= 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 =

 * [http://dlmf.nist.gov/20 DLMF Theta Functions]