Weierstrass zeta and sigma functions

Summary

For a lattice $\Lambda$ in $\C$, the Weierstrass sigma function $\sigma$ is an entire function with a simple zero at each lattice point. The Weierstrass zeta function is its logarithmic derivative $\zeta = \sigma'/\sigma$. They are companions of the Weierstrass elliptic function $\wp$, since $\zeta' = -\wp$. Neither is elliptic: shifting by a period adds a constant to $\zeta$ and multiplies $\sigma$ by an exponential factor. They are the building blocks for writing any elliptic function from its zeros and poles.