POISSON SUMMATION COSH SUM

We want to prove the following transformation formula that appears in this twitter post

\sum_{n=-\infty}^{\infty} \frac{1}{\cosh \pi n x}=\sum_{n=-\infty}^{\infty} \frac{1}{x \cosh \pi n / x}

Click here to see the proof.

We used the result

\int_{0}^{\infty} \frac{\cos 2 b x}{\cosh a x} d x=\frac{\pi}{2 a \cosh \frac{b \pi}{a}}

proved here.

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