I'm looking for resource recommendations on the Sommerfeld-Watson summation method, i.e. the use of the residue theorem to obtain expressions like
$$ \tag 1 \sum_{n \in \mathbb{Z}} g(n) = - \pi \sum_i \underset{\,z =z_i}{\operatorname{Res}} \left[ g(z) \cot(z) \right] $$ with $z_i$ the poles of $g$, which is usually assumed to be some good behaved meromorphic function (but I don't have the details about this). Equation (1) is usually obtained in this context from the integral $$ \tag 2 \oint_{\mathcal C} dz \, g(z) \cot(z) $$ where $\mathcal C$ encircles the real axis.
The web seems to be strangely empty about this argument, which makes me think that I may have given the wrong name and that this technique is actually usually called in some other way. If so please point me to the right direction.
Here are some nicely written notes about the method I'm referring to which I recently discovered: http://www.phys.uconn.edu/phys2400/downloads/sommerfeld-watson.pdf
A long time ago, I used the WST approach in applying Regge Pole theory to theoretical chemistry scattering problems. An infinite sum over partial waves (full quantum mechanical approach) was reduced to a sum over poles of the S-matrix plus an integral (that may have required knowledge of the residues?).
Long answer I know, try looking in the literature for Regge Pole applications.