Functional notation support for common calculus methods

EXAMPLES: We illustrate each of the calculus functional functions.

sage: simplify(x - x)
0
sage: a = var('a')
sage: derivative(x^a + sin(x), x)
a*x^(a - 1) + cos(x)
sage: diff(x^a + sin(x), x)
a*x^(a - 1) + cos(x)
sage: derivative(x^a + sin(x), x)
a*x^(a - 1) + cos(x)
sage: integral(a*x*sin(x), x)
-(x*cos(x) - sin(x))*a
sage: integrate(a*x*sin(x), x)
-(x*cos(x) - sin(x))*a
sage: limit(a*sin(x)/x, x=0)
a
sage: taylor(a*sin(x)/x, x, 0, 4)
1/120*a*x^4 - 1/6*a*x^2 + a
sage: expand((x - a)^3)
-a^3 + 3*a^2*x - 3*a*x^2 + x^3

sage.calculus.functional.derivative(f, *args, **kwds)

The derivative of \(f\).

Repeated differentiation is supported by the syntax given in the examples below.

ALIAS: diff

EXAMPLES: We differentiate a callable symbolic function:

sage: f(x,y) = x*y + sin(x^2) + e^(-x)
sage: f
(x, y) |--> x*y + e^(-x) + sin(x^2)
sage: derivative(f, x)
(x, y) |--> 2*x*cos(x^2) + y - e^(-x)
sage: derivative(f, y)
(x, y) |--> x

We differentiate a polynomial:

sage: t = polygen(QQ, 't')
sage: f = (1-t)^5; f
-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1
sage: derivative(f)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t, t)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, t, 2)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, 2)
-20*t^3 + 60*t^2 - 60*t + 20

We differentiate a symbolic expression:

sage: var('a x')
(a, x)
sage: f = exp(sin(a - x^2))/x
sage: derivative(f, x)
-2*cos(-x^2 + a)*e^(sin(-x^2 + a)) - e^(sin(-x^2 + a))/x^2
sage: derivative(f, a)
cos(-x^2 + a)*e^(sin(-x^2 + a))/x

Syntax for repeated differentiation:

sage: R.<u, v> = PolynomialRing(QQ)
sage: f = u^4*v^5
sage: derivative(f, u)
4*u^3*v^5
sage: f.derivative(u)   # can always use method notation too
4*u^3*v^5

sage: derivative(f, u, u)
12*u^2*v^5
sage: derivative(f, u, u, u)
24*u*v^5
sage: derivative(f, u, 3)
24*u*v^5

sage: derivative(f, u, v)
20*u^3*v^4
sage: derivative(f, u, 2, v)
60*u^2*v^4
sage: derivative(f, u, v, 2)
80*u^3*v^3
sage: derivative(f, [u, v, v])
80*u^3*v^3

We differentiate a scalar field on a manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field(x^2*y, name='f')
sage: derivative(f)
1-form df on the 2-dimensional differentiable manifold M
sage: derivative(f).display()
df = 2*x*y dx + x^2 dy

We differentiate a differentiable form, getting its exterior derivative:

sage: a = M.one_form(-y, x, name='a'); a.display()
a = -y dx + x dy
sage: derivative(a)
2-form da on the 2-dimensional differentiable manifold M
sage: derivative(a).display()
da = 2 dx∧dy

sage.calculus.functional.diff(f, *args, **kwds)

The derivative of \(f\).

Repeated differentiation is supported by the syntax given in the examples below.

ALIAS: diff

EXAMPLES: We differentiate a callable symbolic function:

sage: f(x,y) = x*y + sin(x^2) + e^(-x)
sage: f
(x, y) |--> x*y + e^(-x) + sin(x^2)
sage: derivative(f, x)
(x, y) |--> 2*x*cos(x^2) + y - e^(-x)
sage: derivative(f, y)
(x, y) |--> x

We differentiate a polynomial:

sage: t = polygen(QQ, 't')
sage: f = (1-t)^5; f
-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1
sage: derivative(f)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t, t)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, t, 2)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, 2)
-20*t^3 + 60*t^2 - 60*t + 20

We differentiate a symbolic expression:

sage: var('a x')
(a, x)
sage: f = exp(sin(a - x^2))/x
sage: derivative(f, x)
-2*cos(-x^2 + a)*e^(sin(-x^2 + a)) - e^(sin(-x^2 + a))/x^2
sage: derivative(f, a)
cos(-x^2 + a)*e^(sin(-x^2 + a))/x

Syntax for repeated differentiation:

sage: R.<u, v> = PolynomialRing(QQ)
sage: f = u^4*v^5
sage: derivative(f, u)
4*u^3*v^5
sage: f.derivative(u)   # can always use method notation too
4*u^3*v^5

sage: derivative(f, u, u)
12*u^2*v^5
sage: derivative(f, u, u, u)
24*u*v^5
sage: derivative(f, u, 3)
24*u*v^5

sage: derivative(f, u, v)
20*u^3*v^4
sage: derivative(f, u, 2, v)
60*u^2*v^4
sage: derivative(f, u, v, 2)
80*u^3*v^3
sage: derivative(f, [u, v, v])
80*u^3*v^3

We differentiate a scalar field on a manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field(x^2*y, name='f')
sage: derivative(f)
1-form df on the 2-dimensional differentiable manifold M
sage: derivative(f).display()
df = 2*x*y dx + x^2 dy

We differentiate a differentiable form, getting its exterior derivative:

sage: a = M.one_form(-y, x, name='a'); a.display()
a = -y dx + x dy
sage: derivative(a)
2-form da on the 2-dimensional differentiable manifold M
sage: derivative(a).display()
da = 2 dx∧dy

sage.calculus.functional.expand(x, *args, **kwds)

EXAMPLES:

sage: a = (x-1)*(x^2 - 1); a
(x^2 - 1)*(x - 1)
sage: expand(a)
x^3 - x^2 - x + 1

You can also use expand on polynomial, integer, and other factorizations:

sage: x = polygen(ZZ)
sage: F = factor(x^12 - 1); F
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + 1) * (x^2 + x + 1) * (x^4 - x^2 + 1)
sage: expand(F)
x^12 - 1
sage: F.expand()
x^12 - 1
sage: F = factor(2007); F
3^2 * 223
sage: expand(F)
2007

Note: If you want to compute the expanded form of a polynomial arithmetic operation quickly and the coefficients of the polynomial all lie in some ring, e.g., the integers, it is vastly faster to create a polynomial ring and do the arithmetic there.

sage: x = polygen(ZZ)      # polynomial over a given base ring.
sage: f = sum(x^n for n in range(5))
sage: f*f                  # much faster, even if the degree is huge
x^8 + 2*x^7 + 3*x^6 + 4*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1

sage.calculus.functional.integral(f, *args, **kwds)

The integral of \(f\).

EXAMPLES:

sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x)^2, x, pi, 123*pi/2)
121/4*pi
sage: integral( sin(x), x, 0, pi)
2

We integrate a symbolic function:

sage: f(x,y,z) = x*y/z + sin(z)
sage: integral(f, z)
(x, y, z) |--> x*y*log(z) - cos(z)

sage: var('a,b')
(a, b)
sage: assume(b-a>0)
sage: integral( sin(x), x, a, b)
cos(a) - cos(b)
sage: forget()

sage: integral(x/(x^3-1), x)
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)

sage: integral( exp(-x^2), x )
1/2*sqrt(pi)*erf(x)

We define the Gaussian, plot and integrate it numerically and symbolically:

sage: f(x) = 1/(sqrt(2*pi)) * e^(-x^2/2)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2)
sage: P.show(ymin=0, ymax=0.4)
sage: numerical_integral(f, -4, 4)                    # random output
(0.99993665751633376, 1.1101527003413533e-14)
sage: integrate(f, x)
x |--> 1/2*erf(1/2*sqrt(2)*x)

You can have Sage calculate multiple integrals. For example, consider the function \(exp(y^2)\) on the region between the lines \(x=y\), \(x=1\), and \(y=0\). We find the value of the integral on this region using the command:

sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area
1/2*e - 1/2
sage: float(area)
0.859140914229522...

