Symbolic Computation

AUTHORS:

  • Bobby Moretti and William Stein (2006-2007)

  • Robert Bradshaw (2007-10): minpoly(), numerical algorithm

  • Robert Bradshaw (2008-10): minpoly(), algebraic algorithm

  • Golam Mortuza Hossain (2009-06-15): _limit_latex()

  • Golam Mortuza Hossain (2009-06-22): _laplace_latex(), _inverse_laplace_latex()

  • Tom Coates (2010-06-11): fixed Issue #9217

EXAMPLES:

The basic units of the calculus package are symbolic expressions which are elements of the symbolic expression ring (SR). To create a symbolic variable object in Sage, use the var() function, whose argument is the text of that variable. Note that Sage is intelligent about LaTeXing variable names.

sage: x1 = var('x1'); x1
x1
sage: latex(x1)
x_{1}
sage: theta = var('theta'); theta
theta
sage: latex(theta)
\theta

>>> from sage.all import *
>>> x1 = var('x1'); x1
x1
>>> latex(x1)
x_{1}
>>> theta = var('theta'); theta
theta
>>> latex(theta)
\theta

Sage predefines x to be a global indeterminate. Thus the following works:

sage: x^2
x^2
sage: type(x)
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> x**Integer(2)
x^2
>>> type(x)
<class 'sage.symbolic.expression.Expression'>

More complicated expressions in Sage can be built up using ordinary arithmetic. The following are valid, and follow the rules of Python arithmetic: (The ‘=’ operator represents assignment, and not equality)

sage: var('x,y,z')
(x, y, z)
sage: f = x + y + z/(2*sin(y*z/55))
sage: g = f^f; g
(x + y + 1/2*z/sin(1/55*y*z))^(x + y + 1/2*z/sin(1/55*y*z))

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)
>>> f = x + y + z/(Integer(2)*sin(y*z/Integer(55)))
>>> g = f**f; g
(x + y + 1/2*z/sin(1/55*y*z))^(x + y + 1/2*z/sin(1/55*y*z))

Differentiation and integration are available, but behind the scenes through Maxima:

sage: f = sin(x)/cos(2*y)
sage: f.derivative(y)
2*sin(x)*sin(2*y)/cos(2*y)^2
sage: g = f.integral(x); g
-cos(x)/cos(2*y)

>>> from sage.all import *
>>> f = sin(x)/cos(Integer(2)*y)
>>> f.derivative(y)
2*sin(x)*sin(2*y)/cos(2*y)^2
>>> g = f.integral(x); g
-cos(x)/cos(2*y)

Note that these methods usually require an explicit variable name. If none is given, Sage will try to find one for you.

sage: f = sin(x); f.derivative()
cos(x)

>>> from sage.all import *
>>> f = sin(x); f.derivative()
cos(x)

If the expression is a callable symbolic expression (i.e., the variable order is specified), then Sage can calculate the matrix derivative (i.e., the gradient, Jacobian matrix, etc.) if no variables are specified. In the example below, we use the second derivative test to determine that there is a saddle point at (0,-1/2).

sage: f(x,y) = x^2*y + y^2 + y
sage: f.diff()  # gradient
(x, y) |--> (2*x*y, x^2 + 2*y + 1)
sage: solve(list(f.diff()), [x,y])
[[x == -I, y == 0], [x == I, y == 0], [x == 0, y == (-1/2)]]
sage: H=f.diff(2); H  # Hessian matrix
[(x, y) |--> 2*y (x, y) |--> 2*x]
[(x, y) |--> 2*x   (x, y) |--> 2]
sage: H(x=0, y=-1/2)
[-1  0]
[ 0  2]
sage: H(x=0, y=-1/2).eigenvalues()
[-1, 2]

>>> from sage.all import *
>>> __tmp__=var("x,y"); f = symbolic_expression(x**Integer(2)*y + y**Integer(2) + y).function(x,y)
>>> f.diff()  # gradient
(x, y) |--> (2*x*y, x^2 + 2*y + 1)
>>> solve(list(f.diff()), [x,y])
[[x == -I, y == 0], [x == I, y == 0], [x == 0, y == (-1/2)]]
>>> H=f.diff(Integer(2)); H  # Hessian matrix
[(x, y) |--> 2*y (x, y) |--> 2*x]
[(x, y) |--> 2*x   (x, y) |--> 2]
>>> H(x=Integer(0), y=-Integer(1)/Integer(2))
[-1  0]
[ 0  2]
>>> H(x=Integer(0), y=-Integer(1)/Integer(2)).eigenvalues()
[-1, 2]

Here we calculate the Jacobian for the polar coordinate transformation:

sage: T(r,theta) = [r*cos(theta),r*sin(theta)]
sage: T
(r, theta) |--> (r*cos(theta), r*sin(theta))
sage: T.diff() # Jacobian matrix
[   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
[   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
sage: diff(T) # Jacobian matrix
[   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
[   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
sage: T.diff().det() # Jacobian
(r, theta) |--> r*cos(theta)^2 + r*sin(theta)^2

>>> from sage.all import *
>>> __tmp__=var("r,theta"); T = symbolic_expression([r*cos(theta),r*sin(theta)]).function(r,theta)
>>> T
(r, theta) |--> (r*cos(theta), r*sin(theta))
>>> T.diff() # Jacobian matrix
[   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
[   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
>>> diff(T) # Jacobian matrix
[   (r, theta) |--> cos(theta) (r, theta) |--> -r*sin(theta)]
[   (r, theta) |--> sin(theta)  (r, theta) |--> r*cos(theta)]
>>> T.diff().det() # Jacobian
(r, theta) |--> r*cos(theta)^2 + r*sin(theta)^2

When the order of variables is ambiguous, Sage will raise an exception when differentiating:

sage: f = sin(x+y); f.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.

>>> from sage.all import *
>>> f = sin(x+y); f.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.

Simplifying symbolic sums is also possible, using the sum() command, which also uses Maxima in the background:

sage: k, m = var('k, m')
sage: sum(1/k^4, k, 1, oo)
1/90*pi^4
sage: sum(binomial(m,k), k, 0, m)
2^m

>>> from sage.all import *
>>> k, m = var('k, m')
>>> sum(Integer(1)/k**Integer(4), k, Integer(1), oo)
1/90*pi^4
>>> sum(binomial(m,k), k, Integer(0), m)
2^m

Symbolic matrices can be used as well in various ways, including exponentiation:

sage: M = matrix([[x,x^2],[1/x,x]])
sage: M^2
[x^2 + x   2*x^3]
[      2 x^2 + x]
sage: e^M
[          1/2*(e^(2*sqrt(x)) + 1)*e^(x - sqrt(x)) 1/2*(x*e^(2*sqrt(x)) - x)*sqrt(x)*e^(x - sqrt(x))]
[  1/2*(e^(2*sqrt(x)) - 1)*e^(x - sqrt(x))/x^(3/2)           1/2*(e^(2*sqrt(x)) + 1)*e^(x - sqrt(x))]

>>> from sage.all import *
>>> M = matrix([[x,x**Integer(2)],[Integer(1)/x,x]])
>>> M**Integer(2)
[x^2 + x   2*x^3]
[      2 x^2 + x]
>>> e**M
[          1/2*(e^(2*sqrt(x)) + 1)*e^(x - sqrt(x)) 1/2*(x*e^(2*sqrt(x)) - x)*sqrt(x)*e^(x - sqrt(x))]
[  1/2*(e^(2*sqrt(x)) - 1)*e^(x - sqrt(x))/x^(3/2)           1/2*(e^(2*sqrt(x)) + 1)*e^(x - sqrt(x))]

Complex exponentiation works, but may require a patched version of maxima (Issue #32898) for now:

sage: M = i*matrix([[pi]])
sage: e^M  # not tested, requires patched maxima
[-1]
sage: M = i*matrix([[pi,0],[0,2*pi]])
sage: e^M
[-1  0]
[ 0  1]
sage: M = matrix([[0,pi],[-pi,0]])
sage: e^M
[-1  0]
[ 0 -1]

>>> from sage.all import *
>>> M = i*matrix([[pi]])
>>> e**M  # not tested, requires patched maxima
[-1]
>>> M = i*matrix([[pi,Integer(0)],[Integer(0),Integer(2)*pi]])
>>> e**M
[-1  0]
[ 0  1]
>>> M = matrix([[Integer(0),pi],[-pi,Integer(0)]])
>>> e**M
[-1  0]
[ 0 -1]

Substitution works similarly. We can substitute with a python dict:

sage: f = sin(x*y - z)
sage: f({x: var('t'), y: z})
sin(t*z - z)

>>> from sage.all import *
>>> f = sin(x*y - z)
>>> f({x: var('t'), y: z})
sin(t*z - z)

Also we can substitute with keywords:

sage: f = sin(x*y - z)
sage: f(x=t, y=z)
sin(t*z - z)

>>> from sage.all import *
>>> f = sin(x*y - z)
>>> f(x=t, y=z)
sin(t*z - z)

Another example:

sage: f = sin(2*pi*x/y)
sage: f(x=4)
sin(8*pi/y)

>>> from sage.all import *
>>> f = sin(Integer(2)*pi*x/y)
>>> f(x=Integer(4))
sin(8*pi/y)

It is no longer allowed to call expressions with positional arguments:

sage: f = sin(x)
sage: f(y)
Traceback (most recent call last):
...
TypeError: Substitution using function-call syntax and unnamed
arguments has been removed. You can use named arguments instead, like
EXPR(x=..., y=...)
sage: f(x=pi)
0

>>> from sage.all import *
>>> f = sin(x)
>>> f(y)
Traceback (most recent call last):
...
TypeError: Substitution using function-call syntax and unnamed
arguments has been removed. You can use named arguments instead, like
EXPR(x=..., y=...)
>>> f(x=pi)
0

We can also make a CallableSymbolicExpression, which is a SymbolicExpression that is a function of specified variables in a fixed order. Each SymbolicExpression has a function(...) method that is used to create a CallableSymbolicExpression, as illustrated below:

sage: u = log((2-x)/(y+5))
sage: f = u.function(x, y); f
(x, y) |--> log(-(x - 2)/(y + 5))

>>> from sage.all import *
>>> u = log((Integer(2)-x)/(y+Integer(5)))
>>> f = u.function(x, y); f
(x, y) |--> log(-(x - 2)/(y + 5))

There is an easier way of creating a CallableSymbolicExpression, which relies on the Sage preparser.

sage: f(x,y) = log(x)*cos(y); f
(x, y) |--> cos(y)*log(x)

>>> from sage.all import *
>>> __tmp__=var("x,y"); f = symbolic_expression(log(x)*cos(y)).function(x,y); f
(x, y) |--> cos(y)*log(x)

