Exponential form of Fourier Series
From the equation above,
[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] .
Now according to Euler’s formula,
eiθ= cosθ +isinθ
Using this
f(x) = [Tex]\sum_{n=-\infty}^{\infty} [/Tex] Cneinx.
Here Cn is called decomposition coefficient and is calculated as,
[Tex]C_n = \frac{1}{2T} \int_{-T}^{T}e^{{-in}\frac{\pi t}{T} }f(t) [/Tex] .
Check: Sine Function
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
Contact Us