Analyzing real-valued functions

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')

   

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

2. Let's evaluate this function at 1:

   >>> f.subs(x, 1)

   
3. We can compute the derivative of this function:

   >>> diff(f, x)

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

   >>> limit(f, x, oo)

   
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)

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

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

   
7. SymPy can also compute...