Then we have fixed an order of variables and there is no ambiguity substituting or evaluating:

sage: f(x,y) = log((2-x)/(y+5))
sage: f(7,t)
log(-5/(t + 5))

>>> from sage.all import *
>>> __tmp__=var("x,y"); f = symbolic_expression(log((Integer(2)-x)/(y+Integer(5)))).function(x,y)
>>> f(Integer(7),t)
log(-5/(t + 5))

Some further examples:

sage: f = 5*sin(x)
sage: f
5*sin(x)
sage: f(x=2)
5*sin(2)
sage: f(x=pi)
0
sage: float(f(x=pi))
0.0

>>> from sage.all import *
>>> f = Integer(5)*sin(x)
>>> f
5*sin(x)
>>> f(x=Integer(2))
5*sin(2)
>>> f(x=pi)
0
>>> float(f(x=pi))
0.0

Another example:

sage: f = integrate(1/sqrt(9+x^2), x); f
arcsinh(1/3*x)
sage: f(x=3)
arcsinh(1)
sage: f.derivative(x)
1/sqrt(x^2 + 9)

>>> from sage.all import *
>>> f = integrate(Integer(1)/sqrt(Integer(9)+x**Integer(2)), x); f
arcsinh(1/3*x)
>>> f(x=Integer(3))
arcsinh(1)
>>> f.derivative(x)
1/sqrt(x^2 + 9)

We compute the length of the parabola from 0 to 2:

sage: x = var('x')
sage: y = x^2
sage: dy = derivative(y,x)
sage: z = integral(sqrt(1 + dy^2), x, 0, 2)
sage: z
sqrt(17) + 1/4*arcsinh(4)
sage: n(z,200)
4.6467837624329358733826155674904591885104869874232887508703
sage: float(z)
4.646783762432936

>>> from sage.all import *
>>> x = var('x')
>>> y = x**Integer(2)
>>> dy = derivative(y,x)
>>> z = integral(sqrt(Integer(1) + dy**Integer(2)), x, Integer(0), Integer(2))
>>> z
sqrt(17) + 1/4*arcsinh(4)
>>> n(z,Integer(200))
4.6467837624329358733826155674904591885104869874232887508703
>>> float(z)
4.646783762432936

We test pickling:

sage: x, y = var('x,y')
sage: f = -sqrt(pi)*(x^3 + sin(x/cos(y)))
sage: bool(loads(dumps(f)) == f)
True

>>> from sage.all import *
>>> x, y = var('x,y')
>>> f = -sqrt(pi)*(x**Integer(3) + sin(x/cos(y)))
>>> bool(loads(dumps(f)) == f)
True

Coercion examples:

We coerce various symbolic expressions into the complex numbers:

sage: CC(I)
1.00000000000000*I
sage: CC(2*I)
2.00000000000000*I
sage: ComplexField(200)(2*I)
2.0000000000000000000000000000000000000000000000000000000000*I
sage: ComplexField(200)(sin(I))
1.1752011936438014568823818505956008151557179813340958702296*I
sage: f = sin(I) + cos(I/2); f
cosh(1/2) + I*sinh(1)
sage: CC(f)
1.12762596520638 + 1.17520119364380*I
sage: ComplexField(200)(f)
1.1276259652063807852262251614026720125478471180986674836290
 + 1.1752011936438014568823818505956008151557179813340958702296*I
sage: ComplexField(100)(f)
1.1276259652063807852262251614 + 1.1752011936438014568823818506*I

>>> from sage.all import *
>>> CC(I)
1.00000000000000*I
>>> CC(Integer(2)*I)
2.00000000000000*I
>>> ComplexField(Integer(200))(Integer(2)*I)
2.0000000000000000000000000000000000000000000000000000000000*I
>>> ComplexField(Integer(200))(sin(I))
1.1752011936438014568823818505956008151557179813340958702296*I
>>> f = sin(I) + cos(I/Integer(2)); f
cosh(1/2) + I*sinh(1)
>>> CC(f)
1.12762596520638 + 1.17520119364380*I
>>> ComplexField(Integer(200))(f)
1.1276259652063807852262251614026720125478471180986674836290
 + 1.1752011936438014568823818505956008151557179813340958702296*I
>>> ComplexField(Integer(100))(f)
1.1276259652063807852262251614 + 1.1752011936438014568823818506*I

We illustrate construction of an inverse sum where each denominator has a new variable name:

sage: f = sum(1/var('n%s'%i)^i for i in range(10))
sage: f
1/n1 + 1/n2^2 + 1/n3^3 + 1/n4^4 + 1/n5^5 + 1/n6^6 + 1/n7^7 + 1/n8^8 + 1/n9^9 + 1

>>> from sage.all import *
>>> f = sum(Integer(1)/var('n%s'%i)**i for i in range(Integer(10)))
>>> f
1/n1 + 1/n2^2 + 1/n3^3 + 1/n4^4 + 1/n5^5 + 1/n6^6 + 1/n7^7 + 1/n8^8 + 1/n9^9 + 1

Note that after calling var, the variables are immediately available for use:

sage: (n1 + n2)^5
(n1 + n2)^5

>>> from sage.all import *
>>> (n1 + n2)**Integer(5)
(n1 + n2)^5

We can, of course, substitute:

sage: f(n9=9, n7=n6)
1/n1 + 1/n2^2 + 1/n3^3 + 1/n4^4 + 1/n5^5 + 1/n6^6 + 1/n6^7 + 1/n8^8
 + 387420490/387420489

>>> from sage.all import *
>>> f(n9=Integer(9), n7=n6)
1/n1 + 1/n2^2 + 1/n3^3 + 1/n4^4 + 1/n5^5 + 1/n6^6 + 1/n6^7 + 1/n8^8
 + 387420490/387420489
sage.calculus.calculus.at(ex, *args, **kwds)[source]

Parses at formulations from other systems, such as Maxima. Replaces evaluation ‘at’ a point with substitution method of a symbolic expression.

EXAMPLES:

We do not import at at the top level, but we can use it as a synonym for substitution if we import it:

sage: g = x^3 - 3
sage: from sage.calculus.calculus import at
sage: at(g, x=1)
-2
sage: g.subs(x=1)
-2

>>> from sage.all import *
>>> g = x**Integer(3) - Integer(3)
>>> from sage.calculus.calculus import at
>>> at(g, x=Integer(1))
-2
>>> g.subs(x=Integer(1))
-2

We find a formal Taylor expansion:

sage: h,x = var('h,x')
sage: u = function('u')
sage: u(x + h)
u(h + x)
sage: diff(u(x+h), x)
D[0](u)(h + x)
sage: taylor(u(x+h), h, 0, 4)
1/24*h^4*diff(u(x), x, x, x, x) + 1/6*h^3*diff(u(x), x, x, x)
 + 1/2*h^2*diff(u(x), x, x) + h*diff(u(x), x) + u(x)

>>> from sage.all import *
>>> h,x = var('h,x')
>>> u = function('u')
>>> u(x + h)
u(h + x)
>>> diff(u(x+h), x)
D[0](u)(h + x)
>>> taylor(u(x+h), h, Integer(0), Integer(4))
1/24*h^4*diff(u(x), x, x, x, x) + 1/6*h^3*diff(u(x), x, x, x)
 + 1/2*h^2*diff(u(x), x, x) + h*diff(u(x), x) + u(x)

We compute a Laplace transform:

sage: var('s,t')
(s, t)
sage: f = function('f')(t)
sage: f.diff(t, 2)
diff(f(t), t, t)
sage: f.diff(t,2).laplace(t,s)
s^2*laplace(f(t), t, s) - s*f(0) - D[0](f)(0)

>>> from sage.all import *
>>> var('s,t')
(s, t)
>>> f = function('f')(t)
>>> f.diff(t, Integer(2))
diff(f(t), t, t)
>>> f.diff(t,Integer(2)).laplace(t,s)
s^2*laplace(f(t), t, s) - s*f(0) - D[0](f)(0)

We can also accept a non-keyword list of expression substitutions, like Maxima does (Issue #12796):

sage: from sage.calculus.calculus import at
sage: f = function('f')
sage: at(f(x), [x == 1])
f(1)

>>> from sage.all import *
>>> from sage.calculus.calculus import at
>>> f = function('f')
>>> at(f(x), [x == Integer(1)])
f(1)
sage.calculus.calculus.dummy_diff(*args)[source]

This function is called when ‘diff’ appears in a Maxima string.

EXAMPLES:

sage: from sage.calculus.calculus import dummy_diff
sage: x,y = var('x,y')
sage: dummy_diff(sin(x*y), x, SR(2), y, SR(1))
-x*y^2*cos(x*y) - 2*y*sin(x*y)

>>> from sage.all import *
>>> from sage.calculus.calculus import dummy_diff
>>> x,y = var('x,y')
>>> dummy_diff(sin(x*y), x, SR(Integer(2)), y, SR(Integer(1)))
-x*y^2*cos(x*y) - 2*y*sin(x*y)

Here the function is used implicitly:

sage: a = var('a')
sage: f = function('cr')(a)
sage: g = f.diff(a); g
diff(cr(a), a)

>>> from sage.all import *
>>> a = var('a')
>>> f = function('cr')(a)
>>> g = f.diff(a); g
diff(cr(a), a)
sage.calculus.calculus.dummy_integrate(*args)[source]

This function is called to create formal wrappers of integrals that Maxima can’t compute:

EXAMPLES:

sage: from sage.calculus.calculus import dummy_integrate
sage: f = function('f')
sage: dummy_integrate(f(x), x)
integrate(f(x), x)
sage: a,b = var('a,b')
sage: dummy_integrate(f(x), x, a, b)
integrate(f(x), x, a, b)

>>> from sage.all import *
>>> from sage.calculus.calculus import dummy_integrate
>>> f = function('f')
>>> dummy_integrate(f(x), x)
integrate(f(x), x)
>>> a,b = var('a,b')
>>> dummy_integrate(f(x), x, a, b)
integrate(f(x), x, a, b)
sage.calculus.calculus.dummy_inverse_laplace(*args)[source]

This function is called to create formal wrappers of inverse Laplace transforms that Maxima can’t compute:

EXAMPLES:

sage: from sage.calculus.calculus import dummy_inverse_laplace
sage: s,t = var('s,t')
sage: F = function('F')
sage: dummy_inverse_laplace(F(s), s, t)
ilt(F(s), s, t)

>>> from sage.all import *
>>> from sage.calculus.calculus import dummy_inverse_laplace
>>> s,t = var('s,t')
>>> F = function('F')
>>> dummy_inverse_laplace(F(s), s, t)
ilt(F(s), s, t)
sage.calculus.calculus.dummy_laplace(*args)[source]

This function is called to create formal wrappers of laplace transforms that Maxima can’t compute:

EXAMPLES:

sage: from sage.calculus.calculus import dummy_laplace
sage: s,t = var('s,t')
sage: f = function('f')
sage: dummy_laplace(f(t), t, s)
laplace(f(t), t, s)

>>> from sage.all import *
>>> from sage.calculus.calculus import dummy_laplace
>>> s,t = var('s,t')
>>> f = function('f')
>>> dummy_laplace(f(t), t, s)
laplace(f(t), t, s)
sage.calculus.calculus.dummy_pochhammer(*args)[source]

This function is called to create formal wrappers of Pochhammer symbols.

