What is Fourier Series?

Fourier Series is the expansion of a periodic function in terms of the infinite sum of sines and cosines. Periodic functions often appear in problems in higher mathematics. A way of dealing with these issues is to represent them in terms of basic periodic functions, which have a small range and can have a domain of all real numbers, such as sine and cosine; this leads us to the Fourier series (FS). The Fourier series is a particularly useful tool for dealing with situations involving partial differential equations. 

Suppose we are given a periodic function f(x). Now as the original  function is periodic therefore,

c₁​f₁​(x) + … + c_n​f_n​(x)​

Next consider the infinite series,

[Tex]\begin{array}{l}\frac{1}{2} a_{o}+ \sum_{ n=1}^{\infty}a_{n}\;cos\frac{n\pi x}{L}+b_{n}\; sin\frac{n\pi x}{L}\end{array} [/Tex] ⇢ (1)

Consisting of 2L-periodic functions converges for all x, then the function to which it converges will be periodic of period 2L. Now as seen above we need to represent the function f(x) in such a way that the periodic function f(x) is replaced by functions like sine and cosine. For this the Fourier series is given by,

[Tex]\large f(x)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}a_{n}cos\;nx+\sum_{n=1}^{\infty}b_{n}sin\;nx [/Tex] 

Here,

[Tex]\frac{1}{\pi} \int_{- \pi}^{\pi} f(x) dx [/Tex] 

[Tex]\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)cos\;nx\;dx [/Tex] .

[Tex]\frac{1}{\pi} \int_{-\pi}^{\pi}f(x)sin\;nx\;dx [/Tex] .

n = 1,2,3….
   

Fourier Series is a sum of sine and cosine waves that represents a periodic function. Each wave in the sum, or harmonic, has a frequency that is an integral multiple of the periodic function’s fundamental frequency. Harmonic analysis may be used to identify the phase and amplitude of each harmonic. A Fourier series might have an unlimited number of harmonics. Summing some, but not all, of the harmonics in a function’s Fourier series, yields an approximation to that function. For example, a square wave can be approximated by utilizing the first few harmonics of the Fourier series.

In this article, we will learn about Fourier Series, Fourier Series Formula, Fourier Series Examples, and others in detail.