...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

#include <boost/math/special_functions/ellint_3.hpp>

namespace boost { namespace math { template <class T1, class T2, class T3>calculated-result-typeellint_3(T1 k, T2 n, T3 phi); template <class T1, class T2, class T3, class Policy>calculated-result-typeellint_3(T1 k, T2 n, T3 phi, const Policy&); template <class T1, class T2>calculated-result-typeellint_3(T1 k, T2 n); template <class T1, class T2, class Policy>calculated-result-typeellint_3(T1 k, T2 n, const Policy&); }} // namespaces

These two functions evaluate the incomplete elliptic integral of the third
kind *Π(n, φ, k)* and its complete counterpart *Π(n,
k) = E(n, 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, class T3>calculated-result-typeellint_3(T1 k, T2 n, T3 phi); template <class T1, class T2, class T3, class Policy>calculated-result-typeellint_3(T1 k, T2 n, T3 phi, const Policy&);

Returns the incomplete elliptic integral of the third kind *Π(n,
φ, k)*:

Requires *-1 <= k <= 1* and *n < 1/sin ^{2}(φ)*,
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, class T2>calculated-result-typeellint_3(T1 k, T2 n); template <class T1, class T2, class Policy>calculated-result-typeellint_3(T1 k, T2 n, const Policy&);

Returns the complete elliptic integral of the first kind *Π(n,
k)*:

Requires *-1 <= k <= 1* and *n < 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.

These functions are computed using only basic arithmetic operations, so there isn't much variation in accuracy over differing platforms. Note that only results for the widest floating point type on the system are given as narrower types have effectively zero error. All values are relative errors in units of epsilon.

**Table 43. Errors Rates in the Elliptic Integrals of the
Third Kind**

Significand Size |
Platform and Compiler |
Π(n, φ, k) |
Π(n, k) |
---|---|---|---|

53 |
Win32 / Visual C++ 8.0 |
Peak=29 Mean=2.2 |
Peak=3 Mean=0.8 |

64 |
Red Hat Linux / G++ 3.4 |
Peak=14 Mean=1.3 |
Peak=2.3 Mean=0.8 |

113 |
HP-UX / HP aCC 6 |
Peak=10 Mean=1.4 |
Peak=4.2 Mean=1.1 |

The tests use a mixture of spot test values calculated using the online calculator at functions.wolfram.com, and random test data generated using NTL::RR at 1000-bit precision and this implementation.

The implementation for Π(n, φ, k) first siphons off the special cases:

*Π(0, φ, k) = F(φ, k)*

*Π(n, π/2, k) = Π(n, k)*

and

Then if n < 0 the relations (A&S 17.7.15/16):

are used to shift *n* to the range [0, 1].

Then the relations:

*Π(n, -φ, k) = -Π(n, φ, k)*

*Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k) ; n <= 1*

*Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1*
^{[1]}

are used to move φ to the range [0, π/2].

The functions are then implemented in terms of Carlson's integrals using the relations:

and

^{[1] }
I haven't been able to find a literature reference for this relation,
but it appears to be the convention used by Mathematica. Intuitively
the first *2 * m * Π(n, k)* terms cancel out as the
derivative alternates between +∞ and -∞.