We compute the line integral of \(\sin(x)\) along the arc of the curve \(x=y^4\) from \((1,-1)\) to \((1,1)\):

sage: t = var('t')
sage: (x,y) = (t^4,t)
sage: (dx,dy) = (diff(x,t), diff(y,t))
sage: integral(sin(x)*dx, t,-1, 1)
0
sage: restore('x,y')   # restore the symbolic variables x and y

Sage is now (github issue #27958) able to compute the following integral:

sage: integral(exp(-x^2)*log(x), x)  # long time
1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)

and its value:

sage: integral( exp(-x^2)*ln(x), x, 0, oo)
-1/4*sqrt(pi)*(euler_gamma + 2*log(2))

This definite integral is easy:

sage: integral( ln(x)/x, x, 1, 2)
1/2*log(2)^2

Sage cannot do this elliptic integral (yet):

sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
integrate(1/(sqrt(2*t^2 + 1)*sqrt(t^2 - 2)), t, 2, 3)

A double integral:

sage: y = var('y')
sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
32/5

This illustrates using assumptions:

sage: integral(abs(x), x, 0, 5)
25/2
sage: a = var("a")
sage: integral(abs(x), x, 0, a)
1/2*a*abs(a)
sage: integral(abs(x)*x, x, 0, a)
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?
sage: assume(a>0)
sage: integral(abs(x)*x, x, 0, a)
1/3*a^3
sage: forget()      # forget the assumptions.

We integrate and differentiate a huge mess:

sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2
sage: g = integral(f, x)
sage: h = f - diff(g, x)

sage: [float(h(x=i)) for i in range(5)] #random

[0.0,
 -1.1102230246251565e-16,
 -5.5511151231257827e-17,
 -5.5511151231257827e-17,
 -6.9388939039072284e-17]
sage: h.factor()
0
sage: bool(h == 0)
True

sage.calculus.functional.integrate(f, *args, **kwds)

The integral of \(f\).

EXAMPLES:

sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x)^2, x, pi, 123*pi/2)
121/4*pi
sage: integral( sin(x), x, 0, pi)
2

We integrate a symbolic function:

sage: f(x,y,z) = x*y/z + sin(z)
sage: integral(f, z)
(x, y, z) |--> x*y*log(z) - cos(z)

sage: var('a,b')
(a, b)
sage: assume(b-a>0)
sage: integral( sin(x), x, a, b)
cos(a) - cos(b)
sage: forget()

sage: integral(x/(x^3-1), x)
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)

sage: integral( exp(-x^2), x )
1/2*sqrt(pi)*erf(x)

We define the Gaussian, plot and integrate it numerically and symbolically:

sage: f(x) = 1/(sqrt(2*pi)) * e^(-x^2/2)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2)
sage: P.show(ymin=0, ymax=0.4)
sage: numerical_integral(f, -4, 4)                    # random output
(0.99993665751633376, 1.1101527003413533e-14)
sage: integrate(f, x)
x |--> 1/2*erf(1/2*sqrt(2)*x)

You can have Sage calculate multiple integrals. For example, consider the function \(exp(y^2)\) on the region between the lines \(x=y\), \(x=1\), and \(y=0\). We find the value of the integral on this region using the command:

sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area
1/2*e - 1/2
sage: float(area)
0.859140914229522...

We compute the line integral of \(\sin(x)\) along the arc of the curve \(x=y^4\) from \((1,-1)\) to \((1,1)\):

sage: t = var('t')
sage: (x,y) = (t^4,t)
sage: (dx,dy) = (diff(x,t), diff(y,t))
sage: integral(sin(x)*dx, t,-1, 1)
0
sage: restore('x,y')   # restore the symbolic variables x and y

Sage is now (github issue #27958) able to compute the following integral:

sage: integral(exp(-x^2)*log(x), x)  # long time
1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)

and its value:

sage: integral( exp(-x^2)*ln(x), x, 0, oo)
-1/4*sqrt(pi)*(euler_gamma + 2*log(2))

This definite integral is easy:

sage: integral( ln(x)/x, x, 1, 2)
1/2*log(2)^2

Sage cannot do this elliptic integral (yet):

sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
integrate(1/(sqrt(2*t^2 + 1)*sqrt(t^2 - 2)), t, 2, 3)

A double integral:

sage: y = var('y')
sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
32/5

This illustrates using assumptions:

sage: integral(abs(x), x, 0, 5)
25/2
sage: a = var("a")
sage: integral(abs(x), x, 0, a)
1/2*a*abs(a)
sage: integral(abs(x)*x, x, 0, a)
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?
sage: assume(a>0)
sage: integral(abs(x)*x, x, 0, a)
1/3*a^3
sage: forget()      # forget the assumptions.

