Elliptic Theta Functions

Introduction

All of the elliptic theta functions are functions of two parameters: the argument z and a second argument. For these routines the second argument is the nome q, |q| < 1 . Some references use the the lattice parameter tau, with the imaginary part of tau strictly positive.

The nome and lattice parameter are related by q = exp(%i*%pi*tau).

Theta 1

(elliptic-theta-1 z q)

First theta function defined by

  theta1(z, q) = 2*q^(1/4)*sum((-1)^n*q^(n*(n+1))*sin((2*n+1)*z), n, 0, inf)

Theta 2

(elliptic-theta-2 z q)

Second theta function defined by

  theta2(z, q) = 2*q^(1/4)*sum(q^(n*(n+1))*cos((2*n+1)*z), n, 0, inf)

Theta 3

(elliptic-theta-3 z q)

Third theta function defined by

  theta3(z, q) = 1 + 2 * sum(q^(n^2)*cos(2*n*z), n, 1, inf)

Theta 4

(elliptic-theta-4 z q)

Fourth theta function defined by

  theta4(z, q) = 1 + 2*sum((-1)^n*q^(n^2)*cos(2*n*z), n, 1, inf)

Miscellaneous

(elliptic-theta n z q)

Convenience function where n is 1, 2, 3, or 4 for one of the four theta functions

(elliptic-nome m)

Compute the nome q from the parameter m.

References

• ​DLMF Theta Functions