Sine infinite product
= Sine infinite product
{title2=$\sin z=z\prod_{n\ge1}(1-z^2/(\pi^2n^2))$}
The paired zero factors converge locally uniformly because $\sum n^{-2}$ converges. <Hadamard factorization> for the order-one entire sine function gives a possible exponential multiplier; oddness removes its linear exponent and the derivative at zero fixes its constant. This product turns a <Dirichlet oscillator determinant ratio> into an elementary sine ratio.