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Coding Standards
#include <boost/math/special_functions/jacobi_zeta.hpp>
namespace boost { namespace math { template <class T1, class T2> calculatedresulttype jacobi_zeta(T1 k, T2 phi); template <class T1, class T2, class Policy> calculatedresulttype jacobi_zeta(T1 k, T2 phi, const Policy&); }} // namespaces
This function evaluates the Jacobi Zeta Function Z(φ, k)
Please note the use of φ, and k as the parameters, the function is often defined as Z(φ, m) with m = k^{2}, see for example Weisstein, Eric W. "Jacobi Zeta Function." From MathWorldA Wolfram Web Resource. Or else as Z(x, k) with φ = am(x, k), where am is the Jacobi amplitude function which is equivalent to asin(jacobi_elliptic(k, x)).
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.
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 argumentreduction issues apply.
Table 8.68. Error rates for jacobi_zeta
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Elliptic Integral Jacobi Zeta: Mathworld Data 
Max = 0ε (Mean = 0ε) 
Max = 1.66ε (Mean = 0.48ε) 
Max = 1.66ε (Mean = 0.48ε) 
Max = 1.52ε (Mean = 0.357ε) 
Elliptic Integral Jacobi Zeta: Random Data 
Max = 0ε (Mean = 0ε) 
Max = 2.99ε (Mean = 0.824ε) 
Max = 3.96ε (Mean = 1.06ε) 
Max = 3.89ε (Mean = 0.824ε) 
Elliptic Integral Jacobi Zeta: Large Phi Values 
Max = 0ε (Mean = 0ε) 
Max = 2.92ε (Mean = 0.951ε) 
Max = 3.05ε (Mean = 1.13ε) 
Max = 2.52ε (Mean = 0.977ε) 
The tests use a mixture of spot test values calculated using values calculated at Wolfram Alpha, and random test data generated using MPFR at 1000bit precision and a deliberately naive implementation in terms of the Legendre integrals.
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)
A simple example comparing use of Wolfram Alpha with Boost.Math (including much higher precision using Boost.Multiprecision) is jacobi_zeta_example.cpp.