We integrate and differentiate a huge mess:

sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2
sage: g = integral(f, x)
sage: h = f - diff(g, x)

sage: [float(h(x=i)) for i in range(5)] #random

[0.0,
 -1.1102230246251565e-16,
 -5.5511151231257827e-17,
 -5.5511151231257827e-17,
 -6.9388939039072284e-17]
sage: h.factor()
0
sage: bool(h == 0)
True

sage.calculus.functional.lim(f, dir=None, taylor=False, **argv)

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

limit(expr, x = a)
limit(expr, x = a, dir='above')

INPUT:

• dir - (default: None); dir may have the value

  ‘plus’ (or ‘above’) for a limit from above, ‘minus’ (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

ALIAS: You can also use lim instead of limit.

EXAMPLES:

sage: limit(sin(x)/x, x=0)
1
sage: limit(exp(x), x=oo)
+Infinity
sage: lim(exp(x), x=-oo)
0
sage: lim(1/x, x=0)
Infinity
sage: limit(sqrt(x^2+x+1)+x, taylor=True, x=-oo)
-1/2
sage: limit((tan(sin(x)) - sin(tan(x)))/x^7, taylor=True, x=0)
1/30

Sage does not know how to do this limit (which is 0), so it returns it unevaluated:

sage: lim(exp(x^2)*(1-erf(x)), x=infinity)
-limit((erf(x) - 1)*e^(x^2), x, +Infinity)

sage.calculus.functional.limit(f, dir=None, taylor=False, **argv)

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

limit(expr, x = a)
limit(expr, x = a, dir='above')

INPUT:

• dir - (default: None); dir may have the value

  ‘plus’ (or ‘above’) for a limit from above, ‘minus’ (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

ALIAS: You can also use lim instead of limit.

EXAMPLES:

sage: limit(sin(x)/x, x=0)
1
sage: limit(exp(x), x=oo)
+Infinity
sage: lim(exp(x), x=-oo)
0
sage: lim(1/x, x=0)
Infinity
sage: limit(sqrt(x^2+x+1)+x, taylor=True, x=-oo)
-1/2
sage: limit((tan(sin(x)) - sin(tan(x)))/x^7, taylor=True, x=0)
1/30

Sage does not know how to do this limit (which is 0), so it returns it unevaluated:

sage: lim(exp(x^2)*(1-erf(x)), x=infinity)
-limit((erf(x) - 1)*e^(x^2), x, +Infinity)

sage.calculus.functional.simplify(f, algorithm='maxima', **kwds)

Simplify the expression \(f\).

See the documentation of the simplify() method of symbolic expressions for details on options.

EXAMPLES:

We simplify the expression \(i + x - x\):

sage: f = I + x - x; simplify(f)
I

In fact, printing \(f\) yields the same thing - i.e., the simplified form.

Some simplifications are algorithm-specific:

sage: x, t = var("x, t")
sage: ex = 1/2*I*x + 1/2*I*sqrt(x^2 - 1) + 1/2/(I*x + I*sqrt(x^2 - 1))
sage: simplify(ex)
1/2*I*x + 1/2*I*sqrt(x^2 - 1) + 1/(2*I*x + 2*I*sqrt(x^2 - 1))
sage: simplify(ex, algorithm="giac")
I*sqrt(x^2 - 1)

sage.calculus.functional.taylor(f, *args)

Expands self 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 are also supported.

INPUT:

• *args - the following notation is supported

• x, a, n - variable, point, degree

• (x, a), (y, b), ..., n - variables with points, degree of polynomial

EXAMPLES:

sage: var('x,k,n')
(x, k, n)
sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
-1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1

sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1

sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1

Taylor polynomial in two variables:

sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
(x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3