 Weierstrass sigma function

For a period lattice, the displayed canonical product converges normally because its logarithmic tails are bounded by a constant times $|\omega {|}^{-3}$. It defines an odd entire function with simple zeros exactly at the lattice and derivative one at zero. Its logarithmic derivative is the Weierstrass zeta function, and the derivative of that logarithmic derivative is minus the Weierstrass elliptic function. Sigma is quasi-periodic rather than elliptic; balanced products of its translates construct meromorphic functions with prescribed divisors on a complex torus.