EXAMPLES:

sage: from sage.calculus.calculus import dummy_pochhammer
sage: s,t = var('s,t')
sage: dummy_pochhammer(s, t)
gamma(s + t)/gamma(s)

>>> from sage.all import *
>>> from sage.calculus.calculus import dummy_pochhammer
>>> s,t = var('s,t')
>>> dummy_pochhammer(s, t)
gamma(s + t)/gamma(s)
sage.calculus.calculus.inverse_laplace(ex, s, t, algorithm='maxima')[source]

Return the inverse Laplace transform with respect to the variable \(t\) and transform parameter \(s\), if possible.

If this function cannot find a solution, a formal function is returned. The function that is returned may be viewed as a function of \(t\).

DEFINITION:

The inverse Laplace transform of a function \(F(s)\) is the function \(f(t)\), defined by

\[f(t) = \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma + i\infty} e^{st} F(s) ds,\]

where \(\gamma\) is chosen so that the contour path of integration is in the region of convergence of \(F(s)\).

INPUT:

  • ex – a symbolic expression

  • s – transform parameter

  • t – independent variable

  • algorithm – (default: 'maxima') one of

    • 'maxima' – use Maxima (the default)

    • 'sympy' – use SymPy

    • 'giac' – use Giac (optional)

See also

laplace()

EXAMPLES:

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)
sage: laplace(f, m, w)
1/(w^2 + 10)

sage: f(t) = t*cos(t)
sage: s = var('s')
sage: L = laplace(f, t, s); L
t |--> 2*s^2/(s^2 + 1)^2 - 1/(s^2 + 1)
sage: inverse_laplace(L, s, t)
t |--> t*cos(t)
sage: inverse_laplace(1/(s^3+1), s, t)
1/3*(sqrt(3)*sin(1/2*sqrt(3)*t) - cos(1/2*sqrt(3)*t))*e^(1/2*t) + 1/3*e^(-t)

>>> from sage.all import *
>>> var('w, m')
(w, m)
>>> f = (Integer(1)/(w**Integer(2)+Integer(10))).inverse_laplace(w, m); f
1/10*sqrt(10)*sin(sqrt(10)*m)
>>> laplace(f, m, w)
1/(w^2 + 10)

>>> __tmp__=var("t"); f = symbolic_expression(t*cos(t)).function(t)
>>> s = var('s')
>>> L = laplace(f, t, s); L
t |--> 2*s^2/(s^2 + 1)^2 - 1/(s^2 + 1)
>>> inverse_laplace(L, s, t)
t |--> t*cos(t)
>>> inverse_laplace(Integer(1)/(s**Integer(3)+Integer(1)), s, t)
1/3*(sqrt(3)*sin(1/2*sqrt(3)*t) - cos(1/2*sqrt(3)*t))*e^(1/2*t) + 1/3*e^(-t)

No explicit inverse Laplace transform, so one is returned formally a function ilt:

sage: inverse_laplace(cos(s), s, t)
ilt(cos(s), s, t)

>>> from sage.all import *
>>> inverse_laplace(cos(s), s, t)
ilt(cos(s), s, t)

Transform an expression involving a time-shift, via SymPy:

sage: inverse_laplace(1/s^2*exp(-s), s, t, algorithm='sympy').simplify()
(t - 1)*heaviside(t - 1)

>>> from sage.all import *
>>> inverse_laplace(Integer(1)/s**Integer(2)*exp(-s), s, t, algorithm='sympy').simplify()
(t - 1)*heaviside(t - 1)

The same instance with Giac:

sage: # needs giac
sage: inverse_laplace(1/s^2*exp(-s), s, t, algorithm='giac')
(t - 1)*heaviside(t - 1)

>>> from sage.all import *
>>> # needs giac
>>> inverse_laplace(Integer(1)/s**Integer(2)*exp(-s), s, t, algorithm='giac')
(t - 1)*heaviside(t - 1)

Transform a rational expression:

sage: # needs giac
sage: inverse_laplace((2*s^2*exp(-2*s) - exp(-s))/(s^3+1), s, t,
....:                 algorithm='giac')
-1/3*(sqrt(3)*e^(1/2*t - 1/2)*sin(1/2*sqrt(3)*(t - 1))
 - cos(1/2*sqrt(3)*(t - 1))*e^(1/2*t - 1/2) + e^(-t + 1))*heaviside(t - 1)
 + 2/3*(2*cos(1/2*sqrt(3)*(t - 2))*e^(1/2*t - 1) + e^(-t + 2))*heaviside(t - 2)

sage: inverse_laplace(1/(s - 1), s, x)
e^x

>>> from sage.all import *
>>> # needs giac
>>> inverse_laplace((Integer(2)*s**Integer(2)*exp(-Integer(2)*s) - exp(-s))/(s**Integer(3)+Integer(1)), s, t,
...                 algorithm='giac')
-1/3*(sqrt(3)*e^(1/2*t - 1/2)*sin(1/2*sqrt(3)*(t - 1))
 - cos(1/2*sqrt(3)*(t - 1))*e^(1/2*t - 1/2) + e^(-t + 1))*heaviside(t - 1)
 + 2/3*(2*cos(1/2*sqrt(3)*(t - 2))*e^(1/2*t - 1) + e^(-t + 2))*heaviside(t - 2)

>>> inverse_laplace(Integer(1)/(s - Integer(1)), s, x)
e^x

The inverse Laplace transform of a constant is a delta distribution:

sage: inverse_laplace(1, s, t)
dirac_delta(t)
sage: inverse_laplace(1, s, t, algorithm='sympy')
dirac_delta(t)
sage: inverse_laplace(1, s, t, algorithm='giac')  # needs giac
dirac_delta(t)

>>> from sage.all import *
>>> inverse_laplace(Integer(1), s, t)
dirac_delta(t)
>>> inverse_laplace(Integer(1), s, t, algorithm='sympy')
dirac_delta(t)
>>> inverse_laplace(Integer(1), s, t, algorithm='giac')  # needs giac
dirac_delta(t)
sage.calculus.calculus.laplace(ex, t, s, algorithm='maxima')[source]

Return the Laplace transform with respect to the variable \(t\) and transform parameter \(s\), if possible.

If this function cannot find a solution, a formal function is returned. The function that is returned may be viewed as a function of \(s\).

DEFINITION:

The Laplace transform of a function \(f(t)\), defined for all real numbers \(t \geq 0\), is the function \(F(s)\) defined by

\[F(s) = \int_{0}^{\infty} e^{-st} f(t) dt.\]

INPUT:

  • ex – a symbolic expression

  • t – independent variable

  • s – transform parameter

  • algorithm – (default: 'maxima') one of

    • 'maxima' – use Maxima (the default)

    • 'sympy' – use SymPy

    • 'giac' – use Giac (optional)

Note

The 'sympy' algorithm returns the tuple (\(F\), \(a\), cond) where \(F\) is the Laplace transform of \(f(t)\), \(Re(s)>a\) is the half-plane of convergence, and cond are auxiliary convergence conditions.

See also

inverse_laplace()

EXAMPLES:

We compute a few Laplace transforms:

sage: var('x, s, z, t, t0')
(x, s, z, t, t0)
sage: sin(x).laplace(x, s)
1/(s^2 + 1)
sage: (z + exp(x)).laplace(x, s)
z/s + 1/(s - 1)
sage: log(t/t0).laplace(t, s)
-(euler_gamma + log(s) + log(t0))/s

>>> from sage.all import *
>>> var('x, s, z, t, t0')
(x, s, z, t, t0)
>>> sin(x).laplace(x, s)
1/(s^2 + 1)
>>> (z + exp(x)).laplace(x, s)
z/s + 1/(s - 1)
>>> log(t/t0).laplace(t, s)
-(euler_gamma + log(s) + log(t0))/s

We do a formal calculation:

sage: f = function('f')(x)
sage: g = f.diff(x); g
diff(f(x), x)
sage: g.laplace(x, s)
s*laplace(f(x), x, s) - f(0)

>>> from sage.all import *
>>> f = function('f')(x)
>>> g = f.diff(x); g
diff(f(x), x)
>>> g.laplace(x, s)
s*laplace(f(x), x, s) - f(0)

A BATTLE BETWEEN the X-women and the Y-men (by David Joyner): Solve

\[x' = -16y, x(0)=270, y' = -x + 1, y(0) = 90.\]

This models a fight between two sides, the “X-women” and the “Y-men”, where the X-women have 270 initially and the Y-men have 90, but the Y-men are better at fighting, because of the higher factor of “-16” vs “-1”, and also get an occasional reinforcement, because of the “+1” term.

sage: var('t')
t
sage: t = var('t')
sage: x = function('x')(t)
sage: y = function('y')(t)
sage: de1 = x.diff(t) + 16*y
sage: de2 = y.diff(t) + x - 1
sage: de1.laplace(t, s)
s*laplace(x(t), t, s) + 16*laplace(y(t), t, s) - x(0)
sage: de2.laplace(t, s)
s*laplace(y(t), t, s) - 1/s + laplace(x(t), t, s) - y(0)

>>> from sage.all import *
>>> var('t')
t
>>> t = var('t')
>>> x = function('x')(t)
>>> y = function('y')(t)
>>> de1 = x.diff(t) + Integer(16)*y
>>> de2 = y.diff(t) + x - Integer(1)
>>> de1.laplace(t, s)
s*laplace(x(t), t, s) + 16*laplace(y(t), t, s) - x(0)
>>> de2.laplace(t, s)
s*laplace(y(t), t, s) - 1/s + laplace(x(t), t, s) - y(0)

