Open In App

Fourier Series Formula

Improve
Improve
Like Article
Like
Save
Share
Report

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.

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) + … + cn​fn​(x)​

Next consider the infinite series,

\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}      ʉࢠ(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,

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

Here,

\frac{1}{\pi} \int_{- \pi}^{\pi} f(x) dx       

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

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

n = 1,2,3….
   

Fourier Series Formulas

For any function f(x) with period 2L, the formula of Fourier Series is given as,

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}    .

where,

a0 = \frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)dx

an =  \frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)cos~nx ~dx

bn =  \frac{1}{\pi }\int_{-\Pi }^{\Pi }f(x)sin~nx ~dx

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.

Exponential form of Fourier Series

From the equation above,

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}         .

Now according to Euler’s formula,

eiθ= cosθ +isinθ

Using this

f(x) = \sum_{n=-\infty}^{\infty}         Cneinx.

Here Cn is called decomposition coefficient and is calculated as,

C_n = \frac{1}{2T} \int_{-T}^{T}e^{{-in}\frac{\pi t}{T} }f(t)       .

Conditions for Fourier series

Suppose a function f(x) has a period of 2π and is integrable in a period [-π, π]. Now there are two conditions.

The function f(x) with period 2π is absolutely integrable on [-π, π] so that the following Dirichlet integral of this function is finite:
\int\limits_{ - \pi }^\pi  {\left| {f\left( x \right)} \right|dx}  \lt \infty ;

Next condition is that the function is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).

If conditions 1 and 2 are satisfied, the Fourier series for the function exists and converges to the given function. This means that the sum of the Fourier series of any given function converges back to give the same function. This is the basic definition of the Fourier series expansion. Before further understanding the concept of the Fourier Series we should first understand the concept of odd and even functions and periodic functions.

  • Odd function: Suppose we are given a function y = f(x).   

f(-x) = -f(x) = -y

then the function is said to be odd.

Odd Function Graph for x/(x^2 - 1)

The function is odd in nature and symmetric about the origin

  • Even function: Again consider a function f(x) = y.

If f(-x) = f(x) = y

Then the function is even in nature.

Even Function Graph for x^4

This is an even function whose graph that is symmetric along the y-axis.

  • Periodic functions: Let a function f(x) be periodic with an interval λ. Now consider an element x as a part of the domain of this function. This means that,

f(x) = f(x + λ)

Periodic Function Graph

Graph of function tan x is an example of a periodic function.

Hence periodic functions are those functions that repeat themselves over an interval of values(λ as shown above). The smallest possible positive value of λ is called the period of this function.

Applications of Fourier Series

Fourier Series has many applications in mathematical analysis it is one of the most important series that is used to find the expansion of the periodic function in a closed interval. Some of its application are,

  • Fourier Series is used to solved various functions and find its integral and differential.
  • Fourier Series is used in 3-D Graph Modelling
  • Fourier series is used to draw graph of various functions.
  • Fourier series is used in study of Complex function in Statistics, Astronomy, Biology and others, etc.

Also Check

Fourier Series Solved Examples

Example 1: Find the Fourier series expansion of the function f(x) = ex, within the limits  [– Ï€, Ï€].

Solution:

Using Fourier series expansion.

a_0 =  \dfrac{1}{2\pi}\int_{- \pi}^{\pi}e^x dx \\= \dfrac{e^\pi - e^{-\pi}}{2\pi}       .

a_n = \dfrac{1}{\pi}\int_{- \pi}^{\pi}e^x cos (nx) dx \\= \dfrac{1}{\pi}\dfrac{e^x}{1+n^2}[\cos(nx)+ n\sin(nx)]_{-\pi}^{\pi}\\= \dfrac{1}{\pi(1+n^2)}[e^\pi(-1)^n)-e^{-\pi}(-1)^n)]       .

b_n = \dfrac{1}{\pi}\int e^x sin(nx) dx \\= \dfrac{e^x}{\pi (1+n^2)}[\sin(nx)- n \cos(nx)]_{-\pi}^{\pi}\\= \dfrac{1}{\pi (1+n^2)}[e^\pi(-n(-1)^n) - e^{-\pi}(-n)(-1)^n]       .

The Fourier series for this function is given as, 

\dfrac{e^\pi -e^{-\pi}}{2\pi} + \sum_{n=1}^{\infty}\dfrac{(-1)^n(e^\pi - e^{-\pi})}{\pi(1+n^2)}[cos nx -n sin nx]       .

Example 2: Find the Fourier series expansion of the function f(x) = x , within the limits  [– 1, 1].

Solution:

From Fourier series expansion. Here,

A_{0}=\frac{1}{2 } \cdot \int_{-1}^{1} x d x       .

A_{n}=\frac{1}{1} \cdot \int_{-1}^{1} x \cos \left(\frac{n \pi x}{1}\right) d x, \quad n>0       .

B_{n}=\frac{1}{1} \cdot \int_{-1}^{1} x \sin \left(\frac{n \pi x}{1}\right) d x, \quad n>0       .

f(x)=A_{0}+\sum_{n=1}^{\infty} A_{n} \cdot \cos \left(\frac{n \pi x}{L}\right)+\sum_{n=1}^{\infty} B_{n} \cdot \sin \left(\frac{n \pi x}{L}\right)       .

\mathrm{f}(\mathrm{x})=\frac{1}{2 \cdot 1} \cdot \int_{-1}^{1}\left(x\right) d x+\sum_{n=1}^{\infty} \frac{1}{1} \cdot \int_{-1}^{1}\left(x\right) \cos \left(\frac{n \pi x}{1}\right) d x \cdot \cos \left(\frac{n \pi x}{1}\right)+\sum_{n=1}^{\infty} \frac{1}{1} \cdot \int_{-1}^{1}\left(x\right) \sin \left(\frac{n \pi x}{1}\right) d x \cdot \sin \left(\frac{n \pi x}{1}\right)       .

