [1/(x + 1), -2/(x + 1)^2, (x + 1)^(-3)] sage: (4+f).partial_fraction_decomposition() [1/(x + 1), -2/(x + 1)^2, (x + 1)^(-3), 4]
Notice that the first variable in the expression is used by default:
sage: y = var('y') sage: f = y^2/(y+1)^3 sage: f.partial_fraction_decomposition() [1/(y + 1), -2/(y + 1)^2, (y + 1)^(-3)] sage: f = y^2/(y+1)^3 + x/(x-1)^3 sage: f.partial_fraction_decomposition() [y^2/(y^3 + 3*y^2 + 3*y + 1), (x - 1)^(-2), (x - 1)^(-3)]
You can explicitly specify which variable is used:
sage: f.partial_fraction_decomposition(y) [1/(y + 1), -2/(y + 1)^2, (y + 1)^(-3), x/(x^3 - 3*x^2 + 3*x - 1)]
- plot(*args, **kwds)#
Plot a symbolic expression. All arguments are passed onto the standard plot command.
EXAMPLES:
This displays a straight line:
sage: sin(2).plot((x,0,3)) # needs sage.plot Graphics object consisting of 1 graphics primitive
This draws a red oscillatory curve:
sage: sin(x^2).plot((x,0,2*pi), rgbcolor=(1,0,0)) # needs sage.plot Graphics object consisting of 1 graphics primitive
Another plot using the variable theta:
sage: var('theta') theta sage: (cos(theta) - erf(theta)).plot((theta,-2*pi,2*pi)) # needs sage.plot Graphics object consisting of 1 graphics primitive
A very thick green plot with a frame:
sage: sin(x).plot((x, -4*pi, 4*pi), # needs sage.plot ....: thickness=20, rgbcolor=(0,0.7,0)).show(frame=True)
You can embed 2d plots in 3d space as follows:
sage: plot(sin(x^2), (x, -pi, pi), thickness=2).plot3d(z=1) # long time, needs sage.plot Graphics3d Object
A more complicated family:
sage: G = sum(plot(sin(n*x), (x, -2*pi, 2*pi)).plot3d(z=n) # needs sage.plot ....: for n in [0,0.1,..1]) sage: G.show(frame_aspect_ratio=[1,1,1/2]) # long time (5s on sage.math, 2012), needs sage.plot
A plot involving the floor function:
sage: plot(1.0 - x * floor(1/x), (x,0.00001,1.0)) # needs sage.plot Graphics object consisting of 1 graphics primitive
Sage used to allow symbolic functions with “no arguments”; this no longer works:
sage: plot(2*sin, -4, 4) # needs sage.plot Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for *: 'Integer Ring' and '<class 'sage.functions.trig.Function_sin'>'
You should evaluate the function first:
sage: plot(2*sin(x), -4, 4) # needs sage.plot Graphics object consisting of 1 graphics primitive
- poly(x=None)#
Express this symbolic expression as a polynomial in x. If this is not a polynomial in x, then some coefficients may be functions of x.
Warning
This is different from
polynomial()which returns a Sage polynomial over a given base ring.EXAMPLES:
sage: var('a, x') (a, x) sage: p = expand((x-a*sqrt(2))^2 + x + 1); p -2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1 sage: p.poly(a) -2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1 sage: bool(p.poly(a) == (x-a*sqrt(2))^2 + x + 1) True sage: p.poly(x) 2*a^2 - (2*sqrt(2)*a - 1)*x + x^2 + 1
- polynomial(base_ring=None, ring=None)#
Return this symbolic expression as an algebraic polynomial over the given base ring, if possible.
The point of this function is that it converts purely symbolic polynomials into optimised algebraic polynomials over a given base ring.
You can specify either the base ring (
base_ring) you want the output polynomial to be over, or you can specify the full polynomial ring (ring) you want the output polynomial to be an element of.INPUT:
base_ring- (optional) the base ring for the polynomialring- (optional) the parent for the polynomial
Warning
This is different from
poly()which is used to rewrite self as a polynomial in terms of one of the variables.EXAMPLES:
sage: f = x^2 -2/3*x + 1 sage: f.polynomial(QQ) x^2 - 2/3*x + 1 sage: f.polynomial(GF(19)) x^2 + 12*x + 1
Polynomials can be useful for getting the coefficients of an expression:
sage: g = 6*x^2 - 5 sage: g.coefficients() [[-5, 0], [6, 2]] sage: g.polynomial(QQ).list() [-5, 0, 6] sage: g.polynomial(QQ).dict() {0: -5, 2: 6}
sage: f = x^2*e + x + pi/e sage: f.polynomial(RDF) # abs tol 5e-16 2.718281828459045*x^2 + x + 1.1557273497909217 sage: g = f.polynomial(RR); g 2.71828182845905*x^2 + x + 1.15572734979092 sage: g.parent() Univariate Polynomial Ring in x over Real Field with 53 bits of precision sage: f.polynomial(RealField(100)) 2.7182818284590452353602874714*x^2 + x + 1.1557273497909217179100931833 sage: f.polynomial(CDF) # abs tol 5e-16 2.718281828459045*x^2 + x + 1.1557273497909217 sage: f.polynomial(CC) 2.71828182845905*x^2 + x + 1.15572734979092
We coerce a multivariate polynomial with complex symbolic coefficients:
sage: x, y, n = var('x, y, n') sage: f = pi^3*x - y^2*e - I; f pi^3*x - y^2*e - I sage: f.polynomial(CDF) # abs tol 1e-15 (-2.718281828459045)*y^2 + 31.006276680299816*x - 1.0*I sage: f.polynomial(CC) (-2.71828182845905)*y^2 + 31.0062766802998*x - 1.00000000000000*I sage: f.polynomial(ComplexField(70)) (-2.7182818284590452354)*y^2 + 31.006276680299820175*x - 1.0000000000000000000*I
Another polynomial:
sage: f = sum((e*I)^n*x^n for n in range(5)); f x^4*e^4 - I*x^3*e^3 - x^2*e^2 + I*x*e + 1 sage: f.polynomial(CDF) # abs tol 5e-16 54.598150033144236*x^4 - 20.085536923187668*I*x^3 - 7.38905609893065*x^2 + 2.718281828459045*I*x + 1.0 sage: f.polynomial(CC) 54.5981500331442*x^4 - 20.0855369231877*I*x^3 - 7.38905609893065*x^2 + 2.71828182845905*I*x + 1.00000000000000
A multivariate polynomial over a finite field:
sage: f = (3*x^5 - 5*y^5)^7; f (3*x^5 - 5*y^5)^7 sage: g = f.polynomial(GF(7)); g 3*x^35 + 2*y^35 sage: parent(g) Multivariate Polynomial Ring in x, y over Finite Field of size 7
We check to make sure constants are converted appropriately:
sage: (pi*x).polynomial(SR) pi*x
Using the
ringparameter, you can also create polynomials rings over the symbolic ring where only certain variables are considered generators of the polynomial ring and the others are considered “constants”:sage: a, x, y = var('a,x,y') sage: f = a*x^10*y+3*x sage: B = f.polynomial(ring=SR['x,y']) sage: B.coefficients() [a, 3]
- power(exp, hold=False)#
Return the current expression to the power
exp.To prevent automatic evaluation use the
holdargument.EXAMPLES:
sage: (x^2).power(2) x^4 sage: (x^2).power(2, hold=True) (x^2)^2
To then evaluate again, we use
unhold():sage: a = (x^2).power(2, hold=True); a.unhold() x^4
- power_series(base_ring)#
Return algebraic power series associated to this symbolic expression, which must be a polynomial in one variable, with coefficients coercible to the base ring.
The power series is truncated one more than the degree.
EXAMPLES:
sage: theta = var('theta') sage: f = theta^3 + (1/3)*theta - 17/3 sage: g = f.power_series(QQ); g -17/3 + 1/3*theta + theta^3 + O(theta^4) sage: g^3 -4913/27 + 289/9*theta - 17/9*theta^2 + 2602/27*theta^3 + O(theta^4) sage: g.parent() Power Series Ring in theta over Rational Field
- primitive_part(s)#
Return the primitive polynomial of this expression when considered as a polynomial in
s.See also
unit(),content(), andunit_content_primitive().INPUT:
s– a symbolic expression.
OUTPUT:
The primitive polynomial as a symbolic expression. It is defined as the quotient by the
unit()andcontent()parts (with respect to the variables).EXAMPLES:
sage: (2*x+4).primitive_part(x) x + 2 sage: (2*x+1).primitive_part(x) 2*x + 1 sage: (2*x+1/2).primitive_part(x) 4*x + 1 sage: var('y') y sage: (2*x + 4*sin(y)).primitive_part(sin(y)) x + 2*sin(y)
- prod(*args, **kwds)#
Return the symbolic product \(\prod_{v = a}^b\)
self.This is the product respect to the variable \(v\) with endpoints \(a\) and \(b\).
INPUT:
expression– a symbolic expressionv– a variable or variable namea– lower endpoint of the productb– upper endpoint of the productalgorithm– (default:'maxima') one of'maxima'– use Maxima (the default)'giac'– (optional) use Giac'sympy'– use SymPy
hold– (default:False) ifTrue, don’t evaluate
- pyobject()#
Get the underlying Python object.
OUTPUT:
The Python object corresponding to this expression, assuming this expression is a single numerical value or an infinity representable in Python. Otherwise, a
TypeErroris raised.EXAMPLES:
sage: var('x') x sage: b = -17.3 sage: a = SR(b) sage: a.pyobject() -17.3000000000000 sage: a.pyobject() is b True
Integers and Rationals are converted internally though, so you won’t get back the same object:
sage: b = -17/3 sage: a = SR(b) sage: a.pyobject() -17/3 sage: a.pyobject() is b False
- rational_expand(side=None)#
Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.
EXAMPLES:
We expand the expression \((x-y)^5\) using both method and functional notation.
sage: x,y = var('x,y') sage: a = (x-y)^5 sage: a.expand() x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5 sage: expand(a) x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
We expand some other expressions:
sage: expand((x-1)^3/(y-1)) x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1) sage: expand((x+sin((x+y)^2))^2) x^2 + 2*x*sin(x^2 + 2*x*y + y^2) + sin(x^2 + 2*x*y + y^2)^2
Observe that
expand()also expands function arguments:sage: f(x) = function('f')(x) sage: fx = f(x*(x+1)); fx f((x + 1)*x) sage: fx.expand() f(x^2 + x)
We can expand individual sides of a relation:
sage: a = (16*x-13)^2 == (3*x+5)^2/2 sage: a.expand() 256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2 sage: a.expand('left') 256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2 sage: a.expand('right') (16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2
- rational_simplify(algorithm='full', map=False)#
Simplify rational expressions.
INPUT:
self- symbolic expressionalgorithm- (default: ‘full’) string which switches the algorithm for simplifications. Possible values are‘simple’ (simplify rational functions into quotient of two polynomials),
‘full’ (apply repeatedly, if necessary)
‘noexpand’ (convert to common denominator and add)
map- (default:False) ifTrue, the result is an expression whose leading operator is the same as that of the expressionselfbut whose subparts are the results of applying simplification rules to the corresponding subparts of the expressions.
ALIAS:
rational_simplify()andsimplify_rational()are the sameDETAILS: We call Maxima functions ratsimp, fullratsimp and xthru. If each part of the expression has to be simplified separately, we use Maxima function map.
EXAMPLES:
sage: f = sin(x/(x^2 + x)) sage: f sin(x/(x^2 + x)) sage: f.simplify_rational() sin(1/(x + 1))
sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f -((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1)) sage: f.simplify_rational() -2*sqrt(x - 1)/sqrt(x^2 - 1)
With
map=Trueeach term in a sum is simplified separately and thus the results are shorter for functions which are combination of rational and nonrational functions. In the following example, we use this option if we want not to combine logarithm and the rational function into one fraction:sage: f = (x^2-1)/(x+1)-ln(x)/(x+2) sage: f.simplify_rational() (x^2 + x - log(x) - 2)/(x + 2) sage: f.simplify_rational(map=True) x - log(x)/(x + 2) - 1
Here is an example from the Maxima documentation of where
algorithm='simple'produces an (possibly useful) intermediate step:sage: y = var('y') sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1) sage: g.simplify_rational(algorithm='simple') (x^(2*y) - 2*x^y + 1)/(x^y - 1) sage: g.simplify_rational() x^y - 1
With option
algorithm='noexpand'we only convert to common denominators and add. No expansion of products is performed:sage: f = 1/(x+1)+x/(x+2)^2 sage: f.simplify_rational() (2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4) sage: f.simplify_rational(algorithm='noexpand') ((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))
- real(hold=False)#
Return the real part of this symbolic expression.
EXAMPLES:
sage: x = var('x') sage: x.real_part() real_part(x) sage: SR(2+3*I).real_part() 2 sage: SR(CDF(2,3)).real_part() 2.0 sage: SR(CC(2,3)).real_part() 2.00000000000000 sage: f = log(x) sage: f.real_part() log(abs(x))
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(2).real_part() 2 sage: SR(2).real_part(hold=True) real_part(2)
This also works using functional notation:
sage: real_part(I,hold=True) real_part(I) sage: real_part(I) 0
To then evaluate again, we use
unhold():sage: a = SR(2).real_part(hold=True); a.unhold() 2
- real_part(hold=False)#
Return the real part of this symbolic expression.
