Degree of an elliptic function ($\mathrm{deg}g$)

The degree of a nonconstant elliptic function is the sum of the orders of its poles in a fundamental parallelogram. Equivalently, it is the degree of the induced holomorphic map from its complex torus to the Riemann sphere.

If an elliptic function $g$ has degree $d$ and $r$ distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, soBecause $1\le r\le d$, this gives $d+1\le \mathrm{deg}{g}^{\prime }\le 2d$.