1. ## Fourier series

Define $f(t)=e^{-t} on [-\pi,\pi)$ and extend f to be $2\pi$-periodic.Find the complex Fourier series of f.
Then, apply Parseval's relation to f to evaluate $/sum {1/(1+k^2)}$.
I know how the complex fourier series looks like but I don't know how to extend it periodically

2. Originally Posted by AkilMAI
Define $f(t)=e^{-t} on [-\pi,\pi)$ and extend f to be $2\pi$-periodic.Find the complex Fourier series of f.
Then, apply Parseval's relation to f to evaluate $/sum {1/(1+k^2)}$.
I know how the complex fourier series looks like but I don't know how to extend it periodically
The function $e^{t}$ is not peroidic, but when you find its Fourier series on $[-\pi,\pi)$ and then let $t \in \mathbb{R}$ not just the interval you will be a new function that is periodic on all of $\mathbb{R}$ (It will just be translated copies of the series on the original interval)

3. Originally Posted by AkilMAI
Define $f(t)=e^{-t} on [-\pi,\pi)$ and extend f to be $2\pi$-periodic.Find the complex Fourier series of f.
Then, apply Parseval's relation to f to evaluate $/sum {1/(1+k^2)}$.
I know how the complex fourier series looks like but I don't know how to extend it periodically
http://www.mathhelpforum.com/math-he...em-167764.html

Kind regards

$\chi$ $\sigma$

4. Originally Posted by AkilMAI
Define $f(t)=e^{-t} on [-\pi,\pi)$ and extend f to be $2\pi$-periodic.Find the complex Fourier series of f.
Then, apply Parseval's relation to f to evaluate $/sum {1/(1+k^2)}$.
I know how the complex fourier series looks like but I don't know how to extend it periodically
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