Show that the following function have the indicated Fourier...

Show that the following function have the indicated Fourier series (calculate the Fourier coefficients) Please do it step by step: a) $e^{b\theta} = \frac{sinh \ b\pi}{\pi} \sum_{1}^{\infty} \frac{(-1)^n}{b - in} e^{in\theta}$, for $-\pi < \theta < \pi$

Transcript

00:01 Hi everyone we have this problem in which we need to find out the fourier series representation of f x now the interval is given to us which is what pi to pi so we can say the time period will be what 2 pi so frequency will be 2 pi which is but 1 now it is given that now basically not a fic frequency we can say that what angular frequency you can say omega is but 1 so f of s the forrear series will be what a 0 plus a n summation of n is equal to 1 to infinity cause of n omega x okay now since omega is what 1 so we can just write at n x plus summation this is b n is equal to 1 to infinity and this is what sign of n x we just need to find first a 0 now what is a 0 will be what 1 divide by t this is minus pi to pi fx times d x now for minus pi to zero function is not zero so one divide we just into write time period 2 pi basically this is 2 pi so we have what i can write this is 0 to pi we have what sine x d x now what is the integration of sine x this is minus cos pi so this is minus cos x and the limits is but 0 to pi and if you substitute 0 to pi that becomes 2 so this is 2 divided by 2 5 that comes out to be what 1 divide by pi now talking about the a n now what is a n a n will be 2 divide by t so this is what 1 divide by pi and the this goes from 0 to pi because in minus pi to 0 the function is what a 0 we have a sine x and we have a cost of n x let's just multiply with 2 and divide with 2 so you have basically 1 divide by 2 pi and let's write this thing as what sign a plus b plus a minus b so basically you have what integration 0 to pi we have what sign of 1 plus n x plus sign of 1 minus n x times d x now integration of this thing which is what 1 divide by 2 pi integration of sign 1 plus x which is what cause of 1 plus n x divide by 1 plus n plus same thing cause of 1 minus n divide by minus times x 1 minus this could be what 1 minus 9 limits 0 to pi now if you substitute what pi what happened see carefully 1 divided by 2 pi substituting what substituting pi so we have what cost of 1 plus n pi divide by 1 plus n plus again cost of 1 minus n pi divide by 1 minus pi and if you substitute 0 so what will be happen cause of this thing will be what it will be one so basically cost of zero is what one so one divide by one one plus n and plus again one one divide by one minus n now you can solve this thing what is the value of this thing this thing comes out to be what one minus n this is one minus n plus one plus n now this this can see that so we have only two divided by 1 minus s as well so we have what 1 divide by 2 pi and this is here what cost of 1 plus n pi divide by 1 plus n plus we have again cause of 1 minus n pi divide by 1 minus n and this is minus 2 divide by 1 minus n square this is the…

Question

Show that the following function have the indicated Fourier series (calculate the Fourier coefficients) Please do it step by step: a) $e^{b\theta} = \frac{sinh \ b\pi}{\pi} \sum_{1}^{\infty} \frac{(-1)^n}{b - in} e^{in\theta}$, for $-\pi < \theta < \pi$

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