EXAMPLES:
sage: x = var('x') sage: x.real_part() real_part(x) sage: SR(2+3*I).real_part() 2 sage: SR(CDF(2,3)).real_part() 2.0 sage: SR(CC(2,3)).real_part() 2.00000000000000 sage: f = log(x) sage: f.real_part() log(abs(x))
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(2).real_part() 2 sage: SR(2).real_part(hold=True) real_part(2)
This also works using functional notation:
sage: real_part(I,hold=True) real_part(I) sage: real_part(I) 0
To then evaluate again, we use
unhold():sage: a = SR(2).real_part(hold=True); a.unhold() 2
- rectform()#
Convert this symbolic expression to rectangular form; that is, the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
Note
The name "rectangular" comes from the fact that, in the complex plane, \(a\) and \(bi\) are perpendicular.
INPUT:
self– the expression to convert.
OUTPUT:
A new expression, equivalent to the original, but expressed in the form \(a + bi\).
ALGORITHM:
We call Maxima’s
rectform()and return the result unmodified.EXAMPLES:
The exponential form of \(\sin(x)\):
sage: f = (e^(I*x) - e^(-I*x)) / (2*I) sage: f.rectform() sin(x)
And \(\cos(x)\):
sage: f = (e^(I*x) + e^(-I*x)) / 2 sage: f.rectform() cos(x)
In some cases, this will simplify the given expression. For example, here, \(e^{ik\pi}\), \(\sin(k\pi)=0\) should cancel leaving only \(\cos(k\pi)\) which can then be simplified:
sage: k = var('k') sage: assume(k, 'integer') sage: f = e^(I*pi*k) sage: f.rectform() (-1)^k
However, in general, the resulting expression may be more complicated than the original:
sage: f = e^(I*x) sage: f.rectform() cos(x) + I*sin(x)
- reduce_trig(var=None)#
Combine products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators.
INPUT:
self– a symbolic expressionvar– (default:None) the variable which is used for these transformations. If not specified, all variables are used.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: y = var('y') sage: f = sin(x)*cos(x)^3+sin(y)^2 sage: f.reduce_trig() -1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2
To reduce only the expressions involving x we use optional parameter:
sage: f.reduce_trig(x) sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)
ALIAS:
trig_reduce()andreduce_trig()are the same
- residue(symbol)#
Calculate the residue of
selfwith respect tosymbol.INPUT:
symbol- a symbolic variable or symbolic equality such asx == 5. If an equality is given, the expansion is around the value on the right hand side of the equality, otherwise at0.
OUTPUT:
The residue of
self.Say,
symbolisx == a, then this function calculates the residue ofselfat \(x=a\), i.e., the coefficient of \(1/(x-a)\) of the series expansion ofselfaround \(a\).EXAMPLES:
sage: (1/x).residue(x == 0) 1 sage: (1/x).residue(x == oo) -1 sage: (1/x^2).residue(x == 0) 0 sage: (1/sin(x)).residue(x == 0) 1 sage: var('q, n, z') (q, n, z) sage: (-z^(-n-1)/(1-z/q)^2).residue(z == q).simplify_full() (n + 1)/q^n sage: var('s') s sage: zeta(s).residue(s == 1) 1
We can also compute the residue at more general places, given that the pole is recognized:
sage: k = var('k', domain='integer') sage: (gamma(1+x)/(1 - exp(-x))).residue(x==2*I*pi*k) gamma(2*I*pi*k + 1) sage: csc(x).residue(x==2*pi*k) 1
- resultant(other, var)#
Compute the resultant of this polynomial expression and the first argument with respect to the variable given as the second argument.
EXAMPLES:
sage: _ = var('a b n k u x y') sage: x.resultant(y, x) y sage: (x+y).resultant(x-y, x) -2*y sage: r = (x^4*y^2+x^2*y-y).resultant(x*y-y*a-x*b+a*b+u,x) sage: r.coefficient(a^4) b^4*y^2 - 4*b^3*y^3 + 6*b^2*y^4 - 4*b*y^5 + y^6 sage: x.resultant(sin(x), x) Traceback (most recent call last): ... RuntimeError: resultant(): arguments must be polynomials
- rhs()#
If
selfis a relational expression, return the right hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 <= x^2 - 2*x + 3 sage: eqn.right_hand_side() x^2 - 2*x + 3 sage: eqn.rhs() x^2 - 2*x + 3 sage: eqn.right() x^2 - 2*x + 3
- right()#
If
selfis a relational expression, return the right hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 <= x^2 - 2*x + 3 sage: eqn.right_hand_side() x^2 - 2*x + 3 sage: eqn.rhs() x^2 - 2*x + 3 sage: eqn.right() x^2 - 2*x + 3
- right_hand_side()#
If
selfis a relational expression, return the right hand side of the relation. Otherwise, raise aValueError.EXAMPLES:
sage: x = var('x') sage: eqn = (x-1)^2 <= x^2 - 2*x + 3 sage: eqn.right_hand_side() x^2 - 2*x + 3 sage: eqn.rhs() x^2 - 2*x + 3 sage: eqn.right() x^2 - 2*x + 3
- roots(x=None, explicit_solutions=True, multiplicities=True, ring=None)#
Return roots of
selfthat can be found exactly, possibly with multiplicities. Not all roots are guaranteed to be found.Warning
This is not a numerical solver - use
find_rootto solve for self == 0 numerically on an interval.INPUT:
x- variable to view the function in terms of (use default variable if not given)explicit_solutions– bool (defaultTrue); require that roots be explicit rather than implicitmultiplicities– bool (defaultTrue); whenTrue, return multiplicitiesring– a ring (defaultNone): if notNone, convertselfto a polynomial over ring and find roots over ring
OUTPUT:
A list of pairs
(root, multiplicity)or list of roots.If there are infinitely many roots, e.g., a function like \(\sin(x)\), only one is returned.
EXAMPLES:
sage: var('x, a') (x, a)
A simple example:
sage: ((x^2-1)^2).roots() [(-1, 2), (1, 2)] sage: ((x^2-1)^2).roots(multiplicities=False) [-1, 1]
A complicated example:
sage: f = expand((x^2 - 1)^3*(x^2 + 1)*(x-a)); f -a*x^8 + x^9 + 2*a*x^6 - 2*x^7 - 2*a*x^2 + 2*x^3 + a - x
The default variable is \(a\), since it is the first in alphabetical order:
sage: f.roots() [(x, 1)]
As a polynomial in \(a\), \(x\) is indeed a root:
sage: f.poly(a) x^9 - 2*x^7 + 2*x^3 - (x^8 - 2*x^6 + 2*x^2 - 1)*a - x sage: f(a=x) 0
The roots in terms of \(x\) are what we expect:
sage: f.roots(x) [(a, 1), (-I, 1), (I, 1), (1, 3), (-1, 3)]
Only one root of \(\sin(x) = 0\) is given:
sage: f = sin(x) sage: f.roots(x) [(0, 1)]
Note
It is possible to solve a greater variety of equations using
solve()and the keywordto_poly_solve, but only at the price of possibly encountering approximate solutions. See documentation for f.solve for more details.We derive the roots of a general quadratic polynomial:
sage: var('a,b,c,x') (a, b, c, x) sage: (a*x^2 + b*x + c).roots(x) [(-1/2*(b + sqrt(b^2 - 4*a*c))/a, 1), (-1/2*(b - sqrt(b^2 - 4*a*c))/a, 1)]
By default, all the roots are required to be explicit rather than implicit. To get implicit roots, pass
explicit_solutions=Falseto.roots()sage: var('x') x sage: f = x^(1/9) + (2^(8/9) - 2^(1/9))*(x - 1) - x^(8/9) sage: f.roots() Traceback (most recent call last): ... RuntimeError: no explicit roots found sage: f.roots(explicit_solutions=False) [((2^(8/9) + x^(8/9) - 2^(1/9) - x^(1/9))/(2^(8/9) - 2^(1/9)), 1)]
Another example, but involving a degree 5 poly whose roots do not get computed explicitly:
sage: f = x^5 + x^3 + 17*x + 1 sage: f.roots() Traceback (most recent call last): ... RuntimeError: no explicit roots found sage: f.roots(explicit_solutions=False) [(x^5 + x^3 + 17*x + 1, 1)] sage: f.roots(explicit_solutions=False, multiplicities=False) [x^5 + x^3 + 17*x + 1]
Now let us find some roots over different rings:
sage: f.roots(ring=CC) [(-0.0588115223184..., 1), (-1.331099917875... - 1.52241655183732*I, 1), (-1.331099917875... + 1.52241655183732*I, 1), (1.36050567903502 - 1.51880872209965*I, 1), (1.36050567903502 + 1.51880872209965*I, 1)] sage: (2.5*f).roots(ring=RR) [(-0.058811522318449..., 1)] sage: f.roots(ring=CC, multiplicities=False) [-0.05881152231844..., -1.331099917875... - 1.52241655183732*I, -1.331099917875... + 1.52241655183732*I, 1.36050567903502 - 1.51880872209965*I, 1.36050567903502 + 1.51880872209965*I] sage: f.roots(ring=QQ) [] sage: f.roots(ring=QQbar, multiplicities=False) [-0.05881152231844944?, -1.331099917875796? - 1.522416551837318?*I, -1.331099917875796? + 1.522416551837318?*I, 1.360505679035020? - 1.518808722099650?*I, 1.360505679035020? + 1.518808722099650?*I]
Root finding over finite fields:
sage: f.roots(ring=GF(7^2, 'a')) # needs sage.rings.finite_rings [(3, 1), (4*a + 6, 2), (3*a + 3, 2)]
- round()#
Round this expression to the nearest integer.
EXAMPLES:
sage: u = sqrt(43203735824841025516773866131535024) sage: u.round() 207855083711803945 sage: t = sqrt(Integer('1'*1000)).round(); print(str(t)[-10:]) 3333333333 sage: (-sqrt(110)).round() -10 sage: (-sqrt(115)).round() -11 sage: (sqrt(-3)).round() Traceback (most recent call last): ... ValueError: could not convert sqrt(-3) to a real number
- series(symbol, order=None)#
Return the power series expansion of
selfin terms of the given variable to the given order.INPUT:
symbol– a symbolic variable or symbolic equality such asx == 5; if an equality is given, the expansion is around the value on the right hand side of the equalityorder– an integer; if nothing given, it is set to the global default (20), which can be changed usingset_series_precision()
OUTPUT:
A power series.
To truncate the power series and obtain a normal expression, use the
truncate()command.EXAMPLES:
We expand a polynomial in \(x\) about 0, about \(1\), and also truncate it back to a polynomial:
sage: var('x,y') (x, y) sage: f = (x^3 - sin(y)*x^2 - 5*x + 3); f x^3 - x^2*sin(y) - 5*x + 3 sage: g = f.series(x, 4); g 3 + (-5)*x + (-sin(y))*x^2 + 1*x^3 + Order(x^4) sage: g.truncate() x^3 - x^2*sin(y) - 5*x + 3 sage: g = f.series(x==1, 4); g (-sin(y) - 1) + (-2*sin(y) - 2)*(x - 1) + (-sin(y) + 3)*(x - 1)^2 + 1*(x - 1)^3 + Order((x - 1)^4) sage: h = g.truncate(); h (x - 1)^3 - (x - 1)^2*(sin(y) - 3) - 2*(x - 1)*(sin(y) + 1) - sin(y) - 1 sage: h.expand() x^3 - x^2*sin(y) - 5*x + 3
We computer another series expansion of an analytic function:
sage: f = sin(x)/x^2 sage: f.series(x,7) 1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7) sage: f.series(x) 1*x^(-1) + (-1/6)*x + ... + Order(x^20) sage: f.series(x==1,3) (sin(1)) + (cos(1) - 2*sin(1))*(x - 1) + (-2*cos(1) + 5/2*sin(1))*(x - 1)^2 + Order((x - 1)^3) sage: f.series(x==1,3).truncate().expand() -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
Expressions formed by combining series can be expanded by applying series again:
sage: (1/(1-x)).series(x, 3)+(1/(1+x)).series(x,3) (1 + 1*x + 1*x^2 + Order(x^3)) + (1 + (-1)*x + 1*x^2 + Order(x^3)) sage: _.series(x,3) 2 + 2*x^2 + Order(x^3) sage: (1/(1-x)).series(x, 3)*(1/(1+x)).series(x,3) (1 + 1*x + 1*x^2 + Order(x^3))*(1 + (-1)*x + 1*x^2 + Order(x^3)) sage: _.series(x,3) 1 + 1*x^2 + Order(x^3)
Following the GiNaC tutorial, we use John Machin’s amazing formula \(\pi = 16 \tan^{-1}(1/5) - 4 \tan^{-1}(1/239)\) to compute digits of \(\pi\). We expand the arc tangent around 0 and insert the fractions 1/5 and 1/239.
sage: x = var('x') sage: f = atan(x).series(x, 10); f 1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10) sage: float(16*f.subs(x==1/5) - 4*f.subs(x==1/239)) 3.1415926824043994
- show()#
Pretty-print this symbolic expression.
This typesets it nicely and prints it immediately.
OUTPUT:
This method does not return anything. Like
print, output is sent directly to the screen.Note that the output depends on the display preferences. For details, see
pretty_print().EXAMPLES:
sage: (x^2 + 1).show() x^2 + 1
EXAMPLES:
sage: %display ascii_art # not tested sage: (x^2 + 1).show() 2 x + 1
- simplify(algorithm='maxima', **kwds)#
Return a simplified version of this symbolic expression.
INPUT:
algorithm– one of :maxima: (default) sends the expression tomaximaand converts it back to Sagesympy: converts the expression tosympy, simplifies it (passing any optional keyword(s)), and converts the result to Sagegiac: converts the expression togiac, simplifies it, and converts the result to Sagefricas: converts the expression tofricas, simplifies it, and converts the result to Sage
See also
simplify_full(),simplify_trig(),simplify_rational(),simplify_rectform()simplify_factorial(),simplify_log(),simplify_real(),simplify_hypergeometric(),canonicalize_radical()EXAMPLES:
sage: a = var('a'); f = x*sin(2)/(x^a); f x*sin(2)/x^a sage: f.simplify() x^(-a + 1)*sin(2)
Some simplifications are quite algorithm-specific:
sage: x, t = var("x, t") sage: ex = cos(t).exponentialize() sage: ex = ex.subs((sin(t).exponentialize()==x).solve(t)[0]) sage: ex 1/2*I*x + 1/2*I*sqrt(x^2 - 1) + 1/2/(I*x + I*sqrt(x^2 - 1)) sage: ex.simplify() 1/2*I*x + 1/2*I*sqrt(x^2 - 1) + 1/(2*I*x + 2*I*sqrt(x^2 - 1)) sage: ex.simplify(algorithm="sympy") I*(x^2 + sqrt(x^2 - 1)*x - 1)/(x + sqrt(x^2 - 1)) sage: ex.simplify(algorithm="giac") I*sqrt(x^2 - 1) sage: ex.simplify(algorithm="fricas") # optional - fricas (I*x^2 + I*sqrt(x^2 - 1)*x - I)/(x + sqrt(x^2 - 1))
- simplify_factorial()#
Simplify by combining expressions with factorials, and by expanding binomials into factorials.
ALIAS: factorial_simplify and simplify_factorial are the same
EXAMPLES:
Some examples are relatively clear:
sage: var('n,k') (n, k) sage: f = factorial(n+1)/factorial(n); f factorial(n + 1)/factorial(n) sage: f.simplify_factorial() n + 1
sage: f = factorial(n)*(n+1); f (n + 1)*factorial(n) sage: simplify(f) (n + 1)*factorial(n) sage: f.simplify_factorial() factorial(n + 1)
sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f binomial(n, k)*factorial(k)*factorial(-k + n) sage: f.simplify_factorial() factorial(n)
A more complicated example, which needs further processing:
sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f 1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2) sage: g = f.simplify_factorial(); g 1/2*(x - 1)*x + 1/2*x + 1/2 sage: g.simplify_rational() 1/2*x^2 + 1/2
- simplify_full()#
Apply
simplify_factorial(),simplify_rectform(),simplify_trig(),simplify_rational(), and thenexpand_sum()to self (in that order).ALIAS:
simplify_fullandfull_simplifyare the same.EXAMPLES:
sage: f = sin(x)^2 + cos(x)^2 sage: f.simplify_full() 1
sage: f = sin(x/(x^2 + x)) sage: f.simplify_full() sin(1/(x + 1))
sage: var('n,k') (n, k) sage: f = binomial(n,k)*factorial(k)*factorial(n-k) sage: f.simplify_full() factorial(n)
- simplify_hypergeometric(algorithm='maxima')#
Simplify an expression containing hypergeometric or confluent hypergeometric functions.
INPUT:
algorithm– (default:'maxima') the algorithm to use for for simplification. Implemented are'maxima', which uses Maxima’shgfredfunction, and'sage', which uses an algorithm implemented in the hypergeometric module
ALIAS:
hypergeometric_simplify()andsimplify_hypergeometric()are the sameEXAMPLES:
sage: hypergeometric((5, 4), (4, 1, 2, 3), ....: x).simplify_hypergeometric() 1/144*x^2*hypergeometric((), (3, 4), x) +... 1/3*x*hypergeometric((), (2, 3), x) + hypergeometric((), (1, 2), x) sage: (2*hypergeometric((), (), x)).simplify_hypergeometric() 2*e^x sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1) # not tested, unstable ....: .simplify_hypergeometric()) laguerre(-laguerre(-e^x, x), x) sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1) # not tested, unstable ....: .simplify_hypergeometric(algorithm='sage')) hypergeometric((hypergeometric((e^x,), (1,), x),), (1,), x) sage: hypergeometric_M(1, 3, x).simplify_hypergeometric() -2*((x + 1)*e^(-x) - 1)*e^x/x^2 sage: (2 * hypergeometric_U(1, 3, x)).simplify_hypergeometric() 2*(x + 1)/x^2
- simplify_log(algorithm=None)#
Simplify a (real) symbolic expression that contains logarithms.
The given expression is scanned recursively, transforming subexpressions of the form \(a \log(b) + c \log(d)\) into \(\log(b^{a} d^{c})\) before simplifying within the
log().The user can specify conditions that \(a\) and \(c\) must satisfy before this transformation will be performed using the optional parameter
algorithm.Warning
This is only safe to call if every variable in the given expression is assumed to be real. The simplification it performs is in general not valid over the complex numbers. For example:
sage: x,y = SR.var('x,y') sage: f = log(x*y) - (log(x) + log(y)) sage: f(x=-1, y=i) -2*I*pi sage: f.simplify_log() 0
INPUT:
self- expression to be simplifiedalgorithm- (default: None) optional, governs the condition on \(a\) and \(c\) which must be satisfied to contract expression \(a \log(b) + c \log(d)\). Values areNone(use Maxima default, integers),'one'(1 and -1),'ratios'(rational numbers),'constants'(constants),'all'(all expressions).
ALGORITHM:
This uses the Maxima
logcontract()command.ALIAS:
log_simplify()andsimplify_log()are the same.EXAMPLES:
sage: x,y,t = var('x y t')
Only two first terms are contracted in the following example; the logarithm with coefficient \(\frac{1}{2}\) is not contracted:
sage: f = log(x)+2*log(y)+1/2*log(t) sage: f.simplify_log() log(x*y^2) + 1/2*log(t)
To contract all terms in the previous example, we use the
'ratios'algorithm:sage: f.simplify_log(algorithm='ratios') log(sqrt(t)*x*y^2)
To contract terms with no coefficient (more precisely, with coefficients \(1\) and \(-1\)), we use the
'one'algorithm:sage: f = log(x)+2*log(y)-log(t) sage: f.simplify_log('one') 2*log(y) + log(x/t)
sage: f = log(x)+log(y)-1/3*log((x+1)) sage: f.simplify_log() log(x*y) - 1/3*log(x + 1) sage: f.simplify_log('ratios') log(x*y/(x + 1)^(1/3))
\(\pi\) is an irrational number; to contract logarithms in the following example we have to set
algorithmto'constants'or'all':sage: f = log(x)+log(y)-pi*log((x+1)) sage: f.simplify_log('constants') log(x*y/(x + 1)^pi)
x*log(9)is contracted only ifalgorithmis'all':sage: (x*log(9)).simplify_log() 2*x*log(3) sage: (x*log(9)).simplify_log('all') log(3^(2*x))
AUTHORS:
Robert Marik (11-2009)
- simplify_rational(algorithm='full', map=False)#
Simplify rational expressions.
INPUT:
self- symbolic expressionalgorithm- (default: ‘full’) string which switches the algorithm for simplifications. Possible values are‘simple’ (simplify rational functions into quotient of two polynomials),
‘full’ (apply repeatedly, if necessary)
‘noexpand’ (convert to common denominator and add)
map- (default:False) ifTrue, the result is an expression whose leading operator is the same as that of the expressionselfbut whose subparts are the results of applying simplification rules to the corresponding subparts of the expressions.
ALIAS:
rational_simplify()andsimplify_rational()are the sameDETAILS: We call Maxima functions ratsimp, fullratsimp and xthru. If each part of the expression has to be simplified separately, we use Maxima function map.
EXAMPLES:
sage: f = sin(x/(x^2 + x)) sage: f sin(x/(x^2 + x)) sage: f.simplify_rational() sin(1/(x + 1))
sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f -((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1)) sage: f.simplify_rational() -2*sqrt(x - 1)/sqrt(x^2 - 1)
With
map=Trueeach term in a sum is simplified separately and thus the results are shorter for functions which are combination of rational and nonrational functions. In the following example, we use this option if we want not to combine logarithm and the rational function into one fraction:sage: f = (x^2-1)/(x+1)-ln(x)/(x+2) sage: f.simplify_rational() (x^2 + x - log(x) - 2)/(x + 2) sage: f.simplify_rational(map=True) x - log(x)/(x + 2) - 1
Here is an example from the Maxima documentation of where
algorithm='simple'produces an (possibly useful) intermediate step:sage: y = var('y') sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1) sage: g.simplify_rational(algorithm='simple') (x^(2*y) - 2*x^y + 1)/(x^y - 1) sage: g.simplify_rational() x^y - 1
With option
algorithm='noexpand'we only convert to common denominators and add. No expansion of products is performed:sage: f = 1/(x+1)+x/(x+2)^2 sage: f.simplify_rational() (2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4) sage: f.simplify_rational(algorithm='noexpand') ((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))
- simplify_real()#
Simplify the given expression over the real numbers. This allows the simplification of \(\sqrt{x^{2}}\) into \(\left|x\right|\) and the contraction of \(\log(x) + \log(y)\) into \(\log(xy)\).
INPUT:
self– the expression to convert.
OUTPUT:
A new expression, equivalent to the original one under the assumption that the variables involved are real.
EXAMPLES:
sage: f = sqrt(x^2) sage: f.simplify_real() abs(x)
sage: y = SR.var('y') sage: f = log(x) + 2*log(y) sage: f.simplify_real() log(x*y^2)
- simplify_rectform(complexity_measure='string_length')#
Attempt to simplify this expression by expressing it in the form \(a + bi\) where both \(a\) and \(b\) are real. This transformation is generally not a simplification, so we use the given
complexity_measureto discard non-simplifications.INPUT:
self– the expression to simplify.complexity_measure– (default:sage.symbolic.complexity_measures.string_length) a function taking a symbolic expression as an argument and returning a measure of that expressions complexity. IfNoneis supplied, the simplification will be performed regardless of the result.
OUTPUT:
If the transformation produces a simpler expression (according to
complexity_measure) then that simpler expression is returned. Otherwise, the original expression is returned.ALGORITHM:
We first call
rectform()on the given expression. Then, the supplied complexity measure is used to determine whether or not the result is simpler than the original expression.EXAMPLES:
The exponential form of \(\tan(x)\):
sage: f = ( e^(I*x) - e^(-I*x) ) / ( I*e^(I*x) + I*e^(-I*x) ) sage: f.simplify_rectform() sin(x)/cos(x)
This should not be expanded with Euler’s formula since the resulting expression is longer when considered as a string, and the default
complexity_measureuses string length to determine which expression is simpler:sage: f = e^(I*x) sage: f.simplify_rectform() e^(I*x)
However, if we pass
Noneas our complexity measure, it is:sage: f = e^(I*x) sage: f.simplify_rectform(complexity_measure = None) cos(x) + I*sin(x)
- simplify_trig(expand=True)#
Optionally expand and then employ identities such as \(\sin(x)^2 + \cos(x)^2 = 1\), \(\cosh(x)^2 - \sinh(x)^2 = 1\), \(\sin(x)\csc(x) = 1\), or \(\tanh(x)=\sinh(x)/\cosh(x)\) to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.
INPUT:
self- symbolic expressionexpand- (default:True) if True, expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring inselffirst. For best results,selfshould be expanded. See alsoexpand_trig()to get more controls on this expansion.
ALIAS:
trig_simplify()andsimplify_trig()are the sameEXAMPLES:
sage: f = sin(x)^2 + cos(x)^2; f cos(x)^2 + sin(x)^2 sage: f.simplify() cos(x)^2 + sin(x)^2 sage: f.simplify_trig() 1 sage: h = sin(x)*csc(x) sage: h.simplify_trig() 1 sage: k = tanh(x)*cosh(2*x) sage: k.simplify_trig() (2*sinh(x)^3 + sinh(x))/cosh(x)
In some cases we do not want to expand:
sage: f = tan(3*x) sage: f.simplify_trig() -(4*cos(x)^2 - 1)*sin(x)/(4*cos(x)*sin(x)^2 - cos(x)) sage: f.simplify_trig(False) sin(3*x)/cos(3*x)
- sin(hold=False)#
EXAMPLES:
sage: var('x, y') (x, y) sage: sin(x^2 + y^2) sin(x^2 + y^2) sage: sin(sage.symbolic.constants.pi) 0 sage: sin(SR(1)) sin(1) sage: sin(SR(RealField(150)(1))) 0.84147098480789650665250232163029899962256306
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(0).sin() 0 sage: SR(0).sin(hold=True) sin(0)
This also works using functional notation:
sage: sin(0,hold=True) sin(0) sage: sin(0) 0
To then evaluate again, we use
unhold():sage: a = SR(0).sin(hold=True); a.unhold() 0
- sinh(hold=False)#
Return sinh of self.