Next we form the augmented matrix of the above system:

sage: A = matrix([[s, 16, 270], [1, s, 90+1/s]])
sage: E = A.echelon_form()
sage: xt = E[0,2].inverse_laplace(s,t)
sage: yt = E[1,2].inverse_laplace(s,t)
sage: xt
-91/2*e^(4*t) + 629/2*e^(-4*t) + 1
sage: yt
91/8*e^(4*t) + 629/8*e^(-4*t)
sage: p1 = plot(xt, 0, 1/2, rgbcolor=(1,0,0))                                   # needs sage.plot
sage: p2 = plot(yt, 0, 1/2, rgbcolor=(0,1,0))                                   # needs sage.plot
sage: import tempfile
sage: with tempfile.NamedTemporaryFile(suffix='.png') as f:                     # needs sage.plot
....:     (p1 + p2).save(f.name)

>>> from sage.all import *
>>> A = matrix([[s, Integer(16), Integer(270)], [Integer(1), s, Integer(90)+Integer(1)/s]])
>>> E = A.echelon_form()
>>> xt = E[Integer(0),Integer(2)].inverse_laplace(s,t)
>>> yt = E[Integer(1),Integer(2)].inverse_laplace(s,t)
>>> xt
-91/2*e^(4*t) + 629/2*e^(-4*t) + 1
>>> yt
91/8*e^(4*t) + 629/8*e^(-4*t)
>>> p1 = plot(xt, Integer(0), Integer(1)/Integer(2), rgbcolor=(Integer(1),Integer(0),Integer(0)))                                   # needs sage.plot
>>> p2 = plot(yt, Integer(0), Integer(1)/Integer(2), rgbcolor=(Integer(0),Integer(1),Integer(0)))                                   # needs sage.plot
>>> import tempfile
>>> with tempfile.NamedTemporaryFile(suffix='.png') as f:                     # needs sage.plot
...     (p1 + p2).save(f.name)

Another example:

sage: var('a,s,t')
(a, s, t)
sage: f = exp (2*t + a) * sin(t) * t; f
t*e^(a + 2*t)*sin(t)
sage: L = laplace(f, t, s); L
2*(s - 2)*e^a/(s^2 - 4*s + 5)^2
sage: inverse_laplace(L, s, t)
t*e^(a + 2*t)*sin(t)

>>> from sage.all import *
>>> var('a,s,t')
(a, s, t)
>>> f = exp (Integer(2)*t + a) * sin(t) * t; f
t*e^(a + 2*t)*sin(t)
>>> L = laplace(f, t, s); L
2*(s - 2)*e^a/(s^2 - 4*s + 5)^2
>>> inverse_laplace(L, s, t)
t*e^(a + 2*t)*sin(t)

The Laplace transform of the exponential function:

sage: laplace(exp(x), x, s)
1/(s - 1)

>>> from sage.all import *
>>> laplace(exp(x), x, s)
1/(s - 1)

Dirac’s delta distribution is handled (the output of SymPy is related to a choice that has to be made when defining Laplace transforms of distributions):

sage: laplace(dirac_delta(t), t, s)
1
sage: F, a, cond = laplace(dirac_delta(t), t, s, algorithm='sympy')
sage: a, cond  # random - sympy <1.10 gives (-oo, True)
(0, True)
sage: F        # random - sympy <1.9 includes undefined heaviside(0) in answer
1
sage: laplace(dirac_delta(t), t, s, algorithm='giac')  # needs giac
1

>>> from sage.all import *
>>> laplace(dirac_delta(t), t, s)
1
>>> F, a, cond = laplace(dirac_delta(t), t, s, algorithm='sympy')
>>> a, cond  # random - sympy <1.10 gives (-oo, True)
(0, True)
>>> F        # random - sympy <1.9 includes undefined heaviside(0) in answer
1
>>> laplace(dirac_delta(t), t, s, algorithm='giac')  # needs giac
1

Heaviside step function can be handled with different interfaces. Try with Maxima:

sage: laplace(heaviside(t-1), t, s)
e^(-s)/s

>>> from sage.all import *
>>> laplace(heaviside(t-Integer(1)), t, s)
e^(-s)/s

Try with giac, if it is installed:

sage: # needs giac
sage: laplace(heaviside(t-1), t, s, algorithm='giac')
e^(-s)/s

>>> from sage.all import *
>>> # needs giac
>>> laplace(heaviside(t-Integer(1)), t, s, algorithm='giac')
e^(-s)/s

Try with SymPy:

sage: laplace(heaviside(t-1), t, s, algorithm='sympy')
(e^(-s)/s, 0, True)

>>> from sage.all import *
>>> laplace(heaviside(t-Integer(1)), t, s, algorithm='sympy')
(e^(-s)/s, 0, True)
sage.calculus.calculus.lim(ex, dir=None, taylor=False, algorithm='maxima', **argv)[source]

Return the limit as the variable \(v\) approaches \(a\) from the given direction.

expr.limit(x = a)
expr.limit(x = a, dir='+')

INPUT:

  • dir – (default: None) may have the value 'plus' (or '+' or 'right' or 'above') for a limit from above, 'minus' (or '-' or 'left' or 'below') for a limit from below, or may be omitted (implying a two-sided limit is to be computed).

  • taylor – (default: False) if True, use Taylor series, which allows more limits to be computed (but may also crash in some obscure cases due to bugs in Maxima).

  • **argv – 1 named parameter

Note

The output may also use und (undefined), ind (indefinite but bounded), and infinity (complex infinity).

EXAMPLES:

sage: x = var('x')
sage: f = (1 + 1/x)^x
sage: f.limit(x=oo)
e
sage: f.limit(x=5)
7776/3125

>>> from sage.all import *
>>> x = var('x')
>>> f = (Integer(1) + Integer(1)/x)**x
>>> f.limit(x=oo)
e
>>> f.limit(x=Integer(5))
7776/3125

Domain to real, a regression in 5.46.0, see https://sf.net/p/maxima/bugs/4138

sage: maxima_calculus.eval("domain:real")
...
sage: f.limit(x=1.2).n()
2.06961575467...
sage: maxima_calculus.eval("domain:complex");
...

>>> from sage.all import *
>>> maxima_calculus.eval("domain:real")
...
>>> f.limit(x=RealNumber('1.2')).n()
2.06961575467...
>>> maxima_calculus.eval("domain:complex");
...

Otherwise, it works

sage: f.limit(x=I, taylor=True)
(-I + 1)^I
sage: f(x=1.2)
2.0696157546720...
sage: f(x=I)
(-I + 1)^I
sage: CDF(f(x=I))
2.0628722350809046 + 0.7450070621797239*I
sage: CDF(f.limit(x=I))
2.0628722350809046 + 0.7450070621797239*I

>>> from sage.all import *
>>> f.limit(x=I, taylor=True)
(-I + 1)^I
>>> f(x=RealNumber('1.2'))
2.0696157546720...
>>> f(x=I)
(-I + 1)^I
>>> CDF(f(x=I))
2.0628722350809046 + 0.7450070621797239*I
>>> CDF(f.limit(x=I))
2.0628722350809046 + 0.7450070621797239*I

Notice that Maxima may ask for more information:

sage: var('a')
a
sage: limit(x^a,x=0)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see
`assume?` for more details)
Is a positive, negative or zero?

>>> from sage.all import *
>>> var('a')
a
>>> limit(x**a,x=Integer(0))
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see
`assume?` for more details)
Is a positive, negative or zero?

With this example, Maxima is looking for a LOT of information:

sage: assume(a>0)
sage: limit(x^a,x=0)  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an integer?
sage: assume(a,'integer')
sage: limit(x^a, x=0)  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an even number?
sage: assume(a, 'even')
sage: limit(x^a, x=0)
0
sage: forget()

>>> from sage.all import *
>>> assume(a>Integer(0))
>>> limit(x**a,x=Integer(0))  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an integer?
>>> assume(a,'integer')
>>> limit(x**a, x=Integer(0))  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an even number?
>>> assume(a, 'even')
>>> limit(x**a, x=Integer(0))
0
>>> forget()

More examples:

sage: limit(x*log(x), x=0, dir='+')
0
sage: lim((x+1)^(1/x), x=0)
e
sage: lim(e^x/x, x=oo)
+Infinity
sage: lim(e^x/x, x=-oo)
0
sage: lim(-e^x/x, x=oo)
-Infinity
sage: lim((cos(x))/(x^2), x=0)
+Infinity
sage: lim(sqrt(x^2+1) - x, x=oo)
0
sage: lim(x^2/(sec(x)-1), x=0)
2
sage: lim(cos(x)/(cos(x)-1), x=0)
-Infinity
sage: lim(x*sin(1/x), x=0)
0
sage: limit(e^(-1/x), x=0, dir='right')
0
sage: limit(e^(-1/x), x=0, dir='left')
+Infinity

>>> from sage.all import *
>>> limit(x*log(x), x=Integer(0), dir='+')
0
>>> lim((x+Integer(1))**(Integer(1)/x), x=Integer(0))
e
>>> lim(e**x/x, x=oo)
+Infinity
>>> lim(e**x/x, x=-oo)
0
>>> lim(-e**x/x, x=oo)
-Infinity
>>> lim((cos(x))/(x**Integer(2)), x=Integer(0))
+Infinity
>>> lim(sqrt(x**Integer(2)+Integer(1)) - x, x=oo)
0
>>> lim(x**Integer(2)/(sec(x)-Integer(1)), x=Integer(0))
2
>>> lim(cos(x)/(cos(x)-Integer(1)), x=Integer(0))
-Infinity
>>> lim(x*sin(Integer(1)/x), x=Integer(0))
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left')
+Infinity


sage: f = log(log(x)) / log(x)
sage: forget(); assume(x < -2); lim(f, x=0, taylor=True)
0
sage: forget()

>>> from sage.all import *
>>> f = log(log(x)) / log(x)
>>> forget(); assume(x < -Integer(2)); lim(f, x=Integer(0), taylor=True)
0
>>> forget()

Here ind means “indefinite but bounded”:

sage: lim(sin(1/x), x = 0)
ind

>>> from sage.all import *
>>> lim(sin(Integer(1)/x), x = Integer(0))
ind

We can use other packages than maxima, namely “sympy”, “giac”, “fricas”.

