15.3. Analyzing real-valued functions

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SymPy contains a rich calculus toolbox to analyze real-valued functions: limits, power series, derivatives, integrals, Fourier transforms, and so on. In this recipe, we will show the very basics of these capabilities.

How to do it...

1.  Let's define a few symbols and a function (which is just an expression depending on x):

from sympy import *
init_printing()

var('x z')

(x, z)

f = 1 / (1 + x**2)

2.  Let's evaluate this function at 1:

f.subs(x, 1)

1/2

3.  We can compute the derivative of this function:

diff(f, x)

Output

4.  What is \(f\)'s limit to infinity? (Note the double o (oo) for the infinity symbol):

limit(f, x, oo)

0

5.  Here's how to compute a Taylor series (here, around 0, of order 9). The Big O can be removed with the removeO() method.

series(f, x0=0, n=9)

Output

6.  We can compute definite integrals (here, over the entire real line):

integrate(f, (x, -oo, oo))

Pi

7.  SymPy can also compute indefinite integrals:

integrate(f, x)

atan(x)

8.  Finally, let's compute \(f\)'s Fourier transforms:

fourier_transform(f, x, z)

Output

There's more...

SymPy includes a large number of other integral transforms besides the Fourier transform (http://docs.sympy.org/latest/modules/integrals/integrals.html). However, SymPy will not always be able to find closed-form solutions.

Here are a few general references about real analysis and calculus: