Exercise 2 check class notes for solution prove that if fx...

Exercise 2 (Check class notes for solution): Prove that if f(x) is an odd function, its Fourier series reduces to a Fourier sine series. f(x) = sum_{n=1}^{infty} b_n sin frac{npi x}{L} (f odd) with coefficients b_n = frac{2}{L} int_{0}^{L} f(x) sin frac{npi x}{L} dx a_n=0

Transcript

00:01 Hello everyone we are going to solve a question and this question we have to prove this is we have to prove that f x if f x is odd function then four year series reduces to four year sign series so let's start with the solution here we have four year series of f x is given by f x is equal to a not plus summation and is from one to infinity a n cost of n pi x divided by l plus b times b n sorry sign of n pi n pi i i will write it clearly this is n pi x divided by l.

01:15 So from here this is the fourier series and where the values of a not is equal to 1 divided by 2l integration from minus l to l fx dx and next is value of a .n which is equals to 1 divided by l integration from minus l to l f of x and cos of n pi x divided by l with respect to dx.

01:43 And the next one is bn, which is equal to 1 divided by l integration from minus l to l.

01:50 Here we have f of x sine of n pi x divided by l with respect to dx.

01:58 So now from here we have if f of x is odd function.

02:07 If f of x is odd function, then that means f of minus x is equals to minus of f x so from here now we find the value of a not so here we have a not is equals to 1 divided by 2 l integration from minus l to l fx d x so now this is equals to 1 divided by 2l minus l to l fx d x here f of minus l plus l minus x d x d x here f of minus l plus l minus x d x d x d x and after solving this we have this is equal to 1 divided by 2l integration minus l to l f of minus x d x and f of minus x d x is this is minus 1 divided by 2 l integration minus l 2 l 2.

03:00 Here we have minus l to l minus l to l, f of x d x if we take this as i then this is equals from negative i.

03:14 Here this is our equation one and this is our equation two.

03:20 Now adding 1 and 2 here we have adding 1 and 2.

03:27 And after adding these we have 2 i is equals to 0.

03:32 So that means i is equals to 0.

03:35 So that means a not is equals to 0.

03:38 In short our a not is equals to 0.

03:41 Moving ahead towards a now we find a of n.

03:45 So for finding a of n, this is equals to 1 divided by l, integration minus l to l, f of x, kosh of n pi x divided by l with respect to dx, say this is equals to i, and this is our equation number, roman 1 in this...

Question

Exercise 2 (Check class notes for solution): Prove that if f(x) is an odd function, its Fourier series reduces to a Fourier sine series. f(x) = sum_{n=1}^{infty} b_n sin frac{npi x}{L} (f odd) with coefficients b_n = frac{2}{L} int_{0}^{L} f(x) sin frac{npi x}{L} dx a_n=0

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