A simple evaluation of a theta value and the Kronecker limit formula

We evaluate the classic sum $\sum_{n\in\mathbb{Z}} e^{-πn^2}$. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the $Γ$ function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula.