Calculus

Here are some examples of calculus symbolic computations using Sage.

Differentiation

Differentiation:

sage: var('x k w')
(x, k, w)
sage: f = x^3 * e^(k*x) * sin(w*x); f
x^3*e^(k*x)*sin(w*x)
sage: f.diff(x)
w*x^3*cos(w*x)*e^(k*x) + k*x^3*e^(k*x)*sin(w*x) + 3*x^2*e^(k*x)*sin(w*x)
sage: latex(f.diff(x))
w x^{3} \cos\left(w x\right) e^{\left(k x\right)} + k x^{3} e^{\left(k x\right)} \sin\left(w x\right) + 3 \, x^{2} e^{\left(k x\right)} \sin\left(w x\right)

If you type view(f.diff(x)) another window will open up displaying the compiled output. In the notebook, you can enter

var('x k w')
f = x^3 * e^(k*x) * sin(w*x)
show(f)
show(f.diff(x))

into a cell and press shift-enter for a similar result. You can also differentiate and integrate using the commands

R = PolynomialRing(QQ,"x")
x = R.gen()
p = x^2 + 1
show(p.derivative())
show(p.integral())

in a notebook cell, or

sage: R = PolynomialRing(QQ,"x")
sage: x = R.gen()
sage: p = x^2 + 1
sage: p.derivative()
2*x
sage: p.integral()
1/3*x^3 + x

on the command line. At this point you can also type view(p.derivative()) or view(p.integral()) to open a new window with output typeset by LaTeX.

Critical points

You can find critical points of a piecewise defined function:

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f1 = x^0
sage: f2 = 1-x
sage: f3 = 2*x
sage: f4 = 10*x-x^2
sage: f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)])
sage: f.critical_points()
[5.0]

Power series

Sage offers several ways to construct and work with power series.

To get Taylor series from function expressions use the method .taylor() on the expression:

sage: var('f0 k x')
(f0, k, x)
sage: g = f0/sinh(k*x)^4
sage: g.taylor(x, 0, 3)
-62/945*f0*k^2*x^2 + 11/45*f0 - 2/3*f0/(k^2*x^2) + f0/(k^4*x^4)

Formal power series expansions of functions can be had with the .series() method:

sage: (1/(2-cos(x))).series(x,7)
1 + (-1/2)*x^2 + 7/24*x^4 + (-121/720)*x^6 + Order(x^7)

Certain manipulations on such series are hard to perform at the moment, however. There are two alternatives: either use the Maxima subsystem of Sage for full symbolic functionality:

sage: f = log(sin(x)/x)
sage: f.taylor(x, 0, 10)
-1/467775*x^10 - 1/37800*x^8 - 1/2835*x^6 - 1/180*x^4 - 1/6*x^2
sage: maxima(f).powerseries(x,0)._sage_()
sum(2^(2*i... - 1)*(-1)^i...*x^(2*i...)*bern(2*i...)/(i...*factorial(2*i...)), i..., 1, +Infinity)

Or you can use the formal power series rings for fast computation. These are missing symbolic functions, on the other hand:

sage: R.<w> = QQ[[]]
sage: ps = w + 17/2*w^2 + 15/4*w^4 + O(w^6); ps
w + 17/2*w^2 + 15/4*w^4 + O(w^6)
sage: ps.exp()
1 + w + 9*w^2 + 26/3*w^3 + 265/6*w^4 + 413/10*w^5 + O(w^6)
sage: (1+ps).log()
w + 8*w^2 - 49/6*w^3 - 193/8*w^4 + 301/5*w^5 + O(w^6)
sage: (ps^1000).coefficients()
[1, 8500, 36088875, 102047312625, 1729600092867375/8]

Integration

Numerical integration is discussed in Riemann and trapezoid sums for integrals below.