We have \(\sinh(x) = (e^{x} - e^{-x})/2\).
EXAMPLES:
sage: x.sinh() sinh(x) sage: SR(1).sinh() sinh(1) sage: SR(0).sinh() 0 sage: SR(1.0).sinh() 1.17520119364380 sage: maxima('sinh(1.0)') 1.17520119364380... sinh(1.0000000000000000000000000) sage: SR(1).sinh().n(90) 1.1752011936438014568823819 sage: SR(RIF(1)).sinh() 1.175201193643802?
To prevent automatic evaluation use the
holdargument:sage: arccosh(x).sinh() sqrt(x + 1)*sqrt(x - 1) sage: arccosh(x).sinh(hold=True) sinh(arccosh(x))
This also works using functional notation:
sage: sinh(arccosh(x),hold=True) sinh(arccosh(x)) sage: sinh(arccosh(x)) sqrt(x + 1)*sqrt(x - 1)
To then evaluate again, we use
unhold():sage: a = arccosh(x).sinh(hold=True); a.simplify() sqrt(x + 1)*sqrt(x - 1)
- solve(x, multiplicities=False, solution_dict=False, explicit_solutions=False, to_poly_solve=False, algorithm=None, domain=None)#
Analytically solve the equation
self == 0or a univariate inequality for the variable \(x\).Warning
This is not a numerical solver – use
find_root()to solve forself == 0numerically on an interval.INPUT:
x– variable(s) to solve formultiplicities– bool (default:False); ifTrue, return corresponding multiplicities. This keyword is incompatible withto_poly_solve=Trueand does not make any sense when solving an inequality.solution_dict– bool (default:False); ifTrueor non-zero, return a list of dictionaries containing solutions. Not used when solving an inequality.explicit_solutions– bool (default:False); require that all roots be explicit rather than implicit. Not used when solving an inequality.to_poly_solve– bool (default:False) or string; use Maxima’sto_poly_solverpackage to search for more possible solutions, but possibly encounter approximate solutions. This keyword is incompatible withmultiplicities=Trueand is not used when solving an inequality. Settingto_poly_solveto'force'omits Maxima’s solve command (useful when some solutions of trigonometric equations are lost).
EXAMPLES:
sage: z = var('z') sage: (z^5 - 1).solve(z) [z == 1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4, z == -1/4*sqrt(5) + 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4, z == -1/4*sqrt(5) - 1/4*I*sqrt(-2*sqrt(5) + 10) - 1/4, z == 1/4*sqrt(5) - 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4, z == 1] sage: solve((z^3-1)^3, z, multiplicities=True) ([z == 1/2*I*sqrt(3) - 1/2, z == -1/2*I*sqrt(3) - 1/2, z == 1], [3, 3, 3])
- solve_diophantine(x=None, solution_dict=False)#
Solve a polynomial equation in the integers (a so called Diophantine).
If the argument is just a polynomial expression, equate to zero. If
solution_dict=Truereturn a list of dictionaries instead of a list of tuples.EXAMPLES:
sage: x,y = var('x,y') sage: solve_diophantine(3*x == 4) # needs sympy [] sage: solve_diophantine(x^2 - 9) # needs sympy [-3, 3] sage: sorted(solve_diophantine(x^2 + y^2 == 25)) # needs sympy [(-5, 0), (-4, -3), (-4, 3), (-3, -4), (-3, 4), (0, -5)...
The function is used when
solve()is called with all variables assumed integer:sage: assume(x, 'integer') sage: assume(y, 'integer') sage: sorted(solve(x*y == 1, (x,y))) # needs sympy [(-1, -1), (1, 1)]
You can also pick specific variables, and get the solution as a dictionary:
sage: # needs sympy sage: solve_diophantine(x*y == 10, x) [-10, -5, -2, -1, 1, 2, 5, 10] sage: sorted(solve_diophantine(x*y - y == 10, (x,y))) [(-9, -1), (-4, -2), (-1, -5), (0, -10), (2, 10), (3, 5), (6, 2), (11, 1)] sage: res = solve_diophantine(x*y - y == 10, solution_dict=True) sage: sol = [{y: -5, x: -1}, {y: -10, x: 0}, {y: -1, x: -9}, {y: -2, x: -4}, ....: {y: 10, x: 2}, {y: 1, x: 11}, {y: 2, x: 6}, {y: 5, x: 3}] sage: all(solution in res ....: for solution in sol) and bool(len(res) == len(sol)) True
If the solution is parametrized the parameter(s) are not defined, but you can substitute them with specific integer values:
sage: # needs sympy sage: x,y,z = var('x,y,z') sage: sol = solve_diophantine(x^2-y == 0); sol (t, t^2) sage: [(sol[0].subs(t=t),sol[1].subs(t=t)) for t in range(-3,4)] [(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)] sage: sol = solve_diophantine(x^2 + y^2 == z^2); sol (2*p*q, p^2 - q^2, p^2 + q^2) sage: [(sol[0].subs(p=p,q=q), sol[1].subs(p=p,q=q), sol[2].subs(p=p,q=q)) ....: for p in range(1,4) for q in range(1,4)] [(2, 0, 2), (4, -3, 5), (6, -8, 10), (4, 3, 5), (8, 0, 8), (12, -5, 13), (6, 8, 10), (12, 5, 13), (18, 0, 18)]
Solve Brahmagupta-Pell equations:
sage: sol = sorted(solve_diophantine(x^2 - 2*y^2 == 1), key=str); sol # needs sympy [(-sqrt(2)*(2*sqrt(2) + 3)^t + sqrt(2)*(-2*sqrt(2) + 3)^t - 3/2*(2*sqrt(2) + 3)^t - 3/2*(-2*sqrt(2) + 3)^t,... sage: [(sol[1][0].subs(t=t).simplify_full(), # needs sympy ....: sol[1][1].subs(t=t).simplify_full()) for t in range(-1,5)] [(1, 0), (3, -2), (17, -12), (99, -70), (577, -408), (3363, -2378)]
- sqrt(hold=False)#
Return the square root of this expression
EXAMPLES:
sage: var('x, y') (x, y) sage: SR(2).sqrt() sqrt(2) sage: (x^2+y^2).sqrt() sqrt(x^2 + y^2) sage: (x^2).sqrt() sqrt(x^2)
Immediate simplifications are applied:
sage: sqrt(x^2) sqrt(x^2) sage: x = SR.symbol('x', domain='real') sage: sqrt(x^2) abs(x) sage: forget() sage: assume(x<0) sage: sqrt(x^2) -x sage: sqrt(x^4) x^2 sage: forget() sage: x = SR.symbol('x', domain='real') sage: sqrt(x^4) x^2 sage: sqrt(sin(x)^2) abs(sin(x)) sage: sqrt((x+1)^2) abs(x + 1) sage: forget() sage: assume(x<0) sage: sqrt((x-1)^2) -x + 1 sage: forget()
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(4).sqrt() 2 sage: SR(4).sqrt(hold=True) sqrt(4)
To then evaluate again, we use
unhold():sage: a = SR(4).sqrt(hold=True); a.unhold() 2
To use this parameter in functional notation, you must coerce to the symbolic ring:
sage: sqrt(SR(4),hold=True) sqrt(4) sage: sqrt(4,hold=True) Traceback (most recent call last): ... TypeError: ..._do_sqrt() got an unexpected keyword argument 'hold'
- step(hold=False)#
Return the value of the unit step function, which is 0 for negative x, 1 for 0, and 1 for positive x.
EXAMPLES:
sage: x = var('x') sage: SR(1.5).step() 1 sage: SR(0).step() 1 sage: SR(-1/2).step() 0 sage: SR(float(-1)).step() 0
Using the
holdparameter it is possible to prevent automatic evaluation:sage: SR(2).step() 1 sage: SR(2).step(hold=True) unit_step(2)
- subs(*args, **kwds)#
Substitute the given subexpressions in this expression.
EXAMPLES:
sage: var('x,y,z,a,b,c,d,f,g') (x, y, z, a, b, c, d, f, g) sage: w0 = SR.wild(0); w1 = SR.wild(1) sage: t = a^2 + b^2 + (x+y)^3
Substitute with keyword arguments (works only with symbols):
sage: t.subs(a=c) (x + y)^3 + b^2 + c^2 sage: t.subs(b=19, x=z) (y + z)^3 + a^2 + 361
Substitute with a dictionary argument:
sage: t.subs({a^2: c}) (x + y)^3 + b^2 + c sage: t.subs({w0^2: w0^3}) a^3 + b^3 + (x + y)^3
Substitute with one or more relational expressions:
sage: t.subs(w0^2 == w0^3) a^3 + b^3 + (x + y)^3 sage: t.subs(w0 == w0^2) a^8 + b^8 + (x^2 + y^2)^6 sage: t.subs(a == b, b == c) (x + y)^3 + b^2 + c^2
Any number of arguments is accepted:
sage: t.subs(a=b, b=c) (x + y)^3 + b^2 + c^2 sage: t.subs({a:b}, b=c) (x + y)^3 + b^2 + c^2 sage: t.subs([x == 3, y == 2], a == 2, {b:3}) 138
It can even accept lists of lists:
sage: eqn1 = (a*x + b*y == 0) sage: eqn2 = (1 + y == 0) sage: soln = solve([eqn1, eqn2], [x, y]) sage: soln [[x == b/a, y == -1]] sage: f = x + y sage: f.subs(soln) b/a - 1
Duplicate assignments will throw an error:
sage: t.subs({a:b}, a=c) Traceback (most recent call last): ... ValueError: duplicate substitution for a, got values b and c sage: t.subs([x == 1], a = 1, b = 2, x = 2) Traceback (most recent call last): ... ValueError: duplicate substitution for x, got values 1 and 2
All substitutions are performed at the same time:
sage: t.subs({a:b, b:c}) (x + y)^3 + b^2 + c^2
Substitutions are done term by term, in other words Sage is not able to identify partial sums in a substitution (see github issue #18396):
sage: f = x + x^2 + x^4 sage: f.subs(x = y) y^4 + y^2 + y sage: f.subs(x^2 == y) # one term is fine x^4 + x + y sage: f.subs(x + x^2 == y) # partial sum does not work x^4 + x^2 + x sage: f.subs(x + x^2 + x^4 == y) # whole sum is fine y
Note that it is the very same behavior as in Maxima:
sage: E = 'x^4 + x^2 + x' sage: subs = [('x','y'), ('x^2','y'), ('x^2+x','y'), ('x^4+x^2+x','y')] sage: cmd = '{}, {}={}' sage: for s1,s2 in subs: ....: maxima.eval(cmd.format(E, s1, s2)) 'y^4+y^2+y' 'y+x^4+x' 'x^4+x^2+x' 'y'
Or as in Maple:
sage: cmd = 'subs({}={}, {})' # optional - maple sage: for s1,s2 in subs: # optional - maple ....: maple.eval(cmd.format(s1,s2, E)) 'y^4+y^2+y' 'x^4+x+y' 'x^4+x^2+x' 'y'
But Mathematica does something different on the third example:
sage: cmd = '{} /. {} -> {}' # optional - mathematica sage: for s1,s2 in subs: # optional - mathematica ....: mathematica.eval(cmd.format(E,s1,s2)) 2 4 y + y + y 4 x + x + y 4 x + y y
The same, with formatting more suitable for cut and paste:
sage: for s1,s2 in subs: # optional - mathematica ....: mathematica(cmd.format(E,s1,s2)) y + y^2 + y^4 x + x^4 + y x^4 + y y
Warning
Unexpected results may occur if the left-hand side of some substitution is not just a single variable (or is a “wildcard” variable). For example, the result of
cos(cos(cos(x))).subs({cos(x) : x})isx, because the substitution is applied repeatedly. Such repeated substitutions (and pattern-matching code that may be somewhat unpredictable) are disabled only in the basic case where the left-hand side of every substitution is a variable. In particular, although the result of(x^2).subs({x : sqrt(x)})isx, the result of(x^2).subs({x : sqrt(x), y^2 : y})issqrt(x), because repeated substitution is enabled by the presence of the expressiony^2in the left-hand side of one of the substitutions, even though that particular substitution does not get applied.
- substitute_function(*args, **kwds)#
Substitute the given functions by their replacements in this expression.
EXAMPLES:
sage: x,y = var('x,y') sage: foo = function('foo'); bar = function('bar') sage: f = foo(x) + 1/foo(pi*y)
Substitute with a dictionary:
sage: f.substitute_function({foo: bar}) 1/bar(pi*y) + bar(x) sage: f.substitute_function({foo(x): bar(x)}) 1/bar(pi*y) + bar(x)
If the function expression to be substituted includes its arguments, the right hand side can be an arbitrary symbolic expression:
sage: f.substitute_function({foo(x): x^2}) x^2 + 1/(pi^2*y^2)
Substitute with keyword arguments (works only if no function arguments are given):
sage: f.substitute_function(foo=bar) 1/bar(pi*y) + bar(x)
Substitute with a relational expression:
sage: f.substitute_function(foo(x)==bar(x)) 1/bar(pi*y) + bar(x) sage: f.substitute_function(foo(x)==bar(x+1)) 1/bar(pi*y + 1) + bar(x + 1)
All substitutions are performed at the same time:
sage: g = foo(x) + 1/bar(pi*y) sage: g.substitute_function({foo: bar, bar: foo}) 1/foo(pi*y) + bar(x)
Any number of arguments is accepted:
sage: g.substitute_function({foo: bar}, bar(x) == x^2) 1/(pi^2*y^2) + bar(x)
As well as lists of substitutions:
sage: g.substitute_function([foo(x) == 1, bar(x) == x]) 1/(pi*y) + 1
Alternative syntax:
sage: g.substitute_function(foo, bar) 1/bar(pi*y) + bar(x)
Duplicate assignments will throw an error:
sage: g.substitute_function({foo:bar}, foo(x) == x^2) Traceback (most recent call last): ... ValueError: duplicate substitution for foo, got values bar and x |--> x^2 sage: g.substitute_function([foo(x) == x^2], foo = bar) Traceback (most recent call last): ... ValueError: duplicate substitution for foo, got values x |--> x^2 and bar
- substitution_delayed(pattern, replacement)#
Replace all occurrences of pattern by the result of replacement.
In contrast to
subs(), the pattern may contains wildcards and the replacement can depend on the particular term matched by the pattern.INPUT:
pattern– anExpression, usually containing wildcards.replacement– a function. Its argument is a dictionary mapping the wildcard occurring inpatternto the actual values. If it returnsNone, this occurrence ofpatternis not replaced. Otherwise, it is replaced by the output ofreplacement.
OUTPUT:
An
Expression.EXAMPLES:
sage: var('x y') (x, y) sage: w0 = SR.wild(0) sage: sqrt(1 + 2*x + x^2).substitution_delayed( ....: sqrt(w0), lambda d: sqrt(factor(d[w0])) ....: ) sqrt((x + 1)^2) sage: def r(d): ....: if x not in d[w0].variables(): ....: return cos(d[w0]) sage: (sin(x^2 + x) + sin(y^2 + y)).substitution_delayed(sin(w0), r) cos(y^2 + y) + sin(x^2 + x)
See also
- subtract_from_both_sides(x)#
Return a relation obtained by subtracting
xfrom both sides of this relation.EXAMPLES:
sage: eqn = x*sin(x)*sqrt(3) + sqrt(2) > cos(sin(x)) sage: eqn.subtract_from_both_sides(sqrt(2)) sqrt(3)*x*sin(x) > -sqrt(2) + cos(sin(x)) sage: eqn.subtract_from_both_sides(cos(sin(x))) sqrt(3)*x*sin(x) + sqrt(2) - cos(sin(x)) > 0
- sum(*args, **kwds)#
Return the symbolic sum \(\sum_{v = a}^b\)
selfwith respect to the variable \(v\) with endpoints \(a\) and \(b\).
INPUT:
v– a variable or variable namea– lower endpoint of the sumb– upper endpoint of the sumalgorithm- (default:'maxima') one of'maxima'– use Maxima (the default)'maple'– (optional) use Maple'mathematica'– (optional) use Mathematica'giac'– (optional) use Giac'sympy'– use SymPy
EXAMPLES:
sage: k, n = var('k,n') sage: k.sum(k, 1, n).factor() 1/2*(n + 1)*n
sage: (1/k^4).sum(k, 1, oo) 1/90*pi^4
sage: (1/k^5).sum(k, 1, oo) zeta(5)
A well known binomial identity:
sage: assume(n>=0) sage: binomial(n,k).sum(k, 0, n) 2^n
And some truncations thereof:
sage: binomial(n,k).sum(k,1,n) 2^n - 1 sage: binomial(n,k).sum(k,2,n) 2^n - n - 1 sage: binomial(n,k).sum(k,0,n-1) 2^n - 1 sage: binomial(n,k).sum(k,1,n-1) 2^n - 2
The binomial theorem:
sage: x, y = var('x, y') sage: (binomial(n,k) * x^k * y^(n-k)).sum(k, 0, n) (x + y)^n
sage: (k * binomial(n, k)).sum(k, 1, n) 2^(n - 1)*n
sage: ((-1)^k*binomial(n,k)).sum(k, 0, n) 0
sage: (2^(-k)/(k*(k+1))).sum(k, 1, oo) -log(2) + 1
Summing a hypergeometric term:
sage: (binomial(n, k) * factorial(k) / factorial(n+1+k)).sum(k, 0, n) 1/2*sqrt(pi)/factorial(n + 1/2)
We check a well known identity:
sage: bool((k^3).sum(k, 1, n) == k.sum(k, 1, n)^2) True
A geometric sum:
sage: a, q = var('a, q') sage: (a*q^k).sum(k, 0, n) (a*q^(n + 1) - a)/(q - 1)
The geometric series:
sage: assume(abs(q) < 1) sage: (a*q^k).sum(k, 0, oo) -a/(q - 1)
A divergent geometric series. Do not forget to \(forget\) your assumptions:
sage: forget() sage: assume(q > 1) sage: (a*q^k).sum(k, 0, oo) Traceback (most recent call last): ... ValueError: Sum is divergent.
This summation only Mathematica can perform:
sage: (1/(1+k^2)).sum(k, -oo, oo, algorithm = 'mathematica') # optional - mathematica pi*coth(pi)
Use Giac to perform this summation:
sage: (sum(1/(1+k^2), k, -oo, oo, algorithm = 'giac')).factor() pi*(e^(2*pi) + 1)/((e^pi + 1)*(e^pi - 1))
Use Maple as a backend for summation:
sage: (binomial(n,k)*x^k).sum(k, 0, n, algorithm = 'maple') # optional - maple (x + 1)^n
Note
Sage can currently only understand a subset of the output of Maxima, Maple and Mathematica, so even if the chosen backend can perform the summation the result might not be convertible into a usable Sage expression.
- tan(hold=False)#
EXAMPLES:
sage: var('x, y') (x, y) sage: tan(x^2 + y^2) tan(x^2 + y^2) sage: tan(sage.symbolic.constants.pi/2) Infinity sage: tan(SR(1)) tan(1) sage: tan(SR(RealField(150)(1))) 1.5574077246549022305069748074583601730872508
To prevent automatic evaluation use the
holdargument:sage: (pi/12).tan() -sqrt(3) + 2 sage: (pi/12).tan(hold=True) tan(1/12*pi)
This also works using functional notation:
sage: tan(pi/12,hold=True) tan(1/12*pi) sage: tan(pi/12) -sqrt(3) + 2
To then evaluate again, we use
unhold():sage: a = (pi/12).tan(hold=True); a.unhold() -sqrt(3) + 2
- tanh(hold=False)#
Return tanh of self.
We have \(\tanh(x) = \sinh(x) / \cosh(x)\).
EXAMPLES:
sage: x.tanh() tanh(x) sage: SR(1).tanh() tanh(1) sage: SR(0).tanh() 0 sage: SR(1.0).tanh() 0.761594155955765 sage: maxima('tanh(1.0)') 0.7615941559557649 sage: plot(lambda x: SR(x).tanh(), -1, 1) # needs sage.plot Graphics object consisting of 1 graphics primitive
To prevent automatic evaluation use the
holdargument:sage: arcsinh(x).tanh() x/sqrt(x^2 + 1) sage: arcsinh(x).tanh(hold=True) tanh(arcsinh(x))
This also works using functional notation:
sage: tanh(arcsinh(x),hold=True) tanh(arcsinh(x)) sage: tanh(arcsinh(x)) x/sqrt(x^2 + 1)
To then evaluate again, we use
unhold():sage: a = arcsinh(x).tanh(hold=True); a.unhold() x/sqrt(x^2 + 1)
- taylor(*args)#
Expand this symbolic expression in a truncated Taylor or Laurent series in the variable \(v\) around the point \(a\), containing terms through \((x - a)^n\). Functions in more variables is also supported.
INPUT:
*args– the following notation is supportedx, a, n– variable, point, degree(x, a), (y, b), n– variables with points, degree of polynomial
EXAMPLES:
sage: var('a, x, z') (a, x, z) sage: taylor(a*log(z), z, 2, 3) 1/24*a*(z - 2)^3 - 1/8*a*(z - 2)^2 + 1/2*a*(z - 2) + a*log(2)
sage: taylor(sqrt(sin(x) + a*x + 1), x, 0, 3) 1/48*(3*a^3 + 9*a^2 + 9*a - 1)*x^3 - 1/8*(a^2 + 2*a + 1)*x^2 + 1/2*(a + 1)*x + 1
sage: taylor(sqrt(x + 1), x, 0, 5) 7/256*x^5 - 5/128*x^4 + 1/16*x^3 - 1/8*x^2 + 1/2*x + 1
sage: taylor(1/log(x + 1), x, 0, 3) -19/720*x^3 + 1/24*x^2 - 1/12*x + 1/x + 1/2
sage: taylor(cos(x) - sec(x), x, 0, 5) -1/6*x^4 - x^2
sage: taylor((cos(x) - sec(x))^3, x, 0, 9) -1/2*x^8 - x^6
sage: taylor(1/(cos(x) - sec(x))^3, x, 0, 5) -15377/7983360*x^4 - 6767/604800*x^2 + 11/120/x^2 + 1/2/x^4 - 1/x^6 - 347/15120
- test_relation(ntests=20, domain=None, proof=True)#
Test this relation at several random values, attempting to find a contradiction. If this relation has no variables, it will also test this relation after casting into the domain.
Because the interval fields never return false positives, we can be assured that if True or False is returned (and proof is False) then the answer is correct.
INPUT:
ntests– (default20) the number of iterations to rundomain– (optional) the domain from which to draw the random values defaults toCIFfor equality testing andRIFfor order testingproof– (defaultTrue) ifFalseand the domain is an interval field, regard overlapping (potentially equal) intervals as equal, and returnTrueif all tests succeeded.
OUTPUT:
Boolean or
NotImplemented, meaningTrue– this relation holds in the domain and has no variables.False– a contradiction was found.NotImplemented– no contradiction found.
EXAMPLES:
sage: (3 < pi).test_relation() True sage: (0 >= pi).test_relation() False sage: (exp(pi) - pi).n() 19.9990999791895 sage: (exp(pi) - pi == 20).test_relation() False sage: (sin(x)^2 + cos(x)^2 == 1).test_relation() NotImplemented sage: (sin(x)^2 + cos(x)^2 == 1).test_relation(proof=False) True sage: (x == 1).test_relation() False sage: var('x,y') (x, y) sage: (x < y).test_relation() False
- to_gamma()#
Convert factorial, binomial, and Pochhammer symbol expressions to their gamma function equivalents.
EXAMPLES:
sage: m,n = var('m n', domain='integer') sage: factorial(n).to_gamma() gamma(n + 1) sage: binomial(m,n).to_gamma() gamma(m + 1)/(gamma(m - n + 1)*gamma(n + 1))
- trailing_coeff(s)#
Return the trailing coefficient of s in self, i.e., the coefficient of the smallest power of s in self.
EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100 sage: f.trailing_coefficient(x) 2*sin(x*y) sage: f.trailing_coefficient(y) x sage: f.trailing_coefficient(sin(x*y)) a*x + x*y + x/y + 100
- trailing_coefficient(s)#
Return the trailing coefficient of s in self, i.e., the coefficient of the smallest power of s in self.
EXAMPLES:
sage: var('x,y,a') (x, y, a) sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100 sage: f.trailing_coefficient(x) 2*sin(x*y) sage: f.trailing_coefficient(y) x sage: f.trailing_coefficient(sin(x*y)) a*x + x*y + x/y + 100
- trig_expand(full=False, half_angles=False, plus=True, times=True)#
Expand trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in
self.For best results,
selfshould already be expanded.INPUT:
full– (default:False) To enhance user control of simplification, this function expands only one level at a time by default, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the optional parameter full toTrue.half_angles- (default:False) IfTrue, causes half-angles to be simplified away.plus– (default:True) Controls the sum rule; expansion of sums (e.g. \(\sin(x + y)\)) will take place only ifplusisTrue.times– (default:True) Controls the product rule, expansion of products (e.g. \(\sin(2 x)\)) will take place only iftimesisTrue.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: sin(5*x).expand_trig() 5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5 sage: cos(2*x + var('y')).expand_trig() cos(2*x)*cos(y) - sin(2*x)*sin(y)
We illustrate various options to this function:
sage: f = sin(sin(3*cos(2*x))*x) sage: f.expand_trig() sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x) sage: f.expand_trig(full=True) sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x) sage: sin(2*x).expand_trig(times=False) sin(2*x) sage: sin(2*x).expand_trig(times=True) 2*cos(x)*sin(x) sage: sin(2 + x).expand_trig(plus=False) sin(x + 2) sage: sin(2 + x).expand_trig(plus=True) cos(x)*sin(2) + cos(2)*sin(x) sage: sin(x/2).expand_trig(half_angles=False) sin(1/2*x) sage: sin(x/2).expand_trig(half_angles=True) (-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)
If the expression contains terms which are factored, we expand first:
sage: (x, k1, k2) = var('x, k1, k2') sage: cos((k1-k2)*x).expand().expand_trig() cos(k1*x)*cos(k2*x) + sin(k1*x)*sin(k2*x)
ALIAS:
trig_expand()andexpand_trig()are the same
- trig_reduce(var=None)#
Combine products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators.
