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### Elliptic Integral D - Legendre Form

##### Synopsis

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

    namespace boost { namespace math {

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

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

    template <class T1>
    BOOST_MATH_GPU_ENABLED calculated-result-type ellint_d(T1 k);

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

    }} // namespaces

##### Description

These two functions evaluate the incomplete elliptic integral D(φ, k) and its complete counterpart D(k) = D(π/2, k).

The return type of these functions 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.

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

    template <class T1, class T2, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type ellint_3(T1 k, T2 phi, const Policy&);

Returns the incomplete elliptic integral:

Requires k2sin2(phi) < 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.

    template <class T1>
    BOOST_MATH_GPU_ENABLED calculated-result-type ellint_d(T1 k);

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

Returns the complete elliptic integral D(k) = D(π/2, k)

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.

##### 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 8.66. Error rates for ellint_d (complete)

| 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 E: Mathworld Data | Max = 0.637ε (Mean = 0.368ε) | Max = 1.27ε (Mean = 0.735ε) | Max = 1.27ε (Mean = 0.735ε) | Max = 0.637ε (Mean = 0.368ε)
Elliptic Integral D: Random Data | Max = 0ε (Mean = 0ε) | Max = 1.27ε (Mean = 0.334ε) | Max = 1.27ε (Mean = 0.334ε) | Max = 1.27ε (Mean = 0.355ε)

Table 8.67. Error rates for ellint_d

| 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 E: Mathworld Data | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 0.862ε (Mean = 0.568ε)) | Max = 1.3ε (Mean = 0.813ε) | Max = 1.3ε (Mean = 0.813ε) | Max = 0.862ε (Mean = 0.457ε)
Elliptic Integral D: Random Data | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 3.01ε (Mean = 0.928ε)) | Max = 2.51ε (Mean = 0.883ε) | Max = 2.51ε (Mean = 0.883ε) | Max = 2.87ε (Mean = 0.805ε)

The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at `double` precision, and GCC-7.1/Ubuntu for `long double` and `__float128`.

##### 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 implementation for D(φ, k) first performs argument reduction using the relations:

> D(-φ, k) = -D(φ, k)

and

> D(nπ+φ, k) = 2nD(k) + D(φ, k)

to move φ to the range [0, π/2].

The functions are then implemented in terms of Carlson's integral RD using the relation: