[/
Copyright (c) 2006 Xiaogang Zhang
Copyright (c) 2006 John Maddock
Use, modification and distribution are subject to the
Boost Software License, Version 1.0. (See accompanying file
LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
]

[section:ellint_1 Elliptic Integrals of the First Kind - Legendre Form]

[heading Synopsis]

``
  #include <boost/math/special_functions/ellint_1.hpp>
``

  namespace boost { namespace math {

  template <class T1, class T2>
  ``__sf_result`` ellint_1(T1 k, T2 phi);

  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&);

  template <class T>
  ``__sf_result`` ellint_1(T k);

  template <class T, class ``__Policy``>
  ``__sf_result`` ellint_1(T k, const ``__Policy``&);

  }} // namespaces
  
[heading Description]

These two functions evaluate the incomplete elliptic integral of the first kind
['F([phi], k)] and its complete counterpart ['K(k) = F([pi]/2, k)].

[graph ellint_1]

The return type of these functions is computed using the __arg_pomotion_rules
when T1 and T2 are different types: when they are the same type then the result
is the same type as the arguments.

  template <class T1, class T2>
  ``__sf_result`` ellint_1(T1 k, T2 phi);
  
  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&);
  
Returns the incomplete elliptic integral of the first kind ['F([phi], k)]:

[equation ellint2]

Requires -1 <= k <= 1, otherwise returns the result of __domain_error.

[optional_policy]

  template <class T>
  ``__sf_result`` ellint_1(T k);
  
  template <class T>
  ``__sf_result`` ellint_1(T k, const ``__Policy``&);
  
Returns the complete elliptic integral of the first kind ['K(k)]:

[equation ellint6]

Requires -1 <= k <= 1, otherwise returns the result of __domain_error.

[optional_policy]

[heading Accuracy]

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 __zero_error.  All values
are relative errors in units of epsilon.

[table Errors Rates in the Elliptic Integrals of the First Kind
[[Significand Size] [Platform and Compiler] [F([phi], k)] [K(k)] ]
[[53] [Win32 / Visual C++ 8.0] [Peak=3 Mean=0.8] [Peak=1.8 Mean=0.7] ]
[[64] [Red Hat Linux / G++ 3.4] [Peak=2.6 Mean=1.7] [Peak=2.2 Mean=1.8]  ]
[[113] [HP-UX / HP aCC 6] [Peak=4.6 Mean=1.5] [Peak=3.7 Mean=1.5] ]
]


[heading Testing]

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

[heading Implementation]

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

[equation ellint19]

and

[equation ellint20]


[endsect]

[section:ellint_2 Elliptic Integrals of the Second Kind - Legendre Form]

[heading Synopsis]

``
  #include <boost/math/special_functions/ellint_2.hpp>
``

  namespace boost { namespace math {

  template <class T1, class T2>
  ``__sf_result`` ellint_2(T1 k, T2 phi);

  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&);

  template <class T>
  ``__sf_result`` ellint_2(T k);

  template <class T, class ``__Policy``>
  ``__sf_result`` ellint_2(T k, const ``__Policy``&);

  }} // namespaces
  
[heading Description]

These two functions evaluate the incomplete elliptic integral of the second kind
['E([phi], k)] and its complete counterpart ['E(k) = E([pi]/2, k)].

[graph ellint_2]

The return type of these functions is computed using the __arg_pomotion_rules
when T1 and T2 are different types: when they are the same type then the result
is the same type as the arguments.

  template <class T1, class T2>
  ``__sf_result`` ellint_2(T1 k, T2 phi);
  
  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&);
  
Returns the incomplete elliptic integral of the second kind ['E([phi], k)]:

[equation ellint3]

Requires -1 <= k <= 1, otherwise returns the result of __domain_error.

[optional_policy]

  template <class T>
  ``__sf_result`` ellint_2(T k);
  
  template <class T>
  ``__sf_result`` ellint_2(T k, const ``__Policy``&);
  
Returns the complete elliptic integral of the second kind ['E(k)]:

[equation ellint7]

Requires -1 <= k <= 1, otherwise returns the result of __domain_error.

[optional_policy]

[heading Accuracy]

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 __zero_error.  All values
are relative errors in units of epsilon.

[table Errors Rates in the Elliptic Integrals of the Second Kind
[[Significand Size] [Platform and Compiler] [F([phi], k)] [K(k)] ]
[[53] [Win32 / Visual C++ 8.0] [Peak=4.6 Mean=1.2] [Peak=3.5 Mean=1.0] ]
[[64] [Red Hat Linux / G++ 3.4] [Peak=4.3 Mean=1.1] [Peak=4.6 Mean=1.2]  ]
[[113] [HP-UX / HP aCC 6] [Peak=5.8 Mean=2.2] [Peak=10.8 Mean=2.3] ]
]


[heading Testing]

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

[heading Implementation]

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

[equation ellint21]

and

[equation ellint22]


[endsect]

[section:ellint_3 Elliptic Integrals of the Third Kind - Legendre Form]

[heading Synopsis]

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

  namespace boost { namespace math {

  template <class T1, class T2, class T3>
  ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi);

  template <class T1, class T2, class T3, class ``__Policy``>
  ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&);

  template <class T1, class T2>
  ``__sf_result`` ellint_3(T1 k, T2 n);

  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_3(T1 k, T2 n, const ``__Policy``&);

  }} // namespaces
  
[heading Description]

These two functions evaluate the incomplete elliptic integral of the third kind
['[Pi](n, [phi], k)] and its complete counterpart ['[Pi](n, k) = E(n, [pi]/2, k)].

[graph ellint_3]

The return type of these functions 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.

  template <class T1, class T2, class T3>
  ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi);
  
  template <class T1, class T2, class T3, class ``__Policy``>
  ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&);
  
Returns the incomplete elliptic integral of the third kind ['[Pi](n, [phi], k)]:

[equation ellint4]

Requires ['-1 <= k <= 1] and ['n < 1/sin[super 2]([phi])], otherwise 
returns the result of __domain_error (outside this range the result 
would be complex).

[optional_policy]

  template <class T1, class T2>
  ``__sf_result`` ellint_3(T1 k, T2 n);
  
  template <class T1, class T2, class ``__Policy``>
  ``__sf_result`` ellint_3(T1 k, T2 n, const ``__Policy``&);
  
Returns the complete elliptic integral of the first kind ['[Pi](n, k)]:

[equation ellint8]

Requires ['-1 <= k <= 1] and ['n < 1], otherwise returns the 
result of __domain_error (outside this range the result would be complex).

[optional_policy]

[heading Accuracy]

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 __zero_error.  All values
are relative errors in units of epsilon.

[table Errors Rates in the Elliptic Integrals of the Third Kind
[[Significand Size] [Platform and Compiler] [[Pi](n, [phi], k)] [[Pi](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] ]
]


[heading Testing]

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

[heading Implementation]

The implementation for [Pi](n, [phi], k) first siphons off the special cases:

['[Pi](0, [phi], k) = F([phi], k)]

['[Pi](n, [pi]/2, k) = [Pi](n, k)]

and

[equation ellint23]

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

[equation ellint24]

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

Then the relations:

['[Pi](n, -[phi], k) = -[Pi](n, [phi], k)]

['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) + 2m[Pi](n, k) ; n <= 1]

['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) ; n > 1] 
[footnote 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 * [Pi](n, k)] terms cancel out as the
derivative alternates between +[infin] and -[infin].]

are used to move [phi][space] to the range \[0, [pi]\/2\].

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

[equation ellint25]

and

[equation ellint26]

[endsect]