Plotting Fourier Series coefficients in Python using Simpson's Rule

Question

I want to 1. express Simpson's Rule as a general function for integration in python and 2. use it to compute and plot the Fourier Series coefficients of the function .

I've stolen and adapted this code for Simpson's Rule, which seems to work fine for integrating simple functions such as , or

Given period , the Fourier Series coefficients are computed as:

where k = 1,2,3,...

I am having difficulty figuring out how to express . I'm aware that since this function is odd, but I would like to be able to compute it in general for other functions.

Here's my attempt so far:

import matplotlib.pyplot as plt
from numpy import *

def f(t):
    k = 1
    for k in range(1,10000): #to give some representation of k's span
        k += 1
    return sin(t)*sin(k*t)

def trapezoid(f, a, b, n):
    h = float(b - a) / n
    s = 0.0
    s += f(a)/2.0
    for j in range(1, n):
        s += f(a + j*h)
    s += f(b)/2.0
    return s * h

print trapezoid(f, 0, 2*pi, 100)

This doesn't give the correct answer of 0 at all since it increases as k increases and I'm sure I'm approaching it with tunnel vision in terms of the for loop. My difficulty in particular is with stating the function so that k is read as k = 1,2,3,...

The problem I've been given unfortunately doesn't specify what the coefficients are to be plotted against, but I am assuming it's meant to be against k.

Mateen Ulhaq · Accepted Answer · 2018-04-21T03:44:37.290

Here's one way to do it, if you want to run your own integration or fourier coefficient determination instead of using numpy or scipy's built in methods:

import numpy as np

def integrate(f, a, b, n):
    t = np.linspace(a, b, n)
    return (b - a) * np.sum(f(t)) / n

def a_k(f, k):
    def ker(t): return f(t) * np.cos(k * t)
    return integrate(ker, 0, 2*np.pi, 2**10+1) / np.pi

def b_k(f, k):
    def ker(t): return f(t) * np.sin(k * t)
    return integrate(ker, 0, 2*np.pi, 2**10+1) / np.pi

print(b_k(np.sin, 0))

This gives the result

0.0

On a side note, trapezoid integration is not very useful for uniform time intervals. But if you desire:

def trap_integrate(f, a, b, n):
    t = np.linspace(a, b, n)
    f_t = f(t)
    dt = t[1:] - t[:-1]
    f_ab = f_t[:-1] + f_t[1:]
    return 0.5 * np.sum(dt * f_ab)

There's also np.trapz if you want to use pre-builtin functionality. Similarly, there's also scipy.integrate.trapz

Plotting Fourier Series coefficients in Python using Simpson's Rule

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