INPUT:
self– a symbolic expressionvar– (default:None) the variable which is used for these transformations. If not specified, all variables are used.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: y = var('y') sage: f = sin(x)*cos(x)^3+sin(y)^2 sage: f.reduce_trig() -1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2
To reduce only the expressions involving x we use optional parameter:
sage: f.reduce_trig(x) sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)
ALIAS:
trig_reduce()andreduce_trig()are the same
- trig_simplify(expand=True)#
Optionally expand and then employ identities such as \(\sin(x)^2 + \cos(x)^2 = 1\), \(\cosh(x)^2 - \sinh(x)^2 = 1\), \(\sin(x)\csc(x) = 1\), or \(\tanh(x)=\sinh(x)/\cosh(x)\) to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.
INPUT:
self- symbolic expressionexpand- (default:True) if True, expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring inselffirst. For best results,selfshould be expanded. See alsoexpand_trig()to get more controls on this expansion.
ALIAS:
trig_simplify()andsimplify_trig()are the sameEXAMPLES:
sage: f = sin(x)^2 + cos(x)^2; f cos(x)^2 + sin(x)^2 sage: f.simplify() cos(x)^2 + sin(x)^2 sage: f.simplify_trig() 1 sage: h = sin(x)*csc(x) sage: h.simplify_trig() 1 sage: k = tanh(x)*cosh(2*x) sage: k.simplify_trig() (2*sinh(x)^3 + sinh(x))/cosh(x)
In some cases we do not want to expand:
sage: f = tan(3*x) sage: f.simplify_trig() -(4*cos(x)^2 - 1)*sin(x)/(4*cos(x)*sin(x)^2 - cos(x)) sage: f.simplify_trig(False) sin(3*x)/cos(3*x)
- truncate()#
Given a power series or expression, return the corresponding expression without the big oh.
INPUT:
self– a series as output by theseries()command.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: f = sin(x)/x^2 sage: f.truncate() sin(x)/x^2 sage: f.series(x,7) 1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7) sage: f.series(x,7).truncate() -1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x sage: f.series(x==1,3).truncate().expand() -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
- unhold(exclude=None)#
Evaluates any held operations (with the
holdkeyword) in the expressionINPUT:
self– an expression with held operationsexclude– (default: None) a list of operators to exclude from evaluation. Excluding arithmetic operators does not yet work (see github issue #10169).
OUTPUT:
A new expression with held operations, except those in
exclude, evaluatedEXAMPLES:
sage: a = exp(I * pi, hold=True) sage: a e^(I*pi) sage: a.unhold() -1 sage: b = x.add(x, hold=True) sage: b x + x sage: b.unhold() 2*x sage: (a + b).unhold() 2*x - 1 sage: c = (x.mul(x, hold=True)).add(x.mul(x, hold=True), hold=True) sage: c x*x + x*x sage: c.unhold() 2*x^2 sage: sin(tan(0, hold=True), hold=True).unhold() 0 sage: sin(tan(0, hold=True), hold=True).unhold(exclude=[sin]) sin(0) sage: (e^sgn(0, hold=True)).unhold() 1 sage: (e^sgn(0, hold=True)).unhold(exclude=[exp]) e^0 sage: log(3).unhold() log(3)
- unit(s)#
Return the unit of this expression when considered as a polynomial in
s.See also
content(),primitive_part(), andunit_content_primitive().INPUT:
s– a symbolic expression.
OUTPUT:
The unit part of a polynomial as a symbolic expression. It is defined as the sign of the leading coefficient.
EXAMPLES:
sage: (2*x+4).unit(x) 1 sage: (-2*x+1).unit(x) -1 sage: (2*x+1/2).unit(x) 1 sage: var('y') y sage: (2*x - 4*sin(y)).unit(sin(y)) -1
- unit_content_primitive(s)#
Return the factorization into unit, content, and primitive part.
INPUT:
s– a symbolic expression, usually a symbolic variable. The whole symbolic expressionselfwill be considered as a univariate polynomial ins.
OUTPUT:
A triple (unit, content, primitive polynomial)` containing the
unit,content, andprimitive polynomial. Their product equalsself.EXAMPLES:
sage: var('x,y') (x, y) sage: ex = 9*x^3*y+3*y sage: ex.unit_content_primitive(x) (1, 3*y, 3*x^3 + 1) sage: ex.unit_content_primitive(y) (1, 9*x^3 + 3, y)
- variables()#
Return sorted tuple of variables that occur in this expression.
EXAMPLES:
sage: (x,y,z) = var('x,y,z') sage: (x+y).variables() (x, y) sage: (2*x).variables() (x,) sage: (x^y).variables() (x, y) sage: sin(x+y^z).variables() (x, y, z)
- zeta(hold=False)#
EXAMPLES:
sage: x, y = var('x, y') sage: (x/y).zeta() zeta(x/y) sage: SR(2).zeta() 1/6*pi^2 sage: SR(3).zeta() zeta(3) sage: SR(CDF(0,1)).zeta() # abs tol 1e-16 # needs sage.libs.pari 0.003300223685324103 - 0.4181554491413217*I sage: CDF(0,1).zeta() # abs tol 1e-16 # needs sage.libs.pari 0.003300223685324103 - 0.4181554491413217*I sage: plot(lambda x: SR(x).zeta(), -10,10).show(ymin=-3, ymax=3) # needs sage.plot
To prevent automatic evaluation use the
holdargument:sage: SR(2).zeta(hold=True) zeta(2)
This also works using functional notation:
sage: zeta(2,hold=True) zeta(2) sage: zeta(2) 1/6*pi^2
To then evaluate again, we use
unhold():sage: a = SR(2).zeta(hold=True); a.unhold() 1/6*pi^2
- class sage.symbolic.expression.ExpressionIterator#
Bases:
object
- class sage.symbolic.expression.OperandsWrapper#
Bases:
SageObjectOperands wrapper for symbolic expressions.
EXAMPLES:
sage: x,y,z = var('x,y,z') sage: e = x + x*y + z^y + 3*y*z; e x*y + 3*y*z + x + z^y sage: e.op[1] 3*y*z sage: e.op[1,1] z sage: e.op[-1] z^y sage: e.op[1:] [3*y*z, x, z^y] sage: e.op[:2] [x*y, 3*y*z] sage: e.op[-2:] [x, z^y] sage: e.op[:-2] [x*y, 3*y*z] sage: e.op[-5] Traceback (most recent call last): ... IndexError: operand index out of range, got -5, expect between -4 and 3 sage: e.op[5] Traceback (most recent call last): ... IndexError: operand index out of range, got 5, expect between -4 and 3 sage: e.op[1,1,0] Traceback (most recent call last): ... TypeError: expressions containing only a numeric coefficient, constant or symbol have no operands sage: e.op[:1.5] Traceback (most recent call last): ... TypeError: slice indices must be integers or None or have an __index__ method sage: e.op[:2:1.5] Traceback (most recent call last): ... ValueError: step value must be an integer
- class sage.symbolic.expression.PynacConstant#
Bases:
object- expression()#
Returns this constant as an Expression.
EXAMPLES:
sage: from sage.symbolic.expression import PynacConstant sage: f = PynacConstant('foo', 'foo', 'real') sage: f + 2 Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: '<class 'sage.symbolic.expression.PynacConstant'>' and 'Integer Ring' sage: foo = f.expression(); foo foo sage: foo + 2 foo + 2
- name()#
Returns the name of this constant.
EXAMPLES:
sage: from sage.symbolic.expression import PynacConstant sage: f = PynacConstant('foo', 'foo', 'real') sage: f.name() 'foo'
- serial()#
Returns the underlying Pynac serial for this constant.
EXAMPLES:
sage: from sage.symbolic.expression import PynacConstant sage: f = PynacConstant('foo', 'foo', 'real') sage: f.serial() #random 15
- class sage.symbolic.expression.SubstitutionMap#
Bases:
SageObject- apply_to(expr, options)#
Apply the substitution to a symbolic expression
EXAMPLES:
sage: from sage.symbolic.expression import make_map sage: subs = make_map({x:x+1}) sage: subs.apply_to(x^2, 0) (x + 1)^2
- class sage.symbolic.expression.SymbolicSeries#
Bases:
ExpressionTrivial constructor.
EXAMPLES:
sage: loads(dumps((x+x^3).series(x,2))) 1*x + Order(x^2)
- coefficients(x=None, sparse=True)#
Return the coefficients of this symbolic series as a list of pairs.
INPUT:
x– optional variable.sparse– Boolean. IfFalsereturn a list with as muchentries as the order of the series.
OUTPUT:
Depending on the value of
sparse,A list of pairs
(expr, n), whereexpris a symbolic expression andnis a power (sparse=True, default)A list of expressions where the
n-th element is the coefficient ofx^nwhen self is seen as polynomial inx(sparse=False).
EXAMPLES:
sage: s = (1/(1-x)).series(x,6); s 1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 + Order(x^6) sage: s.coefficients() [[1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]] sage: s.coefficients(x, sparse=False) [1, 1, 1, 1, 1, 1] sage: x,y = var("x,y") sage: s = (1/(1-y*x-x)).series(x,3); s 1 + (y + 1)*x + ((y + 1)^2)*x^2 + Order(x^3) sage: s.coefficients(x, sparse=False) [1, y + 1, (y + 1)^2]
- default_variable()#
Return the expansion variable of this symbolic series.
EXAMPLES:
sage: s = (1/(1-x)).series(x,3); s 1 + 1*x + 1*x^2 + Order(x^3) sage: s.default_variable() x
- is_terminating_series()#
Return True if the series is without order term.
A series is terminating if it can be represented exactly, without requiring an order term. You can explicitly request terminating series by setting the order to positive infinity.
OUTPUT:
Boolean.
Trueif the series has no order term.EXAMPLES:
sage: (x^5+x^2+1).series(x, +oo) 1 + 1*x^2 + 1*x^5 sage: (x^5+x^2+1).series(x,+oo).is_terminating_series() True sage: SR(5).is_terminating_series() False sage: exp(x).series(x,10).is_terminating_series() False
- power_series(base_ring)#
Return the algebraic power series associated to this symbolic series.
The coefficients must be coercible to the base ring.
EXAMPLES:
sage: ex = (gamma(1-x)).series(x,3); ex 1 + euler_gamma*x + (1/2*euler_gamma^2 + 1/12*pi^2)*x^2 + Order(x^3) sage: g = ex.power_series(SR); g 1 + euler_gamma*x + (1/2*euler_gamma^2 + 1/12*pi^2)*x^2 + O(x^3) sage: g.parent() Power Series Ring in x over Symbolic Ring
- truncate()#
Given a power series or expression, return the corresponding expression without the big oh.
OUTPUT:
A symbolic expression.
EXAMPLES:
sage: f = sin(x)/x^2 sage: f.truncate() sin(x)/x^2 sage: f.series(x,7) 1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7) sage: f.series(x,7).truncate() -1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x sage: f.series(x==1,3).truncate().expand() -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
- sage.symbolic.expression.call_registered_function(serial, nargs, args, hold, allow_numeric_result, result_parent)#
Call a function registered with Pynac (GiNaC).
INPUT:
serial- serial number of the functionnargs- declared number of args (0 is variadic)args- a list of arguments to pass to the function; each must be anExpressionhold- whether to leave the call unevaluatedallow_numeric_result- ifTrue, keep numeric results numeric; ifFalse, make all results symbolic expressionsresult_parent- an instance ofSymbolicRing
EXAMPLES:
sage: from sage.symbolic.expression import find_registered_function, call_registered_function sage: s_arctan = find_registered_function('arctan', 1) sage: call_registered_function(s_arctan, 1, [SR(1)], False, True, SR) 1/4*pi sage: call_registered_function(s_arctan, 1, [SR(1)], True, True, SR) arctan(1) sage: call_registered_function(s_arctan, 1, [SR(0)], False, True, SR) 0 sage: call_registered_function(s_arctan, 1, [SR(0)], False, True, SR).parent() Integer Ring sage: call_registered_function(s_arctan, 1, [SR(0)], False, False, SR).parent() Symbolic Ring
- sage.symbolic.expression.doublefactorial(n)#
The double factorial combinatorial function:
n!! == n * (n-2) * (n-4) * … * ({1|2}) with 0!! == (-1)!! == 1.
INPUT:
n – an integer > = 1
EXAMPLES:
sage: from sage.symbolic.expression import doublefactorial sage: doublefactorial(-1) 1 sage: doublefactorial(0) 1 sage: doublefactorial(1) 1 sage: doublefactorial(5) 15 sage: doublefactorial(20) 3715891200 sage: prod( [20,18,..,2] ) 3715891200
- sage.symbolic.expression.find_registered_function(name, nargs)#
Look up a function registered with Pynac (GiNaC).
