Elliptic functions are doubly periodic and so their values everywhere in the complex plane are determined by their values in a fundamental parallelogram. We will assume the parameter *m* is real and between 0 and 1 so that the fundamental parallelograms are rectangles aligned with the real and imaginary axes.

The functions sn, ns, cd, and dc all have a fundamental rectangle with vertices at o and 4*K*(*m*) + 2*K*(1-*m*) *i*.

The functions cn, nc, ds, and sd all have a fundamental rectangle with vertices at o and 4*K*(*m*) + 4*K*(1-*m*) *i*.

The functions cs, sc, dn, and nd all have a fundamental rectangle with vertices at o and 2*K*(*m*) + 4*K*(1-*m*) *i*.

In the plots below *m* = 1/2 so that *m* = 1 – *m*.

Pinwheels of color indicate zeros and white areas indicate poles.

Note that the reciprocal of a function swaps the letters in the name, e.g. ns(*z*) = 1/sn(z). Taking the reciprocal swaps zeros and poles.

## sn

This function was plotted in Mathematica using

K = EllipticK[1/2] ComplexPlot[JacobiSN[z, 1/2], {z, 0, 4 K + 2 I K}, AspectRatio -> Automatic]

and the rest of the plots were made analogously.