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 __arg_pomotion_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.

Table 6.68. Error rates for jacobi_zeta

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral Jacobi Zeta: Mathworld Data	Max = 0.505ε (Mean = 0.119ε)	Max = 0ε (Mean = 0ε)	Max = 1.66ε (Mean = 0.48ε)	Max = 1.66ε (Mean = 0.48ε)
Elliptic Integral Jacobi Zeta: Random Data	Max = 3.89ε (Mean = 0.818ε)	Max = 0ε (Mean = 0ε)	Max = 2.99ε (Mean = 0.824ε)	Max = 3.96ε (Mean = 1.06ε)
Elliptic Integral Jacobi Zeta: Large Phi Values	Max = 2.92ε (Mean = 0.964ε)	Max = 0ε (Mean = 0ε)	Max = 2.92ε (Mean = 0.951ε)	Max = 3.05ε (Mean = 1.13ε)

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 R_J 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)