Compute $\sum_{n=1}^\infty \cos(\frac{n\pi x}{L})\cos(\frac{n\pi x'}{L})$

Asked Nov 04 '15 at 18:42

Active Nov 04 '15 at 18:45

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Let $x$ and $x'$ be in the interval $[0,L]$

Compute:

$$ \sum_{n=1}^\infty \cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{n\pi x'}{L}\right) $$

I remember a lot of time i have done this and if I'm not bat the result is $\delta(x-x')$

I have proved the sum diverges when $x'=x$ but i'm stuck when $x'\neq x$

edited Nov 04 '15 at 18:45

Zhanxiong

asked Nov 04 '15 at 18:42

There is probably a better way to do this, but I was thinking... take fourier transform of the sum and then invert the transformed sum? With dirac delta functions I find dealing with them in integrals is a lot easier. – Ismail Bello Nov 04 '15 at 20:36

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