Raise a
ValueErrorif the function is not registered.OUTPUT:
serial number of the function, for use in
call_registered_function()
EXAMPLES:
sage: from sage.symbolic.expression import find_registered_function sage: find_registered_function('arctan', 1) # random 19 sage: find_registered_function('archenemy', 1) Traceback (most recent call last): ... ValueError: cannot find GiNaC function with name archenemy and 1 arguments
- sage.symbolic.expression.get_fn_serial()#
Return the overall size of the Pynac function registry which corresponds to the last serial value plus one.
EXAMPLES:
sage: from sage.symbolic.expression import get_fn_serial sage: from sage.symbolic.function import get_sfunction_from_serial sage: get_fn_serial() > 125 True sage: print(get_sfunction_from_serial(get_fn_serial())) None sage: get_sfunction_from_serial(get_fn_serial() - 1) is not None True
- sage.symbolic.expression.get_ginac_serial()#
Number of C++ level functions defined by GiNaC. (Defined mainly for testing.)
EXAMPLES:
sage: sage.symbolic.expression.get_ginac_serial() >= 35 True
- sage.symbolic.expression.get_sfunction_from_hash(myhash)#
Return an already created
SymbolicFunctiongiven the hash.EXAMPLES:
sage: from sage.symbolic.expression import get_sfunction_from_hash sage: get_sfunction_from_hash(1) # random
- sage.symbolic.expression.get_sfunction_from_serial(serial)#
Return an already created
SymbolicFunctiongiven the serial.These are stored in the dictionary
sfunction_serial_dict.EXAMPLES:
sage: from sage.symbolic.function import get_sfunction_from_serial sage: get_sfunction_from_serial(65) #random f
- class sage.symbolic.expression.hold_class#
Bases:
objectInstances of this class can be used with Python \(with\).
EXAMPLES:
sage: with hold: ....: tan(1/12*pi) ....: tan(1/12*pi) sage: tan(1/12*pi) -sqrt(3) + 2 sage: with hold: ....: 2^5 ....: 32 sage: with hold: ....: SR(2)^5 ....: 2^5 sage: with hold: ....: t=tan(1/12*pi) ....: sage: t tan(1/12*pi) sage: t.unhold() -sqrt(3) + 2
- start()#
Start a hold context.
EXAMPLES:
sage: hold.start() sage: SR(2)^5 2^5 sage: hold.stop() sage: SR(2)^5 32
- stop()#
Stop any hold context.
EXAMPLES:
sage: hold.start() sage: SR(2)^5 2^5 sage: hold.stop() sage: SR(2)^5 32
- sage.symbolic.expression.init_function_table()#
Initializes the function pointer table in Pynac. This must be called before Pynac is used; otherwise, there will be segfaults.
- sage.symbolic.expression.init_pynac_I()#
Initialize the numeric
Iobject in pynac. We use the generator ofQQ(i).EXAMPLES:
sage: from sage.symbolic.constants import I as symbolic_I sage: symbolic_I I sage: symbolic_I^2 -1
Note that conversions to real fields will give
TypeError:sage: float(symbolic_I) Traceback (most recent call last): ... TypeError: unable to simplify to float approximation sage: gp(symbolic_I) I sage: RR(symbolic_I) Traceback (most recent call last): ... TypeError: unable to convert '1.00000000000000*I' to a real number
We can convert to complex fields:
sage: C = ComplexField(200); C Complex Field with 200 bits of precision sage: C(symbolic_I) 1.0000000000000000000000000000000000000000000000000000000000*I sage: symbolic_I._complex_mpfr_field_(ComplexField(53)) 1.00000000000000*I sage: symbolic_I._complex_double_(CDF) 1.0*I sage: CDF(symbolic_I) 1.0*I sage: z = symbolic_I + symbolic_I; z 2*I sage: C(z) 2.0000000000000000000000000000000000000000000000000000000000*I sage: 1e8*symbolic_I 1.00000000000000e8*I sage: complex(symbolic_I) 1j sage: QQbar(symbolic_I) I sage: abs(symbolic_I) 1 sage: symbolic_I.minpoly() x^2 + 1 sage: maxima(2*symbolic_I) 2*%i
- sage.symbolic.expression.is_SymbolicEquation(x)#
Return True if x is a symbolic equation.
This function is deprecated.
EXAMPLES:
The following two examples are symbolic equations:
sage: from sage.symbolic.expression import is_SymbolicEquation sage: is_SymbolicEquation(sin(x) == x) doctest:warning... DeprecationWarning: is_SymbolicEquation is deprecated; use 'isinstance(x, sage.structure.element.Expression) and x.is_relational()' instead See https://github.com/sagemath/sage/issues/35505 for details. True sage: is_SymbolicEquation(sin(x) < x) True sage: is_SymbolicEquation(x) False
This is not, since
2==3evaluates to the booleanFalse:sage: is_SymbolicEquation(2 == 3) False
However here since both 2 and 3 are coerced to be symbolic, we obtain a symbolic equation:
sage: is_SymbolicEquation(SR(2) == SR(3)) True
- sage.symbolic.expression.make_map(subs_dict)#
Construct a new substitution map
OUTPUT:
A new
SubstitutionMapfor doctestingEXAMPLES:
sage: from sage.symbolic.expression import make_map sage: make_map({x:x+1}) SubsMap
- sage.symbolic.expression.math_sorted(expressions)#
Sort a list of symbolic numbers in the “Mathematics” order
INPUT:
expressions– a list/tuple/iterable of symbolic expressions, or something that can be converted to one.
OUTPUT:
The list sorted by ascending (real) value. If an entry does not define a real value (or plus/minus infinity), or if the comparison is not known, a
ValueErroris raised.EXAMPLES:
sage: from sage.symbolic.expression import math_sorted sage: math_sorted([SR(1), SR(e), SR(pi), sqrt(2)]) [1, sqrt(2), e, pi]
- sage.symbolic.expression.mixed_order(lhs, rhs)#
Comparison in the mixed order
INPUT:
lhs,rhs– two symbolic expressions or something that can be converted to one.
OUTPUT:
Either \(-1\), \(0\), or \(+1\) indicating the comparison. An exception is raised if the arguments cannot be converted into the symbolic ring.
EXAMPLES:
sage: from sage.symbolic.expression import mixed_order sage: mixed_order(1, oo) -1 sage: mixed_order(e, oo) -1 sage: mixed_order(pi, oo) -1 sage: mixed_order(1, sqrt(2)) -1 sage: mixed_order(x + x^2, x*(x+1)) -1
Check that github issue #12967 is fixed:
sage: mixed_order(SR(oo), sqrt(2)) 1
Ensure that github issue #32185 is fixed:
sage: mixed_order(pi, 0) 1 sage: mixed_order(golden_ratio, 0) 1 sage: mixed_order(log2, 0) 1
- sage.symbolic.expression.mixed_sorted(expressions)#
Sort a list of symbolic numbers in the “Mixed” order
INPUT:
expressions– a list/tuple/iterable of symbolic expressions, or something that can be converted to one.
OUTPUT:
In the list the numeric values are sorted by ascending (real) value, and the expressions with variables according to print order. If an entry does not define a real value (or plus/minus infinity), or if the comparison is not known, a
ValueErroris raised.EXAMPLES:
sage: from sage.symbolic.expression import mixed_sorted sage: mixed_sorted([SR(1), SR(e), SR(pi), sqrt(2), x, sqrt(x), sin(1/x)]) [1, sqrt(2), e, pi, sin(1/x), sqrt(x), x]
- sage.symbolic.expression.new_Expression(parent, x)#
Convert
xinto the symbolic expression ringparent.This is the element constructor.
EXAMPLES:
sage: a = SR(-3/4); a -3/4 sage: type(a) <class 'sage.symbolic.expression.Expression'> sage: a.parent() Symbolic Ring sage: K.<a> = QuadraticField(-3) # needs sage.rings.number_field sage: a + sin(x) # needs sage.rings.number_field I*sqrt(3) + sin(x) sage: x = var('x'); y0,y1 = PolynomialRing(ZZ,2,'y').gens() sage: x+y0/y1 x + y0/y1 sage: x.subs(x=y0/y1) y0/y1 sage: x + int(1) x + 1
- sage.symbolic.expression.new_Expression_from_pyobject(parent, x, force=True, recursive=True)#
Wrap the given Python object in a symbolic expression even if it cannot be coerced to the Symbolic Ring.
INPUT:
parent- a symbolic ring.x- a Python object.force- bool, defaultTrue, if True, the Python object is taken as is without attempting coercion or list traversal.recursive- bool, defaultTrue, disables recursive traversal of lists.
EXAMPLES:
sage: t = SR._force_pyobject(QQ); t # indirect doctest Rational Field sage: type(t) <class 'sage.symbolic.expression.Expression'> sage: from sage.symbolic.expression import new_Expression_from_pyobject sage: t = new_Expression_from_pyobject(SR, 17); t 17 sage: type(t) <class 'sage.symbolic.expression.Expression'> sage: t2 = new_Expression_from_pyobject(SR, t, False); t2 17 sage: t2 is t True sage: tt = new_Expression_from_pyobject(SR, t, True); tt 17 sage: tt is t False
- sage.symbolic.expression.new_Expression_symbol(parent, name=None, latex_name=None, domain=None)#
Look up or create a symbol.
EXAMPLES:
sage: t0 = SR.symbol("t0") sage: t0.conjugate() conjugate(t0) sage: t1 = SR.symbol("t1", domain='real') sage: t1.conjugate() t1 sage: t0.abs() abs(t0) sage: t0_2 = SR.symbol("t0", domain='positive') sage: t0_2.abs() t0 sage: bool(t0_2 == t0) True sage: t0.conjugate() t0 sage: SR.symbol() # temporary variable symbol...
- sage.symbolic.expression.new_Expression_wild(parent, n=0)#
Return the n-th wild-card for pattern matching and substitution.
INPUT:
parent- a symbolic ring.n- a nonnegative integer.
OUTPUT:
n-th wildcard expression.
EXAMPLES:
sage: x,y = var('x,y') sage: w0 = SR.wild(0); w1 = SR.wild(1) sage: pattern = sin(x)*w0*w1^2; pattern $1^2*$0*sin(x) sage: f = atan(sin(x)*3*x^2); f arctan(3*x^2*sin(x)) sage: f.has(pattern) True sage: f.subs(pattern == x^2) arctan(x^2)
- sage.symbolic.expression.normalize_index_for_doctests(arg, nops)#
Wrapper function to test
normalize_index.
- sage.symbolic.expression.paramset_from_Expression(e)#
EXAMPLES:
sage: from sage.symbolic.expression import paramset_from_Expression sage: f = function('f') sage: paramset_from_Expression(f(x).diff(x)) [0]
- sage.symbolic.expression.print_order(lhs, rhs)#
Comparison in the print order
INPUT:
lhs,rhs– two symbolic expressions or something that can be converted to one.
OUTPUT:
Either \(-1\), \(0\), or \(+1\) indicating the comparison. An exception is raised if the arguments cannot be converted into the symbolic ring.
EXAMPLES:
sage: from sage.symbolic.expression import print_order sage: print_order(1, oo) 1 sage: print_order(e, oo) -1 sage: print_order(pi, oo) 1 sage: print_order(1, sqrt(2)) 1
Check that github issue #12967 is fixed:
sage: print_order(SR(oo), sqrt(2)) 1
- sage.symbolic.expression.print_sorted(expressions)#
Sort a list in print order
INPUT:
expressions– a list/tuple/iterable of symbolic expressions, or something that can be converted to one.
OUTPUT:
The list sorted by
print_order().EXAMPLES:
sage: from sage.symbolic.expression import print_sorted sage: print_sorted([SR(1), SR(e), SR(pi), sqrt(2)]) [e, sqrt(2), pi, 1]
- sage.symbolic.expression.py_atan2_for_doctests(x, y)#
Wrapper function to test py_atan2.
- sage.symbolic.expression.py_denom_for_doctests(n)#
This function is used to test py_denom().
EXAMPLES:
sage: from sage.symbolic.expression import py_denom_for_doctests sage: py_denom_for_doctests(2/3) 3
- sage.symbolic.expression.py_eval_infinity_for_doctests()#
This function tests py_eval_infinity.
- sage.symbolic.expression.py_eval_neg_infinity_for_doctests()#
This function tests py_eval_neg_infinity.
- sage.symbolic.expression.py_eval_unsigned_infinity_for_doctests()#
This function tests py_eval_unsigned_infinity.
- sage.symbolic.expression.py_exp_for_doctests(x)#
This function tests py_exp.
EXAMPLES:
sage: from sage.symbolic.expression import py_exp_for_doctests sage: py_exp_for_doctests(CC(2)) 7.38905609893065
- sage.symbolic.expression.py_factorial_py(x)#
This function is a python wrapper around py_factorial(). This wrapper is needed when we override the eval() method for GiNaC’s factorial function in sage.functions.other.Function_factorial.
