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What series is this? [closed]

I have a functional series that converges when I am calculating it numerically. As the equation is quite simple, I wonder if it has been studied before and if the converged value has an analytic form.

The series is:

$P(\omega) = \sum_{k=-\infty}^{\infty} \frac{1}{1 + (\omega + c k)^2}$

$\omega, c \in \mathbb{R}$

Does anyone recognize this?

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This can be easily computed with Mathematica. Your question does not seem to fit well in Mathoverflow. – Jon Jul 18 at 13:33
And with Maple. $$\sum_{k = -\infty}^{\infty} \frac{1}{1 + w^{2} + 2 w c k + c^{2} k^{2}} = \frac{\pi \operatorname{cosh} \Bigl(\frac{\pi}{c}\Bigr) \operatorname{sinh} \Bigl(\frac{\pi}{c}\Bigr)}{c \Bigl(\operatorname{cosh} \Bigl(\frac{\pi}{c}\Bigr)^{2} - \operatorname{cos} \Bigl(\frac{\pi w}{c}\Bigr)^{2}\Bigr)}$$ – Gerald Edgar Jul 18 at 13:40
You should look up evaluation of infinite series by residues in complex analysis textbooks. – Michael Renardy Jul 18 at 13:51