Fourier Series

Define Fourier Series

A Fourier series is a series that is used to expand a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier Series uses orthogonal relation of sine and cosine functions.

What is Fourier Series Formula?

Fourier series formula of any function f(x) in the interval [-L, L] is,

f(x) = A₀ + ∑_{{n = 1}}^{∞} A_n cos (nπx/L) + ∑_{{n = 1}}^{∞} B_n sin (nπx/L)

What is Application of Fourier Series?

Fourier series has various applications and some of its important application are,

• It is use to find the expansion of any periodic function.
• It is used in graph modelling.
• It is used to in the study of various curves.

How is the Fourier Series Formula Derived?

The Fourier series formula is derived by decomposing a periodic function into a series of sine and cosine functions. It involves calculating Fourier coefficients, which are determined by integrating the product of the original function and the sine or cosine functions over one period of the function.

What are Types of Fourier Series Formula?

There are two types of Fourier series formulas they are

• Trigonometric Series Formula
• Exponential Series Formula

Can Fourier Series Represent Any Function?

Fourier series can represent any piecewise smooth periodic function. While not every function can be represented by a Fourier series, many practical functions encountered in engineering and physics can be, especially if they are periodic.

What are the Properties of Fourier Series?

The different properties of Fourier Series are Linearity, time shifting, Frequency Shifting, Time Scaling, Time Inversion, Differentiation in Time, Integration, Convolution, Multiplication in Time Domain and Symmetry.

What is Fourier Series of Sine?

The Fourier Sine Series is given as f(x) = [Tex]\sum_{n = 1}^{\infty}b_{n}sin\frac{n\pi x}{L}[/Tex]

What is Fourier Series of Cosine?

Fourier Cosine Series is given as [Tex]\frac{a_{0}}{2}+\sum_{n = 1}^{\infty}a_{n}cos\frac{n\pi x}{L}[/Tex]

What is the Difference between Fourier Series and Fourier Transform?

The difference between Fourier Series and Fourier Transformation is that fourier series expands a periodic function in the form of infinite sum of sine and cosine while fourier transform is used to convert signals from time domain to frequency domain. Another difference between them is that fourier series is applicable to periodic function only and the fourier transform can be applied to the aperiodic function as well.

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.