Plotting Jacobi Elliptic Function

Question

I've written program on Python using pygame library for plotting complex functions phase and modulus graphics.
I'm not programmer and don't have any math background. But now I want to know how I could numerically evaluate Jacobi Elliptic Function value in some point z. I've found definition of the function in Wikipedia Jacobi elliptic function and there was integral but I don't understand how I could use it to evaluate function value in point z of complex plane. I know how to numerically evaluate path integral form some point a to b in complex plane, but there are some theta and phi parameters and I don't understand it.

Could you help me? I don't need Python code (I'll write it myself if I'll understand the principle) but it could be enough if you provide algorithm step by step how to do it.

Answer 1

You could just use mpmath.

from mpmath import ellipfun

print(ellipfun('cd', 1.0 + 2.0j, 0.5))

(1.90652944795345 + 0.225277477847159j)

Answer 2

scipyx, my collection of extensions to SciPy, has support for complex-valued arguments in Jacobi elliptic functions.

Install with

pip install scipyx

and use as

import scipyx as spx

u = 1.0 + 2.0j
m = 0.8
# sn, cn, dn, ph = scipy.special.ellipj(x, m)  # not working
sn, cn, dn, ph = spx.ellipj(u, m)

If you're after plotting those, take a look at cplot (also by me):

Answer 3

Having read the article in Wikipedia Jacobi elliptic function and one at http://mysite.du.edu/~jcalvert/math/jacobi.htm I believe this to be an interpretation.

z is a point in the complex plain then z' is its complementary modulus where z'^2 = 1 - z^2

It seems to be the convention that for the Jacobi elliptic function k is used instead of z and that m is used for k^2 and k is such that k^2 is real and 0<k^2<1

the integral is a function u of two parameters k and phi

u(k,phi) = the integral as given

Note then that instead of starting with a z in the complex plane you are starting with a real m 0<k^2<1 and the results relate to the complex solutions of z^2=m

So for a given m you could numerically integrate for a range of values phi (for example 0 to 6π in steps of π/12) giving u

Now for a given m you have a data set plotting for u against phi

The elliptic function sn is the inverse of this ie given u what phi gives this u

So looking in the u data would give the phi results.

Note for a given u there would be more than one phi.