A very simple proof of the eta transformation formula
The ``very simple proof'' of the title is based upon the behavior of the Weierstrass zeta function with respect to the lattice \(\mathbb Z+\mathbb Zr\) \((\Im\,\tau> 0)\); \[ \zeta(z)= {1\over z}+ \sum_{\substack{ m,n\in\mathbb Z\\ (m,n)\neq (0,0)}} \Biggl({1\over z+ m\tau+ n}- {1\over m\tau+ n}+ {z\over (m\tau+ n)}\Biggr), \] in particular, upon the behavior of \(\varphi_1(\tau)= {i\over 2\pi} \zeta({1\over 2})\) and \(\varphi_2(\tau)= {i\over 2\pi}\zeta(\tau/2)\) under the modular transformation \(\tau\to -1/\tau\). The proof applies Euler's partial fraction expansions of \(\pi\operatorname{cot}\pi z\) and \(\pi^2/(\sin^2\pi z)\). The author points out that his proof is a simplification of one suggested by \textit{H. Petersson} [Modulfunktionen und quadratische Formen, Berlin etc.: Springer-Verlag (1982; Zbl 0493.10033), Appendix D].
scientific article; zbMATH DE number 5115238