With the standard package Giac:

sage: # needs sage.libs.giac
sage: (exp(-x)/(2+sin(x))).limit(x=oo, algorithm='giac')
0
sage: limit(e^(-1/x), x=0, dir='right', algorithm='giac')
0
sage: limit(e^(-1/x), x=0, dir='left', algorithm='giac')
+Infinity
sage: (x / (x+2^x+cos(x))).limit(x=-infinity, algorithm='giac')
1

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> (exp(-x)/(Integer(2)+sin(x))).limit(x=oo, algorithm='giac')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right', algorithm='giac')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left', algorithm='giac')
+Infinity
>>> (x / (x+Integer(2)**x+cos(x))).limit(x=-infinity, algorithm='giac')
1

With the optional package FriCAS:

sage: (x / (x+2^x+cos(x))).limit(x=-infinity, algorithm='fricas')       # optional - fricas
1
sage: limit(e^(-1/x), x=0, dir='right', algorithm='fricas')             # optional - fricas
0
sage: limit(e^(-1/x), x=0, dir='left', algorithm='fricas')              # optional - fricas
+Infinity

>>> from sage.all import *
>>> (x / (x+Integer(2)**x+cos(x))).limit(x=-infinity, algorithm='fricas')       # optional - fricas
1
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right', algorithm='fricas')             # optional - fricas
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left', algorithm='fricas')              # optional - fricas
+Infinity

One can also call Mathematica’s online interface:

sage: limit(pi+log(x)/x,x=oo, algorithm='mathematica_free') # optional - internet
pi

>>> from sage.all import *
>>> limit(pi+log(x)/x,x=oo, algorithm='mathematica_free') # optional - internet
pi
sage.calculus.calculus.limit(ex, dir=None, taylor=False, algorithm='maxima', **argv)[source]

Return the limit as the variable \(v\) approaches \(a\) from the given direction.

expr.limit(x = a)
expr.limit(x = a, dir='+')

INPUT:

  • dir – (default: None) may have the value 'plus' (or '+' or 'right' or 'above') for a limit from above, 'minus' (or '-' or 'left' or 'below') for a limit from below, or may be omitted (implying a two-sided limit is to be computed).

  • taylor – (default: False) if True, use Taylor series, which allows more limits to be computed (but may also crash in some obscure cases due to bugs in Maxima).

  • **argv – 1 named parameter

Note

The output may also use und (undefined), ind (indefinite but bounded), and infinity (complex infinity).

EXAMPLES:

sage: x = var('x')
sage: f = (1 + 1/x)^x
sage: f.limit(x=oo)
e
sage: f.limit(x=5)
7776/3125

>>> from sage.all import *
>>> x = var('x')
>>> f = (Integer(1) + Integer(1)/x)**x
>>> f.limit(x=oo)
e
>>> f.limit(x=Integer(5))
7776/3125

Domain to real, a regression in 5.46.0, see https://sf.net/p/maxima/bugs/4138

sage: maxima_calculus.eval("domain:real")
...
sage: f.limit(x=1.2).n()
2.06961575467...
sage: maxima_calculus.eval("domain:complex");
...

>>> from sage.all import *
>>> maxima_calculus.eval("domain:real")
...
>>> f.limit(x=RealNumber('1.2')).n()
2.06961575467...
>>> maxima_calculus.eval("domain:complex");
...

Otherwise, it works

sage: f.limit(x=I, taylor=True)
(-I + 1)^I
sage: f(x=1.2)
2.0696157546720...
sage: f(x=I)
(-I + 1)^I
sage: CDF(f(x=I))
2.0628722350809046 + 0.7450070621797239*I
sage: CDF(f.limit(x=I))
2.0628722350809046 + 0.7450070621797239*I

>>> from sage.all import *
>>> f.limit(x=I, taylor=True)
(-I + 1)^I
>>> f(x=RealNumber('1.2'))
2.0696157546720...
>>> f(x=I)
(-I + 1)^I
>>> CDF(f(x=I))
2.0628722350809046 + 0.7450070621797239*I
>>> CDF(f.limit(x=I))
2.0628722350809046 + 0.7450070621797239*I

Notice that Maxima may ask for more information:

sage: var('a')
a
sage: limit(x^a,x=0)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see
`assume?` for more details)
Is a positive, negative or zero?

>>> from sage.all import *
>>> var('a')
a
>>> limit(x**a,x=Integer(0))
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation
*may* help (example of legal syntax is 'assume(a>0)', see
`assume?` for more details)
Is a positive, negative or zero?

With this example, Maxima is looking for a LOT of information:

sage: assume(a>0)
sage: limit(x^a,x=0)  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an integer?
sage: assume(a,'integer')
sage: limit(x^a, x=0)  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an even number?
sage: assume(a, 'even')
sage: limit(x^a, x=0)
0
sage: forget()

>>> from sage.all import *
>>> assume(a>Integer(0))
>>> limit(x**a,x=Integer(0))  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an integer?
>>> assume(a,'integer')
>>> limit(x**a, x=Integer(0))  # random - maxima 5.46.0 does not need extra assumption
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before evaluation *may* help
(example of legal syntax is 'assume(a>0)', see `assume?` for
 more details)
Is a an even number?
>>> assume(a, 'even')
>>> limit(x**a, x=Integer(0))
0
>>> forget()

More examples:

sage: limit(x*log(x), x=0, dir='+')
0
sage: lim((x+1)^(1/x), x=0)
e
sage: lim(e^x/x, x=oo)
+Infinity
sage: lim(e^x/x, x=-oo)
0
sage: lim(-e^x/x, x=oo)
-Infinity
sage: lim((cos(x))/(x^2), x=0)
+Infinity
sage: lim(sqrt(x^2+1) - x, x=oo)
0
sage: lim(x^2/(sec(x)-1), x=0)
2
sage: lim(cos(x)/(cos(x)-1), x=0)
-Infinity
sage: lim(x*sin(1/x), x=0)
0
sage: limit(e^(-1/x), x=0, dir='right')
0
sage: limit(e^(-1/x), x=0, dir='left')
+Infinity

>>> from sage.all import *
>>> limit(x*log(x), x=Integer(0), dir='+')
0
>>> lim((x+Integer(1))**(Integer(1)/x), x=Integer(0))
e
>>> lim(e**x/x, x=oo)
+Infinity
>>> lim(e**x/x, x=-oo)
0
>>> lim(-e**x/x, x=oo)
-Infinity
>>> lim((cos(x))/(x**Integer(2)), x=Integer(0))
+Infinity
>>> lim(sqrt(x**Integer(2)+Integer(1)) - x, x=oo)
0
>>> lim(x**Integer(2)/(sec(x)-Integer(1)), x=Integer(0))
2
>>> lim(cos(x)/(cos(x)-Integer(1)), x=Integer(0))
-Infinity
>>> lim(x*sin(Integer(1)/x), x=Integer(0))
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left')
+Infinity


sage: f = log(log(x)) / log(x)
sage: forget(); assume(x < -2); lim(f, x=0, taylor=True)
0
sage: forget()

>>> from sage.all import *
>>> f = log(log(x)) / log(x)
>>> forget(); assume(x < -Integer(2)); lim(f, x=Integer(0), taylor=True)
0
>>> forget()

Here ind means “indefinite but bounded”:

sage: lim(sin(1/x), x = 0)
ind

>>> from sage.all import *
>>> lim(sin(Integer(1)/x), x = Integer(0))
ind

We can use other packages than maxima, namely “sympy”, “giac”, “fricas”.

With the standard package Giac:

sage: # needs sage.libs.giac
sage: (exp(-x)/(2+sin(x))).limit(x=oo, algorithm='giac')
0
sage: limit(e^(-1/x), x=0, dir='right', algorithm='giac')
0
sage: limit(e^(-1/x), x=0, dir='left', algorithm='giac')
+Infinity
sage: (x / (x+2^x+cos(x))).limit(x=-infinity, algorithm='giac')
1

>>> from sage.all import *
>>> # needs sage.libs.giac
>>> (exp(-x)/(Integer(2)+sin(x))).limit(x=oo, algorithm='giac')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right', algorithm='giac')
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left', algorithm='giac')
+Infinity
>>> (x / (x+Integer(2)**x+cos(x))).limit(x=-infinity, algorithm='giac')
1

With the optional package FriCAS:

sage: (x / (x+2^x+cos(x))).limit(x=-infinity, algorithm='fricas')       # optional - fricas
1
sage: limit(e^(-1/x), x=0, dir='right', algorithm='fricas')             # optional - fricas
0
sage: limit(e^(-1/x), x=0, dir='left', algorithm='fricas')              # optional - fricas
+Infinity

>>> from sage.all import *
>>> (x / (x+Integer(2)**x+cos(x))).limit(x=-infinity, algorithm='fricas')       # optional - fricas
1
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='right', algorithm='fricas')             # optional - fricas
0
>>> limit(e**(-Integer(1)/x), x=Integer(0), dir='left', algorithm='fricas')              # optional - fricas
+Infinity

One can also call Mathematica’s online interface:

sage: limit(pi+log(x)/x,x=oo, algorithm='mathematica_free') # optional - internet
pi

>>> from sage.all import *
>>> limit(pi+log(x)/x,x=oo, algorithm='mathematica_free') # optional - internet
pi
sage.calculus.calculus.mapped_opts(v)[source]

Used internally when creating a string of options to pass to Maxima.

INPUT:

  • v – an object

OUTPUT: string

The main use of this is to turn Python bools into lower case strings.

EXAMPLES:

sage: sage.calculus.calculus.mapped_opts(True)
'true'
sage: sage.calculus.calculus.mapped_opts(False)
'false'
sage: sage.calculus.calculus.mapped_opts('bar')
'bar'

>>> from sage.all import *
>>> sage.calculus.calculus.mapped_opts(True)
'true'
>>> sage.calculus.calculus.mapped_opts(False)
'false'
>>> sage.calculus.calculus.mapped_opts('bar')
'bar'
sage.calculus.calculus.maxima_options(**kwds)[source]

Used internally to create a string of options to pass to Maxima.

EXAMPLES:

sage: sage.calculus.calculus.maxima_options(an_option=True, another=False, foo='bar')
'an_option=true,another=false,foo=bar'

>>> from sage.all import *
>>> sage.calculus.calculus.maxima_options(an_option=True, another=False, foo='bar')
'an_option=true,another=false,foo=bar'
sage.calculus.calculus.minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0)[source]

Return the minimal polynomial of self, if possible.

INPUT:

  • var – polynomial variable name (default: 'x')

  • algorithm – 'algebraic' or 'numerical' (default both, but with numerical first)

  • bits – the number of bits to use in numerical approx

  • degree – the expected algebraic degree

  • epsilon – return without error as long as f(self) epsilon, in the case that the result cannot be proven

    All of the above parameters are optional, with epsilon=0, bits and degree tested up to 1000 and 24 by default respectively. The numerical algorithm will be faster if bits and/or degree are given explicitly. The algebraic algorithm ignores the last three parameters.

OUTPUT: the minimal polynomial of self. If the numerical algorithm is used, then it is proved symbolically when epsilon=0 (default).

If the minimal polynomial could not be found, two distinct kinds of errors are raised. If no reasonable candidate was found with the given bits/degree parameters, a ValueError will be raised. If a reasonable candidate was found but (perhaps due to limits in the underlying symbolic package) was unable to be proved correct, a NotImplementedError will be raised.

