This formula was shown in A Course in Modern Analysis - Whittaker, Watson as an exercise
How could this be proven? My guess is that it involves summing the partial fraction version of $\wp$ over all multiples of $\omega_1$.
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2$\wp$ is defined by a double sum. This is what one gets if one sums over one of the variables – Angina Seng Jul 26 '20 at 09:54
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Thanks! How would that summation be performed? How would we end up with cosec^2 and -1/3 and so on. – Jul 26 '20 at 10:00
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There are well-known formulae for sums like $\sum_{n=-\infty}^\infty 1/(w-n)^2$. – Angina Seng Jul 26 '20 at 10:07
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How are they derived? – Jul 26 '20 at 10:09
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1$1/(\sin^2 z)=-(\cot z)'$ and the partial fraction decomposition of the tangent is well known, so just differentiate term by term – Conrad Jul 26 '20 at 13:48
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Look for questions like this one. – metamorphy Jul 26 '20 at 13:53
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What is the meaning of the dash at the 2nd summation? – emacs drives me nuts Feb 15 '21 at 20:59