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A folklore result I have seen used in evaluations of infinite sums is the following clever use of the residue theorem:

$$\begin{align*}\sum_{1}^\infty f(k)&=\frac1{2\pi i}\oint f(z)\pi\cot\pi z\,\mathrm dz\\\sum_{1}^\infty (-1)^k f(k)&=\frac1{2\pi i}\oint f(z)\pi\csc\pi z\,\mathrm dz\end{align*}$$

where $f(z)$ is analytic and the contour surrounds the appropriate subset of poles.

I have seen some references ascribe this to Mittag-Leffler, but as far as I aware, his only contribution was showing that the functions used in the contour integrals above can be decomposed into their poles.

The fact that these results do not have a name attached to them makes it hard to search for more information; my limited searching shows me that these results are often quoted in handbooks or textbooks as a mere consequence of the residue theorem.

So I must ask: is there any history behind these summation formulae (who came up with it, first appearance in the literature, etc.)? References where I can read more on this and other contour integral summation techniques (e.g. I have seen the variants that use tangent, secant, or even the gamma function itself) are welcome.

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Whittaker and Watson credit this method to Mittag-Leffler, Acta Soc. Sci. Fenn., XI 1880 273-293, and Acta Math., IV 1884 1-79.

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  • $\begingroup$ Thank you, I will try to dig up these references. For completeness, and so I can look at my copy of the book when I am able, where in Whittaker and Watson is this mentioned? $\endgroup$ – kolobok Apr 14 at 1:29
  • $\begingroup$ Volume I, section 7.4 (I use 1927 edition, I suppose there was no later changes). $\endgroup$ – Alexandre Eremenko Apr 14 at 23:59

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