## Fourier Series

A Fourier series is an enlargement of a periodic function f(x) with respect to an infinite sum of sines and cosines. The Fourier series considers the orthogonality links of the sine and cosine functions. The study and measure of Fourier series is referred to harmonic analysis and is tremendously useful to break up an arbitrary periodic function into a set of simple terms, which can be plugged in, solved separately, and then recombined to gain the solution to the actual problem or estimation to it to whatever appropriateness is desired or practical.

### Fourier Series Formula

Following is the Fourier series formula:-

f(x) = 1/2a0 + ∑∞n = 1a_{n }cosnx + ∑∞n=1 b_{n }sinnx

Where,

a0 = 1/π∫^{π}_{−}_{p}i f(x)dx

a_{n} = 1/π∫^{π}_{−π} f(x)sin nx dx

b_{n} = 1/π∫^{π}_{−π} f(x)sin nx dx

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### Finding the Coefficients Using Fourier Series Coefficients Formula

How did we know to implement sin (5x)/5, sin (7x)/7, etc?

There are formulas! Formulas like Fourier coefficients formula.

Below is a series of sines and cosines having a name for all coefficients:

f(x) = a0 + ∞n=1an cos(nxπ/L) + ∞n=1 bn sin(nxπ/L)

Where:

f(x) = the function we desire (such as a square wave)

L = half of the period of the function

a0, an and bn = coefficients that we require to calculate!

To calculate the coefficients a0, an and bn we use the given below Fourier series formula list:

a0 = 1/2L ∫L−L f(x) dx

an = 1/L ∫L−L f(x) cos(nxπ/L) dx

bn = 1L ∫L−L f(x) sin(nxπ/L) dx

### Solved Examples

Example:

Identify the Fourier series for f(x)=x f(x)=x on −L≤x≤L−L≤x≤L.

Solution:

Let’s begin with the integrals for An

A0=1/2L∫L−Lf(x)dx=1/2L∫L−Lxdx=0

An=1/L∫L−Lf(x)cos(nπx/L)dx=1/L∫L−Lxcos(nπx/L)dx=0

In both cases remember that we are integrating an odd function (x is odd and cosine is even, thus the product is odd) over the interval [−L, L] and thus we know that both of these integrals will be zero (0).

Next, is the integral for Bn

Bn=1/L∫L−Lf(x)sin(nπx/L)dx=1/L∫L−Lxsin(nπx/L)dx=2/L∫L0xsin(nπx/L)dx

In this case we are integrating an even function (x and sine are both odd thus the product is even) on the interval [−L,L] and thus we can also “simplify” the integral as above. Now, Using the previous result we get,

Bn=(−1)n+12L/nπ n=1,2,3…

In the case the Fourier series is,

f(x) = ∞∑n =0A_{n}cos(nπx/L) + ∞∑n=1Bnsin(nπx/L) = ∞∑n=1(−1)n+12L/nπ sin(nπx/L)

1. What is the Use of Fourier Series Formula?

Answer: A wide array of the phenomena studied in the field of Science and Engineering are periodic in nature. For instance voltage and current that exist in an alternating current circuit. We are able to evaluate these periodic functions into their constituent components with the help of a process called Fourier analysis.

2. What is Fourier Analysis for Periodic Functions?

Answer: The Fourier series depiction of analytic functions is a derivation from the Laurent expansions. We use the fundamental complex analysis in order to derive additional fundamental outcomes in the harmonic analysis that includes representation of the periodic functions by the Fourier series.

The trigonometric functions sin x and cos x are case examples of periodic functions having fundamental period 2π and tan x is periodic with fundamental period.

Thus, a Fourier series is a technique to depict a periodic function as a sum of sine and cosine functions feasibly till infinity. It is in correspondence to the popular Taylor series that depicts functions as possibly infinite sums of monomial terms.