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### Heuman Lambda Function

##### Synopsis

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

    namespace boost { namespace math {

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

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

    }} // namespaces

##### Description

This function evaluates the Heuman Lambda Function Λ0(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 1 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 8.69. Error rates for heuman_lambda

| GNU C++ version 7.1.0 linux double | GNU C++ version 7.1.0 linux long double | Sun compiler version 0x5150 Sun Solaris long double | Microsoft Visual C++ version 14.1 Win32 double
--- | --- | --- | --- | ---
Elliptic Integral Jacobi Zeta: Mathworld Data | Max = 0ε (Mean = 0ε) | Max = 1.89ε (Mean = 0.887ε) | Max = 1.89ε (Mean = 0.887ε) | Max = 1.08ε (Mean = 0.734ε)
Elliptic Integral Heuman Lambda: Random Data | Max = 0ε (Mean = 0ε) | Max = 3.82ε (Mean = 0.609ε) | Max = 3.82ε (Mean = 0.608ε) | Max = 2.12ε (Mean = 0.588ε)

##### Testing

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

##### Implementation

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

This relation fails for |φ| >= π/2 in which case the definition in terms of the Jacobi Zeta is used.