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### Jacobi Zeta Function

##### Synopsis

    #include <boost/math/special_functions/jacobi_zeta.hpp>

    namespace boost { namespace math {

    template <class T1, class T2>
    calculated-result-type jacobi_zeta(T1 k, T2 phi);

    template <class T1, class T2, class Policy>
    calculated-result-type jacobi_zeta(T1 k, T2 phi, const Policy&);

    }} // namespaces

##### Description

This function evaluates the Jacobi Zeta Function Z(φ, k)

The return type of this function is computed using the result type calculation rules when the arguments are of different types: when they are the same type then the result is the same type as the arguments.

Requires -1 <= k <= 1, otherwise returns the result of domain_error (outside this range the result would be complex).

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

Note that there is no complete analogue of this function (where φ = π / 2) as this takes the value 0 for all k.

##### Accuracy

These functions are trivially computed in terms of other elliptic integrals and generally have very low error rates (a few epsilon) unless parameter φ is very large, in which case the usual trigonometric function argument-reduction issues apply.

##### Testing

The tests use a mixture of spot test values calculated using values calculated at wolframalpha.com, and random test data generated using MPFR at 1000-bit precision and a deliberately naive implementation in terms of the Legendre integrals.

##### Implementation

The implementation for Z(φ, k) first makes the argument φ positive using:

Z(-φ, k) = -Z(φ, k)

The function is then implemented in terms of Carlson's integral RJ using the relation:

There is one special case where the above relation fails: when k = 1, in that case the function simplifies to

Z(φ, 1) = sign(cos(φ)) sin(φ)