Chapter 5 Fourier Series
The Fourier series is a mathematical tool useful in representing a periodic function as a sum of simple sine and cosine functions (harmonics). Any periodic function with a well-defined period T can be expressed as an infinite sum of sine and cosine functions with different frequencies. The frequencies are integer multiples of the fundamental frequency, which is the reciprocal of the period (1/T). A note from various musical instruments or other sources are the complicated periodic function with the period of pure tone. This complicated function may be expressed as a series of sines and cosines in the form of
\begin{equation*}
f(x)= \frac{a_{o}}{2}+a_{1}\cos x + a_{2}\cos 2x + \cdots + a_{n}\cos nx + \cdots
\end{equation*}
\begin{equation*}
+ b_{1}\sin x +b_{2}\sin 2x + \cdots + b_{n}\sin nx + \cdots.
\end{equation*}
\begin{equation*}
\therefore \quad f(x) = \frac{a_{o}}{2} + \sum\limits_{n=1}^{\infty}\left[a_{n}\cos nx + b_{n}\sin nx\right].
\end{equation*}
This infinite series is called Fourier series where \(a_{o} {,}a_{1} {,} a_{2}
\cdots{,} a_{n} {,}\cdots {,} b_{1} {,} b_{2} {,} \cdots{,} b_{n}{,} \cdots, \) etc. are called Fourier coefficients.
The Fourier series has numerous applications in various fields, including signal processing, image compression, physics, engineering, and many areas of applied mathematics. It provides a powerful tool for decomposing complex periodic functions into simpler components and understanding their behavior.
for python code visit
https://docs.sympy.org/latest/modules/series/fourier.html
