= 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]