- sage.symbolic.expression.py_float_for_doctests(n, kwds)#
This function is for testing py_float.
EXAMPLES:
sage: from sage.symbolic.expression import py_float_for_doctests sage: py_float_for_doctests(pi, {'parent':RealField(80)}) 3.1415926535897932384626 sage: py_float_for_doctests(I, {'parent':RealField(80)}) 1.0000000000000000000000*I sage: py_float_for_doctests(I, {'parent':float}) 1j sage: py_float_for_doctests(pi, {'parent':complex}) (3.141592653589793+0j)
- sage.symbolic.expression.py_imag_for_doctests(x)#
Used for doctesting py_imag.
- sage.symbolic.expression.py_is_cinteger_for_doctest(x)#
Returns True if pynac should treat this object as an element of \(\ZZ(i)\).
- sage.symbolic.expression.py_is_crational_for_doctest(x)#
Return True if pynac should treat this object as an element of \(\QQ(i)\).
- sage.symbolic.expression.py_is_integer_for_doctests(x)#
Used internally for doctesting purposes.
- sage.symbolic.expression.py_latex_fderivative_for_doctests(id, params, args)#
Used internally for writing doctests for certain cdef’d functions.
EXAMPLES:
sage: from sage.symbolic.expression import py_latex_fderivative_for_doctests as py_latex_fderivative, get_ginac_serial, get_fn_serial sage: var('x,y,z') (x, y, z) sage: from sage.symbolic.function import get_sfunction_from_serial sage: foo = function('foo', nargs=2) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_fderivative(i, (0, 1, 0, 1), (x, y^z)) \mathrm{D}_{0, 1, 0, 1}\left({\rm foo}\right)\left(x, y^{z}\right)
Test latex_name:
sage: foo = function('foo', nargs=2, latex_name=r'\mathrm{bar}') sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_fderivative(i, (0, 1, 0, 1), (x, y^z)) \mathrm{D}_{0, 1, 0, 1}\left(\mathrm{bar}\right)\left(x, y^{z}\right)
Test custom func:
sage: def my_print(self, *args): return "func_with_args(" + ', '.join(map(repr, args)) +')' sage: foo = function('foo', nargs=2, print_latex_func=my_print) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_fderivative(i, (0, 1, 0, 1), (x, y^z)) \mathrm{D}_{0, 1, 0, 1}func_with_args(x, y^z)
- sage.symbolic.expression.py_latex_function_pystring(id, args, fname_paren=False)#
Return a string with the latex representation of the symbolic function specified by the given id applied to args.
See documentation of py_print_function_pystring for more information.
EXAMPLES:
sage: from sage.symbolic.expression import py_latex_function_pystring, get_ginac_serial, get_fn_serial sage: from sage.symbolic.function import get_sfunction_from_serial sage: var('x,y,z') (x, y, z) sage: foo = function('foo', nargs=2) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_function_pystring(i, (x,y^z)) '{\\rm foo}\\left(x, y^{z}\\right)' sage: py_latex_function_pystring(i, (x,y^z), True) '\\left({\\rm foo}\\right)\\left(x, y^{z}\\right)' sage: py_latex_function_pystring(i, (int(0),x)) '{\\rm foo}\\left(0, x\\right)'
Test latex_name:
sage: foo = function('foo', nargs=2, latex_name=r'\mathrm{bar}') sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_function_pystring(i, (x,y^z)) '\\mathrm{bar}\\left(x, y^{z}\\right)'
Test custom func:
sage: def my_print(self, *args): return "my args are: " + ', '.join(map(repr, args)) sage: foo = function('foo', nargs=2, print_latex_func=my_print) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_latex_function_pystring(i, (x,y^z)) 'my args are: x, y^z'
- sage.symbolic.expression.py_latex_variable_for_doctests(x)#
Internal function used so we can doctest a certain cdef’d method.
EXAMPLES:
sage: sage.symbolic.expression.py_latex_variable_for_doctests('x') x sage: sage.symbolic.expression.py_latex_variable_for_doctests('sigma') \sigma
- sage.symbolic.expression.py_lgamma_for_doctests(x)#
This function tests py_lgamma.
EXAMPLES:
sage: from sage.symbolic.expression import py_lgamma_for_doctests sage: py_lgamma_for_doctests(CC(I)) -0.650923199301856 - 1.87243664726243*I
- sage.symbolic.expression.py_li2_for_doctests(x)#
This function is a python wrapper so py_psi2 can be tested. The real tests are in the docstring for py_psi2.
EXAMPLES:
sage: from sage.symbolic.expression import py_li2_for_doctests sage: py_li2_for_doctests(-1.1) -0.890838090262283
- sage.symbolic.expression.py_li_for_doctests(x, n, parent)#
This function is a python wrapper so py_li can be tested. The real tests are in the docstring for py_li.
EXAMPLES:
sage: from sage.symbolic.expression import py_li_for_doctests sage: py_li_for_doctests(0,2,float) 0.000000000000000
- sage.symbolic.expression.py_log_for_doctests(x)#
This function tests py_log.
EXAMPLES:
sage: from sage.symbolic.expression import py_log_for_doctests sage: py_log_for_doctests(CC(e)) 1.00000000000000
- sage.symbolic.expression.py_mod_for_doctests(x, n)#
This function is a python wrapper so py_mod can be tested. The real tests are in the docstring for py_mod.
EXAMPLES:
sage: from sage.symbolic.expression import py_mod_for_doctests sage: py_mod_for_doctests(5, 2) 1
- sage.symbolic.expression.py_numer_for_doctests(n)#
This function is used to test py_numer().
EXAMPLES:
sage: from sage.symbolic.expression import py_numer_for_doctests sage: py_numer_for_doctests(2/3) 2
- sage.symbolic.expression.py_print_fderivative_for_doctests(id, params, args)#
Used for testing a cdef’d function.
EXAMPLES:
sage: from sage.symbolic.expression import py_print_fderivative_for_doctests as py_print_fderivative, get_ginac_serial, get_fn_serial sage: var('x,y,z') (x, y, z) sage: from sage.symbolic.function import get_sfunction_from_serial sage: foo = function('foo', nargs=2) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_print_fderivative(i, (0, 1, 0, 1), (x, y^z)) D[0, 1, 0, 1](foo)(x, y^z)
Test custom print function:
sage: def my_print(self, *args): return "func_with_args(" + ', '.join(map(repr, args)) +')' sage: foo = function('foo', nargs=2, print_func=my_print) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_print_fderivative(i, (0, 1, 0, 1), (x, y^z)) D[0, 1, 0, 1]func_with_args(x, y^z)
- sage.symbolic.expression.py_print_function_pystring(id, args, fname_paren=False)#
Return a string with the representation of the symbolic function specified by the given id applied to args.
INPUT:
id – serial number of the corresponding symbolic function
params – Set of parameter numbers with respect to which to take the derivative.
args – arguments of the function.
EXAMPLES:
sage: from sage.symbolic.expression import py_print_function_pystring, get_ginac_serial, get_fn_serial sage: from sage.symbolic.function import get_sfunction_from_serial sage: var('x,y,z') (x, y, z) sage: foo = function('foo', nargs=2) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_print_function_pystring(i, (x,y)) 'foo(x, y)' sage: py_print_function_pystring(i, (x,y), True) '(foo)(x, y)' sage: def my_print(self, *args): return "my args are: " + ', '.join(map(repr, args)) sage: foo = function('foo', nargs=2, print_func=my_print) sage: for i in range(get_ginac_serial(), get_fn_serial()): ....: if get_sfunction_from_serial(i) == foo: break sage: get_sfunction_from_serial(i) == foo True sage: py_print_function_pystring(i, (x,y)) 'my args are: x, y'
- sage.symbolic.expression.py_psi2_for_doctests(n, x)#
This function is a python wrapper so py_psi2 can be tested. The real tests are in the docstring for py_psi2.
EXAMPLES:
sage: from sage.symbolic.expression import py_psi2_for_doctests sage: py_psi2_for_doctests(1, 2) 0.644934066848226
- sage.symbolic.expression.py_psi_for_doctests(x)#
This function is a python wrapper so py_psi can be tested. The real tests are in the docstring for py_psi.
EXAMPLES:
sage: from sage.symbolic.expression import py_psi_for_doctests sage: py_psi_for_doctests(2) 0.422784335098467
- sage.symbolic.expression.py_real_for_doctests(x)#
Used for doctesting py_real.
- sage.symbolic.expression.py_stieltjes_for_doctests(x)#
This function is for testing py_stieltjes().
EXAMPLES:
sage: from sage.symbolic.expression import py_stieltjes_for_doctests sage: py_stieltjes_for_doctests(0.0) 0.577215664901533
- sage.symbolic.expression.py_tgamma_for_doctests(x)#
This function is for testing py_tgamma().
- sage.symbolic.expression.py_zeta_for_doctests(x)#
This function is for testing py_zeta().
EXAMPLES:
sage: from sage.symbolic.expression import py_zeta_for_doctests sage: py_zeta_for_doctests(CC.0) 0.00330022368532410 - 0.418155449141322*I
- sage.symbolic.expression.register_or_update_function(self, name, latex_name, nargs, evalf_params_first, update)#
Register the function
selfwith Pynac (GiNaC).OUTPUT:
serial number of the function, for use in
call_registered_function()
EXAMPLES:
sage: from sage.symbolic.function import BuiltinFunction sage: class Archosaurian(BuiltinFunction): ....: def __init__(self): ....: BuiltinFunction.__init__(self, 'archsaur', nargs=1) ....: def _eval_(self, x): ....: return x * exp(x) sage: archsaur = Archosaurian() # indirect doctest sage: archsaur(2) 2*e^2
- sage.symbolic.expression.restore_op_wrapper(expr)#
- sage.symbolic.expression.solve_diophantine(f, *args, **kwds)#
Solve a Diophantine equation.
The argument, if not given as symbolic equation, is set equal to zero. It can be given in any form that can be converted to symbolic. Please see
Expression.solve_diophantine()for a detailed synopsis.EXAMPLES:
sage: R.<a,b> = PolynomialRing(ZZ); R Multivariate Polynomial Ring in a, b over Integer Ring sage: solve_diophantine(a^2 - 3*b^2 + 1) [] sage: sorted(solve_diophantine(a^2 - 3*b^2 + 2), key=str) [(-1/2*sqrt(3)*(sqrt(3) + 2)^t + 1/2*sqrt(3)*(-sqrt(3) + 2)^t - 1/2*(sqrt(3) + 2)^t - 1/2*(-sqrt(3) + 2)^t, -1/6*sqrt(3)*(sqrt(3) + 2)^t + 1/6*sqrt(3)*(-sqrt(3) + 2)^t - 1/2*(sqrt(3) + 2)^t - 1/2*(-sqrt(3) + 2)^t), (1/2*sqrt(3)*(sqrt(3) + 2)^t - 1/2*sqrt(3)*(-sqrt(3) + 2)^t + 1/2*(sqrt(3) + 2)^t + 1/2*(-sqrt(3) + 2)^t, 1/6*sqrt(3)*(sqrt(3) + 2)^t - 1/6*sqrt(3)*(-sqrt(3) + 2)^t + 1/2*(sqrt(3) + 2)^t + 1/2*(-sqrt(3) + 2)^t)]
- sage.symbolic.expression.test_binomial(n, k)#
The Binomial coefficients. It computes the binomial coefficients. For integer n and k and positive n this is the number of ways of choosing k objects from n distinct objects. If n is negative, the formula binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result.
INPUT:
n, k – integers, with k >= 0.
OUTPUT:
integer
EXAMPLES:
sage: import sage.symbolic.expression sage: sage.symbolic.expression.test_binomial(5,2) 10 sage: sage.symbolic.expression.test_binomial(-5,3) -35 sage: -sage.symbolic.expression.test_binomial(3-(-5)-1, 3) -35
- sage.symbolic.expression.tolerant_is_symbol(a)#
Utility function to test if something is a symbol.
Returns False for arguments that do not have an is_symbol attribute. Returns the result of calling the is_symbol method otherwise.
EXAMPLES:
sage: from sage.symbolic.expression import tolerant_is_symbol sage: tolerant_is_symbol(var("x")) True sage: tolerant_is_symbol(None) False sage: None.is_symbol() Traceback (most recent call last): ... AttributeError: 'NoneType' object has no attribute 'is_symbol'...
- sage.symbolic.expression.unpack_operands(ex)#
EXAMPLES:
sage: from sage.symbolic.expression import unpack_operands sage: t = SR._force_pyobject((1, 2, x, x+1, x+2)) sage: unpack_operands(t) (1, 2, x, x + 1, x + 2) sage: type(unpack_operands(t)) <... 'tuple'> sage: list(map(type, unpack_operands(t))) [<class 'sage.rings.integer.Integer'>, <class 'sage.rings.integer.Integer'>, <class 'sage.symbolic.expression.Expression'>, <class 'sage.symbolic.expression.Expression'>, <class 'sage.symbolic.expression.Expression'>] sage: u = SR._force_pyobject((t, x^2)) sage: unpack_operands(u) ((1, 2, x, x + 1, x + 2), x^2) sage: type(unpack_operands(u)[0]) <... 'tuple'>