[Python] One frequency is absent from fourier transform representation of the addition of two...

I have wrote a general python code in Sage-math to find out the Fourier transform of the addition of two different frequency cosine wave in order to plot the frequency domain in a graph. The function :

f(t) = cos(2*pi*f1*t)+cos(2*pi*f2*t) # integrate it from -infinity to +infinity


To find the integral, I have used Sympy pacakge. Below is a sample of the resultant piecewise functions I am getting after performing integration of the Fourier transform frequency function:



Then in order to check which frequencies are present in the resultant function, I search for existing frequency between -10 to +10 on the Fourier transform function .

Here f1 and f2 are respective frequency of the two cosine wave. When f1 = 8 and f2 = 4, I get the following graph ( both frequency (8 & 4) is present in the graph as expected):



But if I change any one of f1 or f2 to 7, Then only 7 is present on the graph. Here you can see that frequency 4 is absent from the graph. I have found that this issue only appears if frequency 7 is present in any one of the two cosine functions.



Here is the code:

from sage.all import *
import numpy as np
import sympy as sy
from sympy import lambdify,re, im
import matplotlib.pyplot as plt

t = sy.Symbol('t')
f = sy.Symbol('f')

# Find Fourier Transform
F = sy.integrate((cos(2*pi*7*t)+cos(2*pi*4*t))*(e^(-i*2*pi*f*t)), (t,-oo,oo))

func = F.args[0][0] # extract 1st function from piecewise F

#lambdify to evaluate function at many points
sympy_lambda = lambdify([f],func)

freq_arr = []
amplitude_arr = []

# here l (L) is frequency in x-coordinate, testing for frequency -10 to 10
for l in np.arange(-10, 10, 0.01):
sympy_complex = sympy_lambda(l) # get sympy complex number
# get real part of sympy complex number
real_part = re(sympy_complex)

# get imaginary part of sympy complex number
imag_part = im(sympy_complex)

# make a sage complex number using from sympy real and imaginary
sage_complex = ComplexNumber(real_part, imag_part)

freq_arr.append(l)
amplitude_arr.append(abs(sage_complex))

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