ALGORITHM: Two distinct algorithms are used, depending on the algorithm parameter. By default, the numerical algorithm is attempted first, then the algebraic one.

Algebraic: Attempt to evaluate this expression in QQbar, using cyclotomic fields to resolve exponential and trig functions at rational multiples of \(\pi\), field extensions to handle roots and rational exponents, and computing compositums to represent the full expression as an element of a number field where the minimal polynomial can be computed exactly. The bits, degree, and epsilon parameters are ignored.

Numerical: Computes a numerical approximation of self and use PARI’s pari:algdep to get a candidate minpoly \(f\). If \(f(\mathtt{self})\), evaluated to a higher precision, is close enough to 0 then evaluate \(f(\mathtt{self})\) symbolically, attempting to prove vanishing. If this fails, and epsilon is nonzero, return \(f\) if and only if \(f(\mathtt{self}) < \mathtt{epsilon}\). Otherwise raise a ValueError (if no suitable candidate was found) or a NotImplementedError (if a likely candidate was found but could not be proved correct).

EXAMPLES: First some simple examples:

sage: sqrt(2).minpoly()
x^2 - 2
sage: minpoly(2^(1/3))
x^3 - 2
sage: minpoly(sqrt(2) + sqrt(-1))
x^4 - 2*x^2 + 9
sage: minpoly(sqrt(2)-3^(1/3))
x^6 - 6*x^4 + 6*x^3 + 12*x^2 + 36*x + 1

>>> from sage.all import *
>>> sqrt(Integer(2)).minpoly()
x^2 - 2
>>> minpoly(Integer(2)**(Integer(1)/Integer(3)))
x^3 - 2
>>> minpoly(sqrt(Integer(2)) + sqrt(-Integer(1)))
x^4 - 2*x^2 + 9
>>> minpoly(sqrt(Integer(2))-Integer(3)**(Integer(1)/Integer(3)))
x^6 - 6*x^4 + 6*x^3 + 12*x^2 + 36*x + 1

Works with trig and exponential functions too.

sage: sin(pi/3).minpoly()
x^2 - 3/4
sage: sin(pi/7).minpoly()
x^6 - 7/4*x^4 + 7/8*x^2 - 7/64
sage: minpoly(exp(I*pi/17))
x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8
 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1

>>> from sage.all import *
>>> sin(pi/Integer(3)).minpoly()
x^2 - 3/4
>>> sin(pi/Integer(7)).minpoly()
x^6 - 7/4*x^4 + 7/8*x^2 - 7/64
>>> minpoly(exp(I*pi/Integer(17)))
x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8
 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1

Here we verify it gives the same result as the abstract number field.

sage: (sqrt(2) + sqrt(3) + sqrt(6)).minpoly()
x^4 - 22*x^2 - 48*x - 23
sage: K.<a,b> = NumberField([x^2-2, x^2-3])
sage: (a+b+a*b).absolute_minpoly()
x^4 - 22*x^2 - 48*x - 23

>>> from sage.all import *
>>> (sqrt(Integer(2)) + sqrt(Integer(3)) + sqrt(Integer(6))).minpoly()
x^4 - 22*x^2 - 48*x - 23
>>> K = NumberField([x**Integer(2)-Integer(2), x**Integer(2)-Integer(3)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> (a+b+a*b).absolute_minpoly()
x^4 - 22*x^2 - 48*x - 23

The minpoly() function is used implicitly when creating number fields:

sage: x = var('x')
sage: eqn =  x^3 + sqrt(2)*x + 5 == 0
sage: a = solve(eqn, x)[0].rhs()
sage: QQ[a]
Number Field in a with defining polynomial x^6 + 10*x^3 - 2*x^2 + 25
 with a = 0.7185272465828846? - 1.721353471724806?*I

>>> from sage.all import *
>>> x = var('x')
>>> eqn =  x**Integer(3) + sqrt(Integer(2))*x + Integer(5) == Integer(0)
>>> a = solve(eqn, x)[Integer(0)].rhs()
>>> QQ[a]
Number Field in a with defining polynomial x^6 + 10*x^3 - 2*x^2 + 25
 with a = 0.7185272465828846? - 1.721353471724806?*I

Here we solve a cubic and then recover it from its complicated radical expansion.

sage: f = x^3 - x + 1
sage: a = f.solve(x)[0].rhs(); a
-1/2*(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1)
 - 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)
sage: a.minpoly()
x^3 - x + 1

>>> from sage.all import *
>>> f = x**Integer(3) - x + Integer(1)
>>> a = f.solve(x)[Integer(0)].rhs(); a
-1/2*(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1)
 - 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(23)*sqrt(3) - 1/2)^(1/3)
>>> a.minpoly()
x^3 - x + 1

Note that simplification may be necessary to see that the minimal polynomial is correct.

sage: a = sqrt(2)+sqrt(3)+sqrt(5)
sage: f = a.minpoly(); f
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
sage: f(a)
(sqrt(5) + sqrt(3) + sqrt(2))^8 - 40*(sqrt(5) + sqrt(3) + sqrt(2))^6
 + 352*(sqrt(5) + sqrt(3) + sqrt(2))^4 - 960*(sqrt(5) + sqrt(3) + sqrt(2))^2
 + 576
sage: f(a).expand()
0

>>> from sage.all import *
>>> a = sqrt(Integer(2))+sqrt(Integer(3))+sqrt(Integer(5))
>>> f = a.minpoly(); f
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
>>> f(a)
(sqrt(5) + sqrt(3) + sqrt(2))^8 - 40*(sqrt(5) + sqrt(3) + sqrt(2))^6
 + 352*(sqrt(5) + sqrt(3) + sqrt(2))^4 - 960*(sqrt(5) + sqrt(3) + sqrt(2))^2
 + 576
>>> f(a).expand()
0


sage: a = sin(pi/7)
sage: f = a.minpoly(algorithm='numerical'); f
x^6 - 7/4*x^4 + 7/8*x^2 - 7/64
sage: f(a).horner(a).numerical_approx(100)
0.00000000000000000000000000000

>>> from sage.all import *
>>> a = sin(pi/Integer(7))
>>> f = a.minpoly(algorithm='numerical'); f
x^6 - 7/4*x^4 + 7/8*x^2 - 7/64
>>> f(a).horner(a).numerical_approx(Integer(100))
0.00000000000000000000000000000

The degree must be high enough (default tops out at 24).

sage: a = sqrt(3) + sqrt(2)
sage: a.minpoly(algorithm='numerical', bits=100, degree=3)
Traceback (most recent call last):
...
ValueError: Could not find minimal polynomial (100 bits, degree 3).
sage: a.minpoly(algorithm='numerical', bits=100, degree=10)
x^4 - 10*x^2 + 1

>>> from sage.all import *
>>> a = sqrt(Integer(3)) + sqrt(Integer(2))
>>> a.minpoly(algorithm='numerical', bits=Integer(100), degree=Integer(3))
Traceback (most recent call last):
...
ValueError: Could not find minimal polynomial (100 bits, degree 3).
>>> a.minpoly(algorithm='numerical', bits=Integer(100), degree=Integer(10))
x^4 - 10*x^2 + 1


sage: cos(pi/33).minpoly(algorithm='algebraic')
x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5
 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024
sage: cos(pi/33).minpoly(algorithm='numerical')
x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5
 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024

>>> from sage.all import *
>>> cos(pi/Integer(33)).minpoly(algorithm='algebraic')
x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5
 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024
>>> cos(pi/Integer(33)).minpoly(algorithm='numerical')
x^10 + 1/2*x^9 - 5/2*x^8 - 5/4*x^7 + 17/8*x^6 + 17/16*x^5
 - 43/64*x^4 - 43/128*x^3 + 3/64*x^2 + 3/128*x + 1/1024

Sometimes it fails, as it must given that some numbers aren’t algebraic:

sage: sin(1).minpoly(algorithm='numerical')
Traceback (most recent call last):
...
ValueError: Could not find minimal polynomial (1000 bits, degree 24).

>>> from sage.all import *
>>> sin(Integer(1)).minpoly(algorithm='numerical')
Traceback (most recent call last):
...
ValueError: Could not find minimal polynomial (1000 bits, degree 24).

Note

Of course, failure to produce a minimal polynomial does not necessarily indicate that this number is transcendental.

sage.calculus.calculus.mma_free_limit(expression, v, a, dir=None)[source]

Limit using Mathematica’s online interface.

INPUT:

  • expression – symbolic expression

  • v – variable

  • a – value where the variable goes to

  • dir – '+', '-' or None (default: None)

EXAMPLES:

sage: from sage.calculus.calculus import mma_free_limit
sage: mma_free_limit(sin(x)/x, x, a=0) # optional - internet
1

>>> from sage.all import *
>>> from sage.calculus.calculus import mma_free_limit
>>> mma_free_limit(sin(x)/x, x, a=Integer(0)) # optional - internet
1

Another simple limit:

sage: mma_free_limit(e^(-x), x, a=oo) # optional - internet
0

>>> from sage.all import *
>>> mma_free_limit(e**(-x), x, a=oo) # optional - internet
0
sage.calculus.calculus.nintegral(ex, x, a, b, desired_relative_error='1e-8', maximum_num_subintervals=200)[source]

Return a floating point machine precision numerical approximation to the integral of self from \(a\) to \(b\), computed using floating point arithmetic via maxima.

INPUT:

  • x – variable to integrate with respect to

  • a – lower endpoint of integration

  • b – upper endpoint of integration

  • desired_relative_error – (default: 1e-8) the desired relative error

  • maximum_num_subintervals – (default: 200) maximal number of subintervals

OUTPUT: float; approximation to the integral

  • float: estimated absolute error of the approximation

  • the number of integrand evaluations

  • an error code:

    • 0 – no problems were encountered

    • 1 – too many subintervals were done

    • 2 – excessive roundoff error

    • 3 – extremely bad integrand behavior

    • 4 – failed to converge

    • 5 – integral is probably divergent or slowly convergent

    • 6 – the input is invalid; this includes the case of desired_relative_error being too small to be achieved

ALIAS: nintegrate() is the same as nintegral()

REMARK: There is also a function numerical_integral() that implements numerical integration using the GSL C library. It is potentially much faster and applies to arbitrary user defined functions.

