3.5.1.3.49 Jacobian_theta

Definition:

The y = jacobian_theta(k, x, q) computes the value of one of the Jacobian theta functions \(\theta _0 (x, q)\), \(\theta _1 (x, q)\), \(\theta _2 (x, q)\), \(\theta _3 (x, q)\) or \(\theta _4 (x, q)\) for a real argument x and non-negative q ≤ 1.

The routine evaluates an approximation to the Jacobian theta functions \(\theta _0 (x, q)\), \(\theta _1 (x, q)\), \(\theta _2 (x, q)\), \(\theta _3 (x, q)\) and \(\theta _4 (x, q)\) given by

\(\theta _0\left( x,q\right) =1+2\sum_{n=1}^\infty \left( -1\right) ^nq^{n^2}\cos (2n\pi x)\),

\(\theta _1\left( x,q\right) =2\sum_{n=0}^\infty \left( -1\right)^nq^{(n+0.5)^2}\sin((2n+1)\pi x)\),

\(\theta _2\left( x,q\right) =2\sum_{n=0}^\infty q^{(n+0.5)^2}\cos ((2n+1)\pi x)\),

\(\theta _3\left( x,q\right) =1+2\sum_{n=1}^\infty q^{n^2}\cos (2n\pi x)\),

\(\theta _4\left( x,q\right) =\theta _0\left( x,q\right)\),

where x and q are real with 0 ≤ q ≤ 1. Note that \(\theta _1 (x-0.5, 1)\) is undefined if \((x-0.5)\) is an integer, as is \(\theta _2 (x, 1)\) if x is an integer. Otherwise, \(\theta _i (x, 1)=0\), for \(i=0, 1, ..., 4\).

For more information please refer to the s21ccc function in the NAG document.

Parameters:

k (input, integer)

The function \(\theta _k\) (x,q) to be evaluated. Note that k=4 is equivalent to k=0.

Constraint: 0 ≤ k ≤ 4.

x (input, double)

The argument x of the function.

Constraints: x must not be an integer when q=1.0 and k=2; (x-0.5) must not be an integer when q=1.0 and k=1.

q (input, double)

The argument q of the function.

Constraint: 0.0 ≤ q ≤ 1.0.

y (output, double)

The return value of one of the Jacobian theta functions.