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The Weierstrass zeta function satisfies ${\zeta }^{\prime }=-\wp$ and is quasi-periodic: for every period $\omega$, the difference $\zeta (z+\omega )-\zeta (z)$ is constant. It has a simple pole of residue one at each lattice point. Consequently, if ${\sum }_j{c}_j=0$, thenis an elliptic function.

It is the quasi-periodic zeta member of the Weierstrass functions.