# Jacobi elliptic functions with complex variables

1. May 3, 2015

### karlzr

I am trying to solve a Duffing's equation $\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0$ where $\alpha$ is a complex number with $Re \alpha<0$ and $\beta>0$. The solution can be written as Jacobi elliptic function $cn(\omega t,k)$. Then both $\omega$ and $k$ are complex. The solution to $\omega$ being complex can found from textbooks. So my question is how to deal with $k$ being complex?

How to deal with $cn(z_1+iz_2, k_1+i k_2)$ where $z_i$ and $k_i$ are real numbers?

In the case of harmonic oscillator, imaginary part of the frequency square will change the amplitude of oscillator just like what friction terms do. I am wondering whether we have similar equivalence in this case?

I think you want to use the definition of elliptic functions in term of theta functions. You must use a series inversion described there to convert the elliptic modulus to the $\tau$ modulus.