Also, there are limits to the precision to which Maxima can compute the integral due to limitations in quadpack. In the following example, remark that the last value of the returned tuple is 6, indicating that the input was invalid, in this case because of a too high desired precision.

sage: f = x
sage: f.nintegral(x, 0, 1, 1e-14)
(0.0, 0.0, 0, 6)

>>> from sage.all import *
>>> f = x
>>> f.nintegral(x, Integer(0), Integer(1), RealNumber('1e-14'))
(0.0, 0.0, 0, 6)

EXAMPLES:

sage: f(x) = exp(-sqrt(x))
sage: f.nintegral(x, 0, 1)
(0.5284822353142306, 4.163...e-11, 231, 0)

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(exp(-sqrt(x))).function(x)
>>> f.nintegral(x, Integer(0), Integer(1))
(0.5284822353142306, 4.163...e-11, 231, 0)

We can also use the numerical_integral() function, which calls the GSL C library.

sage: numerical_integral(f, 0, 1)
(0.528482232253147, 6.83928460...e-07)

>>> from sage.all import *
>>> numerical_integral(f, Integer(0), Integer(1))
(0.528482232253147, 6.83928460...e-07)

Note that in exotic cases where floating point evaluation of the expression leads to the wrong value, then the output can be completely wrong:

sage: f = exp(pi*sqrt(163)) - 262537412640768744

>>> from sage.all import *
>>> f = exp(pi*sqrt(Integer(163))) - Integer(262537412640768744)

Despite appearance, \(f\) is really very close to 0, but one gets a nonzero value since the definition of float(f) is that it makes all constants inside the expression floats, then evaluates each function and each arithmetic operation using float arithmetic:

sage: float(f)
-480.0

>>> from sage.all import *
>>> float(f)
-480.0

Computing to higher precision we see the truth:

sage: f.n(200)
-7.4992740280181431112064614366622348652078895136533593355718e-13
sage: f.n(300)
-7.49927402801814311120646143662663009137292462589621789352095066181709095575681963967103004e-13

>>> from sage.all import *
>>> f.n(Integer(200))
-7.4992740280181431112064614366622348652078895136533593355718e-13
>>> f.n(Integer(300))
-7.49927402801814311120646143662663009137292462589621789352095066181709095575681963967103004e-13

Now numerically integrating, we see why the answer is wrong:

sage: f.nintegrate(x,0,1)
(-480.000000000000..., 5.32907051820075...e-12, 21, 0)

>>> from sage.all import *
>>> f.nintegrate(x,Integer(0),Integer(1))
(-480.000000000000..., 5.32907051820075...e-12, 21, 0)

It is just because every floating point evaluation of \(f\) returns \(-480.0\) in floating point.

Important note: using PARI/GP one can compute numerical integrals to high precision:

sage: gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829176211363206621657 E-127'
sage: old_prec = gp.set_real_precision(50)
sage: gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829173563961304957417746108003917 E-127'
sage: gp.set_real_precision(old_prec)
57

>>> from sage.all import *
>>> gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829176211363206621657 E-127'
>>> old_prec = gp.set_real_precision(Integer(50))
>>> gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829173563961304957417746108003917 E-127'
>>> gp.set_real_precision(old_prec)
57

Note that the input function above is a string in PARI syntax.

sage.calculus.calculus.nintegrate(ex, x, a, b, desired_relative_error='1e-8', maximum_num_subintervals=200)[source]

Return a floating point machine precision numerical approximation to the integral of self from \(a\) to \(b\), computed using floating point arithmetic via maxima.

INPUT:

  • x – variable to integrate with respect to

  • a – lower endpoint of integration

  • b – upper endpoint of integration

  • desired_relative_error – (default: 1e-8) the desired relative error

  • maximum_num_subintervals – (default: 200) maximal number of subintervals

OUTPUT: float; approximation to the integral

  • float: estimated absolute error of the approximation

  • the number of integrand evaluations

  • an error code:

    • 0 – no problems were encountered

    • 1 – too many subintervals were done

    • 2 – excessive roundoff error

    • 3 – extremely bad integrand behavior

    • 4 – failed to converge

    • 5 – integral is probably divergent or slowly convergent

    • 6 – the input is invalid; this includes the case of desired_relative_error being too small to be achieved

ALIAS: nintegrate() is the same as nintegral()

REMARK: There is also a function numerical_integral() that implements numerical integration using the GSL C library. It is potentially much faster and applies to arbitrary user defined functions.

Also, there are limits to the precision to which Maxima can compute the integral due to limitations in quadpack. In the following example, remark that the last value of the returned tuple is 6, indicating that the input was invalid, in this case because of a too high desired precision.

sage: f = x
sage: f.nintegral(x, 0, 1, 1e-14)
(0.0, 0.0, 0, 6)

>>> from sage.all import *
>>> f = x
>>> f.nintegral(x, Integer(0), Integer(1), RealNumber('1e-14'))
(0.0, 0.0, 0, 6)

EXAMPLES:

sage: f(x) = exp(-sqrt(x))
sage: f.nintegral(x, 0, 1)
(0.5284822353142306, 4.163...e-11, 231, 0)

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(exp(-sqrt(x))).function(x)
>>> f.nintegral(x, Integer(0), Integer(1))
(0.5284822353142306, 4.163...e-11, 231, 0)

We can also use the numerical_integral() function, which calls the GSL C library.

sage: numerical_integral(f, 0, 1)
(0.528482232253147, 6.83928460...e-07)

>>> from sage.all import *
>>> numerical_integral(f, Integer(0), Integer(1))
(0.528482232253147, 6.83928460...e-07)

Note that in exotic cases where floating point evaluation of the expression leads to the wrong value, then the output can be completely wrong:

sage: f = exp(pi*sqrt(163)) - 262537412640768744

>>> from sage.all import *
>>> f = exp(pi*sqrt(Integer(163))) - Integer(262537412640768744)

Despite appearance, \(f\) is really very close to 0, but one gets a nonzero value since the definition of float(f) is that it makes all constants inside the expression floats, then evaluates each function and each arithmetic operation using float arithmetic:

sage: float(f)
-480.0

>>> from sage.all import *
>>> float(f)
-480.0

Computing to higher precision we see the truth:

sage: f.n(200)
-7.4992740280181431112064614366622348652078895136533593355718e-13
sage: f.n(300)
-7.49927402801814311120646143662663009137292462589621789352095066181709095575681963967103004e-13

>>> from sage.all import *
>>> f.n(Integer(200))
-7.4992740280181431112064614366622348652078895136533593355718e-13
>>> f.n(Integer(300))
-7.49927402801814311120646143662663009137292462589621789352095066181709095575681963967103004e-13

Now numerically integrating, we see why the answer is wrong:

sage: f.nintegrate(x,0,1)
(-480.000000000000..., 5.32907051820075...e-12, 21, 0)

>>> from sage.all import *
>>> f.nintegrate(x,Integer(0),Integer(1))
(-480.000000000000..., 5.32907051820075...e-12, 21, 0)

It is just because every floating point evaluation of \(f\) returns \(-480.0\) in floating point.

Important note: using PARI/GP one can compute numerical integrals to high precision:

sage: gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829176211363206621657 E-127'
sage: old_prec = gp.set_real_precision(50)
sage: gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829173563961304957417746108003917 E-127'
sage: gp.set_real_precision(old_prec)
57

>>> from sage.all import *
>>> gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829176211363206621657 E-127'
>>> old_prec = gp.set_real_precision(Integer(50))
>>> gp.eval('intnum(x=17,42,exp(-x^2)*log(x))')
'2.5657285005610514829173563961304957417746108003917 E-127'
>>> gp.set_real_precision(old_prec)
57

Note that the input function above is a string in PARI syntax.

sage.calculus.calculus.symbolic_expression_from_maxima_string(x, equals_sub=False, maxima=Maxima_lib)[source]

Given a string representation of a Maxima expression, parse it and return the corresponding Sage symbolic expression.

INPUT:

  • x – string

  • equals_sub – boolean (default: False); if True, replace ‘=’ by ‘==’ in self

  • maxima – (default: the calculus package’s copy of Maxima) the Maxima interpreter to use

EXAMPLES:

sage: from sage.calculus.calculus import symbolic_expression_from_maxima_string as sefms
sage: sefms('x^%e + %e^%pi + %i + sin(0)')
x^e + e^pi + I
sage: f = function('f')(x)
sage: sefms('?%at(f(x),x=2)#1')
f(2) != 1
sage: a = sage.calculus.calculus.maxima("x#0"); a
x # 0
sage: a.sage()
x != 0

>>> from sage.all import *
>>> from sage.calculus.calculus import symbolic_expression_from_maxima_string as sefms
>>> sefms('x^%e + %e^%pi + %i + sin(0)')
x^e + e^pi + I
>>> f = function('f')(x)
>>> sefms('?%at(f(x),x=2)#1')
f(2) != 1
>>> a = sage.calculus.calculus.maxima("x#0"); a
x # 0
>>> a.sage()
x != 0
sage.calculus.calculus.symbolic_expression_from_string(s, syms, accept_sequence=None, parser=False)[source]

Given a string, (attempt to) parse it and return the corresponding Sage symbolic expression. Normally used to return Maxima output to the user.

INPUT:

  • s – string

  • syms – (default: {}) dictionary of strings to be regarded as symbols or functions; keys are pairs (string, number of arguments)

  • accept_sequence – boolean (default: False); controls whether to allow a (possibly nested) set of lists and tuples as input

  • parser – (default: SR_parser) parser for internal use

EXAMPLES:

sage: from sage.calculus.calculus import symbolic_expression_from_string
sage: y = var('y')
sage: symbolic_expression_from_string('[sin(0)*x^2,3*spam+e^pi]',
....:                                 syms={('spam',0): y}, accept_sequence=True)
[0, 3*y + e^pi]

>>> from sage.all import *
>>> from sage.calculus.calculus import symbolic_expression_from_string
>>> y = var('y')
>>> symbolic_expression_from_string('[sin(0)*x^2,3*spam+e^pi]',
...                                 syms={('spam',Integer(0)): y}, accept_sequence=True)
[0, 3*y + e^pi]
sage.calculus.calculus.symbolic_product(expression, v, a, b, algorithm='maxima', hold=False)[source]

Return the symbolic product \(\prod_{v = a}^b expression\) with respect to the variable \(v\) with endpoints \(a\) and \(b\).

