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The heart of the problem is finding a fourier series in its complex form for:

$\displaystyle\sum _{k=-\infty }^{\infty } e^{-4|t-k|}$

The form I know of is $\displaystyle\sum_{k=-\infty}^{\infty} C_ne^{ikt\omega}$

My problem is that this is not a periodic and I dont know how to proceed.

A direction will be helpful. Thank you.

SteelSoul
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  • Your function is in fact periodic with period $1$. Use the substitution $\ell = k-1$ in the calculation $f(t + 1) = \sum_{k=-\infty}^{\infty} e^{-4 |(t+1) - k|} = \sum_{k=-\infty}^{\infty} e^{-4 |t - (k-1)|} = \sum_{\ell= -\infty}^{\infty} e^{-4 |t - \ell|} = f(t)$. On each compact intervall your series converges uniformly. You can thus calculate the Fourier coefficients by interchaning the integral and the sum in $\int_0^1 f(x) e^{-2\pi inx} dx$. – PhoemueX May 18 '14 at 10:18
  • Nice spot. Thanks. But how do you integrate over this sum with absolute value? – SteelSoul May 18 '14 at 10:43
  • For $k \in \mathbb{Z}$ the sign of $t-k$ is fixed for $t \in [0,1]$. This should allow you to calculate each term individually. – PhoemueX May 18 '14 at 10:44
  • That is a good explanation. Thank you – SteelSoul May 18 '14 at 11:11

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