Submitted by :- Rucha Pandya ( ) Branch:Civil engineering Under the guidance of Asst.Prof. Reen Patel Gujarat Technological University, Ahmedabad

INDEX  Fourier Series  General Fourier  Discontinuous Functions  Change Of Interval Method  Even And Odd Functions  Half Range Fourier Cosine & Sine Series

FOURIER SERIES A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.periodic functionsinescosines

General Formula For Fourier Series Where,

Formulas To Solve Examples 2SC = S + S 2CS = S – S 2CC = C + C 2SS = cos(α-β) –cos(α+β) Even*Odd = Odd Even*Even = Even Odd*Odd = Even Odd*Even = Odd

Where, u, u’, u”, u’’’,_ _ _ _ are denoted by derivatives. And V1,v2,v3,v4,_ _ _ _ _ are denoted by integral.

Discontinuous Type Functions In the interval The function is discontinuous at x =x 0 f(x)

So Fourier series formula is

Change Of Interval Method In this method, function has period P=2L, where L is any integer number. In interval 0<x<2L Then l = L/2 When interval starts from 0 then l = L/2 In the interval –L < X < L Then l = L For discontinuous function, Take l = C where C is constant.

General Fourier series formula in interval Where,

Even Function The graph of even function is symmetrical about Y – axis. Examples :

Fourier series for even function 1. In the interval

Fourier series for even function (conti.) 2. In the interval

Odd Function The graph of odd function is passing through origin. Examples:-

Fourier series for odd function 1. In the interval

Fourier series for odd function (conti.) In the interval

Half Range Fourier Cosine Series  In this method, we have 0 < x < π or 0 < x < l type interval.  In this method, we find only a 0 and a n.  b n = 0

Half Range Fourier Cosine Series 1.In the interval 0 < x < π

Half Range Fourier Cosine Series(conti.) 2. In the interval 0 < x < l Take l = L

Half Range Fourier Sine Series In this method, we find only b n a n =0 a 0 =0

Half Range Fourier Sine Series 1. In interval 0 < x < π

Half Range Fourier Sine Series (conti.) 2. In the interval 0 < x < l

Thank you!!!