Sage can integrate some simple functions on its own:

sage: f = x^3
sage: f.integral(x)
1/4*x^4
sage: integral(x^3,x)
1/4*x^4
sage: f = x*sin(x^2)
sage: integral(f,x)
-1/2*cos(x^2)

Sage can also compute symbolic definite integrals involving limits.

sage: var('x, k, w')
(x, k, w)
sage: f = x^3 * e^(k*x) * sin(w*x)
sage: f.integrate(x)
((24*k^3*w - 24*k*w^3 - (k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 + 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 - 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*cos(w*x)*e^(k*x) - (6*k^4 - 36*k^2*w^2 + 6*w^4 - (k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 + 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 - 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)
sage: integrate(1/x^2, x, 1, infinity)
1

Convolution

You can find the convolution of any piecewise defined function with another (off the domain of definition, they are assumed to be zero). Here is \(f\), \(f*f\), and \(f*f*f\), where \(f(x)=1\), \(0<x<1\):

sage: x = PolynomialRing(QQ, 'x').gen()
sage: f = piecewise([((0,1),1*x^0)])
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: set_verbose(-1)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))

To view this, type show(P+Q+R).

Riemann and trapezoid sums for integrals

Regarding numerical approximation of \(\int_a^bf(x)\, dx\), where \(f\) is a piecewise defined function, can

  • compute (for plotting purposes) the piecewise linear function defined by the trapezoid rule for numerical integration based on a subdivision into \(N\) subintervals
  • the approximation given by the trapezoid rule,
  • compute (for plotting purposes) the piecewise constant function defined by the Riemann sums (left-hand, right-hand, or midpoint) in numerical integration based on a subdivision into \(N\) subintervals,
  • the approximation given by the Riemann sum approximation.
sage: f1(x) = x^2
sage: f2(x) = 5-x^2
sage: f = piecewise([[[0,1], f1], [RealSet.open_closed(1,2), f2]])
sage: t = f.trapezoid(2); t
piecewise(x|-->1/2*x on (0, 1/2), x|-->3/2*x - 1/2 on (1/2, 1), x|-->7/2*x - 5/2 on (1, 3/2), x|-->-7/2*x + 8 on (3/2, 2); x)
sage: t.integral()
piecewise(x|-->1/4*x^2 on (0, 1/2), x|-->3/4*x^2 - 1/2*x + 1/8 on (1/2, 1), x|-->7/4*x^2 - 5/2*x + 9/8 on (1, 3/2), x|-->-7/4*x^2 + 8*x - 27/4 on (3/2, 2); x)
sage: t.integral(definite=True)
9/4

Laplace transforms

If you have a piecewise-defined polynomial function then there is a “native” command for computing Laplace transforms. This calls Maxima but it’s worth noting that Maxima cannot handle (using the direct interface illustrated in the last few examples) this type of computation.

sage: var('x s')
(x, s)
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f = piecewise([((0,1),f1), ((1,2),f2)])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2

For other “reasonable” functions, Laplace transforms can be computed using the Maxima interface:

sage: var('k, s, t')
(k, s, t)
sage: f = 1/exp(k*t)
sage: f.laplace(t,s)
1/(k + s)

is one way to compute LT’s and

sage: var('s, t')
(s, t)
sage: f = t^5*exp(t)*sin(t)
sage: L = laplace(f, t, s); L
3840*(s - 1)^5/(s^2 - 2*s + 2)^6 - 3840*(s - 1)^3/(s^2 - 2*s + 2)^5 +
720*(s - 1)/(s^2 - 2*s + 2)^4

is another way.

Ordinary differential equations

Symbolically solving ODEs can be done using Sage interface with Maxima. See

sage:desolvers?

for available commands. Numerical solution of ODEs can be done using Sage interface with Octave (an experimental package), or routines in the GSL (Gnu Scientific Library).

