Question

Section - B
(5 mari
Q2)
Obtain the Fourier series for the function F(x)=Cosh ax in the interval - 1<x<1.
1 - e-21​

Answer 1

Answer:

Prove that in the range −π<x<π,

cosh(ax)=2a2sinh(aπ)π(12a2+∑n=1∞(−1)n1n2+a2cos(nx))

Now, I have tried to get the Fourier series of cosh(ax).

I got

a0=2sinh(aπ)πaan=−2asinh(aπ)π(n2−a2)

Probably I got

f(x)=sinh(aπ)π(1a−2a∑n=1∞(−1)n1n2−a2cosnx)

I tried twice and again I got this answer which is not matching with the question. Please tell me what is my fault and how to solve this?

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Jul 24 '17 at 18:46

user466871

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Dec 6 '17 at 9:21

rubik

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Hey, welcome to stackexchange! Please use the formatting tutorial! – Piotr Benedysiuk Jul 24 '17 at 18:51

I think the formula in the yellow box is wrong, I'm using Introduction To Calculus And Analysis by Courant & John and it's actually 2aπ in the front. They might have gotten it wrong , though. There's a couple typos in the book. – Divide1918 Jan 9 '20 at 2:49

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Note that we have

∫π−πcosh(ax)cos(nx)dx=2Re(∫π0cosh(ax)einxdx)=Re(∫π0(e(a+in)x+e−(a−in)x)dx)=Re((−1)n(eaπa+in−e−aπa−in))=(−1)n2asinh(aπ)a2+n2