Fourier Series Formulas
For any function f(x) with period 2L, the formula of Fourier Series is given as,
[Tex]f(x)~=~\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] .
where,
a0 = [Tex]\frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)dx [/Tex]
an = [Tex] \frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)cos~nx ~dx [/Tex]
bn = [Tex] \frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)sin~nx ~dx [/Tex]
Coefficient of Fourier Series
In the above formula of Fourier Series, the terms a0, an and bn are called coefficient of fourier series. The value of these coefficients defines the fourier series of a given periodic function. The value of a0 represents the average value of the function and an and bn represent the amplitude of the sinusoidal functions.
Fourier Series Formula
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.
Table of Content
- What is Fourier Series?
- Fourier Series Formulas
- Exponential form of Fourier Series
- Conditions for Fourier series
- Applications of Fourier Series
- Solved Examples