Fourier series

Once again we will by using the sympy module.

Compute fourier expansion

sympy.fourier_series(f, (x, a, b)) can be used to calculate the fourier series of the function f for x ranging from a to b.

For example, $f\left(x\right)=x^3, \ \ for\ \left[ -1,1\right]$ for n=5 can be computed with:

from sympy import fourier_series, pi
from sympy.abc import x
f = x**3
s = fourier_series(f, (x, -1, 1))
s1 = s.truncate(n=5) # only return first 5 nonzero terms of the series

Plot fourier series

Sympy can also be used to plot a fourier series, here is a step-by-step guide on how to do this.

1. Import module

    from sympy import fourier_series, pi, plot
    from sympy.abc import x
   

2. Define function

   Here we create the function f(x)=x

    f = x
   

3. Calculate fourier series

   We can now calculate the fourier series for x between -pi and pi

    s = fourier_series(f, (x, -pi, pi))
   

4. Calculate approximations

   Let’s calculate the 3th, 5th and 7th term approximations

    s1 = s.truncate(n = 3)
    s2 = s.truncate(n = 5)
    s3 = s.truncate(n = 7)
   

5. Plot the graphs

    p = plot(f, s1, s2, s3, (x, -pi, pi), show=False, legend=True)
   
    p[0].line_color = (0, 0, 0)
    p[0].label = 'x'
    p[1].line_color = (0.7, 0.7, 0.7)
    p[1].label = 'n=3'
    p[2].line_color = (0.5, 0.5, 0.5)
    p[2].label = 'n=5'
    p[3].line_color = (0.3, 0.3, 0.3)
    p[3].label = 'n=7'
   
    p.show()
   

Try it out

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