Find the sum $ \sum_{-\infty}^{\infty} \frac{1}{1 + (x + n)^2} $

Asked Mar 04 '23 at 20:36

Active Mar 04 '23 at 20:36

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More general problem than this $$\sum_{-\infty}^{\infty}\frac{1}{1+n^2}$$ I've seen some cases where it was solved by calculating $$\sum_{n=-\infty}^{\infty}\frac{1}{a^2+n^2}$$ here
or using the Fourier series of $e^x \, x \in [-\pi, \pi]$ here

But I have never seen this problem generalized to my case $$ \sum_{n \in \mathbb{Z}} \frac{1}{1 + (x + n)^2} $$

Is it possible to use the same methods here or it seems to be something new?

asked Mar 04 '23 at 20:36

avoidcatenjoyers

1

Does this answer your question? Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$ - found using an Approach0 search. There are also other at least very similar questions, e.g., Can we give a closed form expression for $\sum_{k=-\infty}^\infty\frac1{a+(k+x)^2}$? (which has a restriction that $x\in(-1,1)$), etc. – John Omielan Mar 04 '23 at 21:04

Find the sum $ \sum_{-\infty}^{\infty} \frac{1}{1 + (x + n)^2} $

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