My question is from Arfken & Weber (Ed. 7) 19.2.2:
In the first part, the question asks for Fourier series expansion of $\delta(x)$. I have found $$\delta(x)=1/2\pi + 1/\pi\sum^{\infty}_{n=1} cos(nx)$$ Then by using the identity $$\sum^{N}_{n=1} cos(nx)=\frac{sin(Nx/2)}{sin(x/2)}cos\left[(N+\frac{1}{2})\frac{x}{2}\right]$$, we need to find a Fourier representation which is consistent with $$\delta(x-t)=\sum^{\infty}_{n=0}\phi^{*}_{n}(t)\phi_{n}(x)$$. I have tried expanding the trigonometric functions with exponential terms countless times, however could not obtain a sufficient result. Any help is appreciated.
Cheers
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