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Jacobi elliptic functions with complex variables

  1. May 3, 2015 #1
    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?

    Thanks for your time!
     
  2. jcsd
  3. May 3, 2015 #2

    fzero

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    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.
     
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