INPUT:

  • expression – a symbolic expression

  • v – a variable or variable name

  • a – lower endpoint of the product

  • b – upper endpoint of the prduct

  • algorithm – (default: 'maxima') one of

    • 'maxima' – use Maxima (the default)

    • 'giac' – use Giac (optional)

    • 'sympy' – use SymPy

    • 'mathematica' – (optional) use Mathematica

  • hold – boolean (default: False); if True, don’t evaluate

EXAMPLES:

sage: i, k, n = var('i,k,n')
sage: from sage.calculus.calculus import symbolic_product
sage: symbolic_product(k, k, 1, n)
factorial(n)
sage: symbolic_product(x + i*(i+1)/2, i, 1, 4)
x^4 + 20*x^3 + 127*x^2 + 288*x + 180
sage: symbolic_product(i^2, i, 1, 7)
25401600
sage: f = function('f')
sage: symbolic_product(f(i), i, 1, 7)
f(7)*f(6)*f(5)*f(4)*f(3)*f(2)*f(1)
sage: symbolic_product(f(i), i, 1, n)
product(f(i), i, 1, n)
sage: assume(k>0)
sage: symbolic_product(integrate (x^k, x, 0, 1), k, 1, n)
1/factorial(n + 1)
sage: symbolic_product(f(i), i, 1, n).log().log_expand()
sum(log(f(i)), i, 1, n)

>>> from sage.all import *
>>> i, k, n = var('i,k,n')
>>> from sage.calculus.calculus import symbolic_product
>>> symbolic_product(k, k, Integer(1), n)
factorial(n)
>>> symbolic_product(x + i*(i+Integer(1))/Integer(2), i, Integer(1), Integer(4))
x^4 + 20*x^3 + 127*x^2 + 288*x + 180
>>> symbolic_product(i**Integer(2), i, Integer(1), Integer(7))
25401600
>>> f = function('f')
>>> symbolic_product(f(i), i, Integer(1), Integer(7))
f(7)*f(6)*f(5)*f(4)*f(3)*f(2)*f(1)
>>> symbolic_product(f(i), i, Integer(1), n)
product(f(i), i, 1, n)
>>> assume(k>Integer(0))
>>> symbolic_product(integrate (x**k, x, Integer(0), Integer(1)), k, Integer(1), n)
1/factorial(n + 1)
>>> symbolic_product(f(i), i, Integer(1), n).log().log_expand()
sum(log(f(i)), i, 1, n)
sage.calculus.calculus.symbolic_sum(expression, v, a, b, algorithm='maxima', hold=False)[source]

Return the symbolic sum \(\sum_{v = a}^b expression\) with respect to the variable \(v\) with endpoints \(a\) and \(b\).

INPUT:

  • expression – a symbolic expression

  • v – a variable or variable name

  • a – lower endpoint of the sum

  • b – upper endpoint of the sum

  • algorithm – (default: 'maxima') one of

    • 'maxima' – use Maxima (the default)

    • 'maple' – (optional) use Maple

    • 'mathematica' – (optional) use Mathematica

    • 'giac' – (optional) use Giac

    • 'sympy' – use SymPy

  • hold – boolean (default: False); if True, don’t evaluate

EXAMPLES:

sage: k, n = var('k,n')
sage: from sage.calculus.calculus import symbolic_sum
sage: symbolic_sum(k, k, 1, n).factor()
1/2*(n + 1)*n

>>> from sage.all import *
>>> k, n = var('k,n')
>>> from sage.calculus.calculus import symbolic_sum
>>> symbolic_sum(k, k, Integer(1), n).factor()
1/2*(n + 1)*n


sage: symbolic_sum(1/k^4, k, 1, oo)
1/90*pi^4

>>> from sage.all import *
>>> symbolic_sum(Integer(1)/k**Integer(4), k, Integer(1), oo)
1/90*pi^4


sage: symbolic_sum(1/k^5, k, 1, oo)
zeta(5)

>>> from sage.all import *
>>> symbolic_sum(Integer(1)/k**Integer(5), k, Integer(1), oo)
zeta(5)

A well known binomial identity:

sage: symbolic_sum(binomial(n,k), k, 0, n)
2^n

>>> from sage.all import *
>>> symbolic_sum(binomial(n,k), k, Integer(0), n)
2^n

And some truncations thereof:

sage: assume(n>1)
sage: symbolic_sum(binomial(n,k), k, 1, n)
2^n - 1
sage: symbolic_sum(binomial(n,k), k, 2, n)
2^n - n - 1
sage: symbolic_sum(binomial(n,k), k, 0, n-1)
2^n - 1
sage: symbolic_sum(binomial(n,k), k, 1, n-1)
2^n - 2

>>> from sage.all import *
>>> assume(n>Integer(1))
>>> symbolic_sum(binomial(n,k), k, Integer(1), n)
2^n - 1
>>> symbolic_sum(binomial(n,k), k, Integer(2), n)
2^n - n - 1
>>> symbolic_sum(binomial(n,k), k, Integer(0), n-Integer(1))
2^n - 1
>>> symbolic_sum(binomial(n,k), k, Integer(1), n-Integer(1))
2^n - 2

The binomial theorem:

sage: x, y = var('x, y')
sage: symbolic_sum(binomial(n,k) * x^k * y^(n-k), k, 0, n)
(x + y)^n

>>> from sage.all import *
>>> x, y = var('x, y')
>>> symbolic_sum(binomial(n,k) * x**k * y**(n-k), k, Integer(0), n)
(x + y)^n


sage: symbolic_sum(k * binomial(n, k), k, 1, n)
2^(n - 1)*n

>>> from sage.all import *
>>> symbolic_sum(k * binomial(n, k), k, Integer(1), n)
2^(n - 1)*n


sage: symbolic_sum((-1)^k*binomial(n,k), k, 0, n)
0

>>> from sage.all import *
>>> symbolic_sum((-Integer(1))**k*binomial(n,k), k, Integer(0), n)
0


sage: symbolic_sum(2^(-k)/(k*(k+1)), k, 1, oo)
-log(2) + 1

>>> from sage.all import *
>>> symbolic_sum(Integer(2)**(-k)/(k*(k+Integer(1))), k, Integer(1), oo)
-log(2) + 1

Summing a hypergeometric term:

sage: symbolic_sum(binomial(n, k) * factorial(k) / factorial(n+1+k), k, 0, n)
1/2*sqrt(pi)/factorial(n + 1/2)

>>> from sage.all import *
>>> symbolic_sum(binomial(n, k) * factorial(k) / factorial(n+Integer(1)+k), k, Integer(0), n)
1/2*sqrt(pi)/factorial(n + 1/2)

We check a well known identity:

sage: bool(symbolic_sum(k^3, k, 1, n) == symbolic_sum(k, k, 1, n)^2)
True

>>> from sage.all import *
>>> bool(symbolic_sum(k**Integer(3), k, Integer(1), n) == symbolic_sum(k, k, Integer(1), n)**Integer(2))
True

A geometric sum:

sage: a, q = var('a, q')
sage: symbolic_sum(a*q^k, k, 0, n)
(a*q^(n + 1) - a)/(q - 1)

>>> from sage.all import *
>>> a, q = var('a, q')
>>> symbolic_sum(a*q**k, k, Integer(0), n)
(a*q^(n + 1) - a)/(q - 1)

For the geometric series, we will have to assume the right values for the sum to converge:

sage: assume(abs(q) < 1)
sage: symbolic_sum(a*q^k, k, 0, oo)
-a/(q - 1)

>>> from sage.all import *
>>> assume(abs(q) < Integer(1))
>>> symbolic_sum(a*q**k, k, Integer(0), oo)
-a/(q - 1)

A divergent geometric series. Don’t forget to forget your assumptions:

sage: forget()
sage: assume(q > 1)
sage: symbolic_sum(a*q^k, k, 0, oo)
Traceback (most recent call last):
...
ValueError: Sum is divergent.
sage: forget()
sage: assumptions()  # check the assumptions were really forgotten
[]

>>> from sage.all import *
>>> forget()
>>> assume(q > Integer(1))
>>> symbolic_sum(a*q**k, k, Integer(0), oo)
Traceback (most recent call last):
...
ValueError: Sum is divergent.
>>> forget()
>>> assumptions()  # check the assumptions were really forgotten
[]

A summation performed by Mathematica:

sage: symbolic_sum(1/(1+k^2), k, -oo, oo, algorithm='mathematica')     # optional - mathematica
pi*coth(pi)

>>> from sage.all import *
>>> symbolic_sum(Integer(1)/(Integer(1)+k**Integer(2)), k, -oo, oo, algorithm='mathematica')     # optional - mathematica
pi*coth(pi)

An example of this summation with Giac:

sage: # needs giac
sage: symbolic_sum(1/(1+k^2), k, -oo, oo, algorithm='giac').factor()
pi*(e^(2*pi) + 1)/((e^pi + 1)*(e^pi - 1))

>>> from sage.all import *
>>> # needs giac
>>> symbolic_sum(Integer(1)/(Integer(1)+k**Integer(2)), k, -oo, oo, algorithm='giac').factor()
pi*(e^(2*pi) + 1)/((e^pi + 1)*(e^pi - 1))

The same summation is solved by SymPy:

sage: symbolic_sum(1/(1+k^2), k, -oo, oo, algorithm='sympy')
pi/tanh(pi)

>>> from sage.all import *
>>> symbolic_sum(Integer(1)/(Integer(1)+k**Integer(2)), k, -oo, oo, algorithm='sympy')
pi/tanh(pi)

SymPy and Maxima 5.39.0 can do the following (see Issue #22005):

sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity, algorithm='sympy')
1/64*pi^2
sage: sum(1/((2*n+1)^2-4)^2, n, 0, Infinity)
1/64*pi^2

>>> from sage.all import *
>>> sum(Integer(1)/((Integer(2)*n+Integer(1))**Integer(2)-Integer(4))**Integer(2), n, Integer(0), Infinity, algorithm='sympy')
1/64*pi^2
>>> sum(Integer(1)/((Integer(2)*n+Integer(1))**Integer(2)-Integer(4))**Integer(2), n, Integer(0), Infinity)
1/64*pi^2

Use Maple as a backend for summation:

sage: symbolic_sum(binomial(n,k)*x^k, k, 0, n, algorithm='maple')      # optional - maple
(x + 1)^n

>>> from sage.all import *
>>> symbolic_sum(binomial(n,k)*x**k, k, Integer(0), n, algorithm='maple')      # optional - maple
(x + 1)^n

If you don’t want to evaluate immediately give the hold keyword:

sage: s = sum(n, n, 1, k, hold=True); s
sum(n, n, 1, k)
sage: s.unhold()
1/2*k^2 + 1/2*k
sage: s.subs(k == 10)
sum(n, n, 1, 10)
sage: s.subs(k == 10).unhold()
55
sage: s.subs(k == 10).n()
55.0000000000000

>>> from sage.all import *
>>> s = sum(n, n, Integer(1), k, hold=True); s
sum(n, n, 1, k)
>>> s.unhold()
1/2*k^2 + 1/2*k
>>> s.subs(k == Integer(10))
sum(n, n, 1, 10)
>>> s.subs(k == Integer(10)).unhold()
55
>>> s.subs(k == Integer(10)).n()
55.0000000000000

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 Sage expression.