An example, how to solve ODE’s symbolically in Sage using the Maxima interface (do not type the ....:):

sage: y=function('y')(x); desolve(diff(y,x,2) + 3*x == y, dvar = y, ics = [1,1,1])
3*x - 2*e^(x - 1)
sage: desolve(diff(y,x,2) + 3*x == y, dvar = y)
_K2*e^(-x) + _K1*e^x + 3*x
sage: desolve(diff(y,x) + 3*x == y, dvar = y)
(3*(x + 1)*e^(-x) + _C)*e^x
sage: desolve(diff(y,x) + 3*x == y, dvar = y, ics = [1,1]).expand()
3*x - 5*e^(x - 1) + 3

sage: f=function('f')(x); desolve_laplace(diff(f,x,2) == 2*diff(f,x)-f, dvar = f, ics = [0,1,2])
x*e^x + e^x

sage: desolve_laplace(diff(f,x,2) == 2*diff(f,x)-f, dvar = f)
-x*e^x*f(0) + x*e^x*D[0](f)(0) + e^x*f(0)

If you have Octave and gnuplot installed,

sage: octave.de_system_plot(['x+y','x-y'], [1,-1], [0,2]) # optional - octave

yields the two plots \((t,x(t)), (t,y(t))\) on the same graph (the \(t\)-axis is the horizontal axis) of the system of ODEs

\[x' = x+y, x(0) = 1; y' = x-y, y(0) = -1,\]

for \(0 <= t <= 2\). The same result can be obtained by using desolve_system_rk4:

sage: x, y, t = var('x y t')
sage: P=desolve_system_rk4([x+y, x-y], [x,y], ics=[0,1,-1], ivar=t, end_points=2)
sage: p1 = list_plot([[i,j] for i,j,k in P], plotjoined=True)
sage: p2 = list_plot([[i,k] for i,j,k in P], plotjoined=True, color='red')
sage: p1+p2
Graphics object consisting of 2 graphics primitives

Another way this system can be solved is to use the command desolve_system.

sage: t=var('t'); x=function('x',t); y=function('y',t)
sage: des = [diff(x,t) == x+y, diff(y,t) == x-y]
sage: desolve_system(des, [x,y], ics = [0, 1, -1])
[x(t) == cosh(sqrt(2)*t), y(t) == sqrt(2)*sinh(sqrt(2)*t) - cosh(sqrt(2)*t)]

The output of this command is not a pair of functions.

Finally, can solve linear DEs using power series:

sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
sage: a = 2 - 3*t + 4*t^2 + O(t^10)
sage: b = 3 - 4*t^2 + O(t^7)
sage: f = a.solve_linear_de(prec=5, b=b, f0=3/5)
sage: f
3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
sage: f.derivative() - a*f - b
O(t^4)

Fourier series of periodic functions

Let \(f\) be a real-valued periodic function of period \(2L\). The Fourier series of \(f\) is

\[S(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n\cos\left(\frac{n\pi x}{L}\right) + b_n\sin\left(\frac{n\pi x}{L}\right)\right]\]

where

\[a_n = \frac{1}{L}\int_{-L}^L f(x)\cos\left(\frac{n\pi x}{L}\right) dx,\]

and

\[b_n = \frac{1}{L}\int_{-L}^L f(x)\sin\left(\frac{n\pi x}{L}\right) dx,\]

The Fourier coefficients \(a_n\) and \(b_n\) are computed by declaring \(f\) as a piecewise-defined function over one period and invoking the methods fourier_series_cosine_coefficient and fourier_series_sine_coefficient, while the partial sums are obtained via fourier_series_partial_sum:

sage: f = piecewise([((0,pi/2), -1), ((pi/2,pi), 2)])
sage: f.fourier_series_cosine_coefficient(0)
1
sage: f.fourier_series_sine_coefficient(5)
-6/5/pi
sage: s5 = f.fourier_series_partial_sum(5); s5
-6/5*sin(10*x)/pi - 2*sin(6*x)/pi - 6*sin(2*x)/pi + 1/2
sage: plot(f, (0,pi)) + plot(s5, (x,0,pi), color='red')
Graphics object consisting of 2 graphics primitives
_images/calculus-1.png