Om solving the integrals we get even functions and one odd function. Therefore,

f(x) = \sum _{n=1}^{\infty \:}-\frac{2\left(-1\right)^n\sin \left(\pi nx\right)}{\pi n}      .

Example 3: Suppose a function f(x) = tanx find its Fourier expansion within the limits [-π, π].

Solution:

 a_0 =  \dfrac{1}{\pi}\int_{- \pi}^{\pi} tanx dx

a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} tanx cosnxdx          

b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} tanx sinnxdx          

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

Now the integral of tanxâ‹…sinnx and tanxâ‹…cosnx cannot be found.

Therefore the Fourier series for this function f(x) = tanx is undefined.

Example 4: Find the Fourier series of the function f(x) = 1 for limits  [– Ï€, Ï€] .

Solution:

Comparing with general Fourier series expansion we get,

 a_0 =  \dfrac{1}{\pi}\int_{- \pi}^{\pi}1  dx          

a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 1 \cdot cosnxdx          

b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 1 \cdot sinnxdx .\\ \large f(x)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}a_{n}cos\;nx+\sum_{n=1}^{\infty}b_{n}sin\;nx           

f(x) = π + 0 + 0

f(x) = π

Example 5: Consider a function f(x) = x2 for the limits  [– Ï€, Ï€]. Find its Fourier series expansion.

Solution:

Comparing with general Fourier series expansion we get,

 a_0 =  \dfrac{1}{\pi}\int_{- \pi}^{\pi}x^2  dx . \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cdot cosnxdx . \\ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cdot sinnxdx . \\ \large f(x)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}a_{n}cos\;nx+\sum_{n=1}^{\infty}b_{n}sin\;nx \\ f(x) = \frac{{\pi}^{3}}{3} + \sum_{n=1}^{\infty}a_{n}cos\;nx +0 .\\ f(x) =  \frac{{\pi}^{3}}{3} + \sum_{n=1}^{\infty} \frac{4πcosnπcosnx}{n2}.

Example 6: Find Fourier series expansion of the function f(x) = 4-3x for the limits  [– 1, 1].

Solution:

Comparing with general Fourier series expansion we get,

a_0 =  \dfrac{1}{\pi}\int_{- \pi}^{\pi}4-3x  dx . \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 4-3x \cdot cosnxdx . \\ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 4 -3x\cdot sinnxdx .\\ \large f(x)=\frac{1}{2}a_{0}+\sum_{n=1}^{\infty}a_{n}cos\;nx+\sum_{n=1}^{\infty}b_{n}sin\;nx .\\ f(x) =\frac{1}{2\cdot \:1}\cdot \:8+\sum _{n=1}^{\infty \:}\frac{1}{1}\cdot \:0\cdot \cos \left(\frac{n\pi x}{1}\right)+\sum _{n=1}^{\infty \:}\frac{1}{1}\left(\frac{6\left(-1\right)^n}{\pi n}\right)\sin \left(\frac{n\pi x}{1}\right).\\ f(x) = 4+\sum _{n=1}^{\infty \:}\frac{6\left(-1\right)^n\sin \left(\pi nx\right)}{\pi n} .

Example 7: Find the expansion of the function 1- \frac{x}{\pi}      . For the limits  [– Ï€, Ï€].

Solution:

Comparing with general Fourier series expansion we get,

a_0 =  \dfrac{1}{\pi}\int_{- \pi}^{\pi}4-3x  dx .\\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 4-3x \cdot cosnxdx .\\ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 4 -3x\cdot sinnxdx .\\ f(x) = \frac{1}{2\pi }\cdot \int _{-\pi }^{\pi }\left(1-\frac{x}{\pi }\right)dx+\sum _{n=1}^{\infty \:}\frac{1}{\pi }\cdot \int _{-\pi }^{\pi }\left(1-\frac{x}{\pi }\right)\cos \left(\frac{n\pi x}{\pi }\right)dx\cdot \cos \left(\frac{n\pi x}{\pi }\right)+\sum _{n=1}^{\infty \:}\frac{1}{\pi }\cdot \int _{-\pi }^{\pi }\left(1-\frac{x}{\pi }\right)\sin \left(\frac{n\pi x}{\pi }\right)dx\cdot \sin \left(\frac{n\pi x}{\pi }\right) .\\ f(x) = =\frac{1}{2\pi }\cdot \:2\pi +\sum _{n=1}^{\infty \:}\frac{1}{\pi }\cdot \:0\cdot \cos \left(\frac{n\pi x}{\pi }\right)+\sum _{n=1}^{\infty \:}\frac{1}{\pi }\left(\frac{2\left(-1\right)^n}{n}\right)\sin \left(\frac{n\pi x}{\pi }\right).\\ f(x) = =1+\sum _{n=1}^{\infty \:}\frac{2\left(-1\right)^n\sin \left(nx\right)}{\pi n} .

FAQs on Fourier Series

1. 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.

2. What is Fourier Series Formula?

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

f(x) = A0 + ∑{n = 1}{∞} An cos (nπx/L) + ∑{n = 1}{∞} Bn sin (nπx/L)

3. 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.

4. What are Types of Fourier Series Formula?

There are two types of Fourier series formulas they are

  • Trigonometric Series Formula
  • Exponential Series Formula

5. 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.

6. What is Fourier Series of Sine?

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

7. What is Fourier Series of Cosine?

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

8. 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.



Last Updated : 08 Sep, 2023
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads