[ceph.git] / ceph / src / boost / libs / math / doc / sf / bessel_spherical.qbk


[section:sph_bessel Spherical Bessel Functions of the First and Second Kinds]

[h4 Synopsis]

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

   template <class T1, class T2>
   ``__sf_result`` sph_bessel(unsigned v, T2 x);

   template <class T1, class T2, class ``__Policy``>
   ``__sf_result`` sph_bessel(unsigned v, T2 x, const ``__Policy``&);

   template <class T1, class T2>
   ``__sf_result`` sph_neumann(unsigned v, T2 x);
   
   template <class T1, class T2, class ``__Policy``>
   ``__sf_result`` sph_neumann(unsigned v, T2 x, const ``__Policy``&);
   
[h4 Description]

The functions __sph_bessel and __sph_neumann return the result of the
Spherical Bessel functions of the first and second kinds respectively:

sph_bessel(v, x) = j[sub v](x)

sph_neumann(v, x) = y[sub v](x) = n[sub v](x)

where:

[equation sbessel2]

The return type of these functions is computed using the __arg_promotion_rules
for the single argument type T.

[optional_policy]

The functions return the result of __domain_error whenever the result is
undefined or complex: this occurs when `x < 0`.

The j[sub v][space] function is cyclic like J[sub v][space] but differs
in its behaviour at the origin:

[graph sph_bessel]

Likewise y[sub v][space] is also cyclic for large x, but tends to -[infin][space]
for small /x/:

[graph sph_neumann]

[h4 Testing]

There are two sets of test values: spot values calculated using
[@http://functions.wolfram.com/ functions.wolfram.com],
and a much larger set of tests computed using
a simplified version of this implementation
(with all the special case handling removed).

[h4 Accuracy]

[table_sph_bessel]

[table_sph_neumann]

[h4 Implementation]

Other than error handling and a couple of special cases these functions
are implemented directly in terms of their definitions:

[equation sbessel2]

The special cases occur for:

j[sub 0][space]= __sinc_pi(x) = sin(x) / x

and for small ['x < 1], we can use the series:

[equation sbessel5]

which neatly avoids the problem of calculating 0/0 that can occur with the
main definition as x [rarr] 0.

[endsect]

[/ 
  Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
  Distributed under 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).
]
Commit	Line	Data
7c673cae FG	1
	2	[section:sph_bessel Spherical Bessel Functions of the First and Second Kinds]
	3
	4	[h4 Synopsis]
	5
	6	`#include <boost/math/special_functions/bessel.hpp>`
	7
	8	template <class T1, class T2>
	9	``__sf_result`` sph_bessel(unsigned v, T2 x);
	10
	11	template <class T1, class T2, class ``__Policy``>
	12	``__sf_result`` sph_bessel(unsigned v, T2 x, const ``__Policy``&);
	13
	14	template <class T1, class T2>
	15	``__sf_result`` sph_neumann(unsigned v, T2 x);
	16
	17	template <class T1, class T2, class ``__Policy``>
	18	``__sf_result`` sph_neumann(unsigned v, T2 x, const ``__Policy``&);
	19
	20	[h4 Description]
	21
	22	The functions __sph_bessel and __sph_neumann return the result of the
	23	Spherical Bessel functions of the first and second kinds respectively:
	24
	25	sph_bessel(v, x) = j[sub v](x)
	26
	27	sph_neumann(v, x) = y[sub v](x) = n[sub v](x)
	28
	29	where:
	30
	31	[equation sbessel2]
	32
	33	The return type of these functions is computed using the __arg_promotion_rules
	34	for the single argument type T.
	35
	36	[optional_policy]
	37
	38	The functions return the result of __domain_error whenever the result is
	39	undefined or complex: this occurs when `x < 0`.
	40
	41	The j[sub v][space] function is cyclic like J[sub v][space] but differs
	42	in its behaviour at the origin:
	43
	44	[graph sph_bessel]
	45
	46	Likewise y[sub v][space] is also cyclic for large x, but tends to -[infin][space]
	47	for small /x/:
	48
	49	[graph sph_neumann]
	50
	51	[h4 Testing]
	52
	53	There are two sets of test values: spot values calculated using
	54	[@http://functions.wolfram.com/ functions.wolfram.com],
	55	and a much larger set of tests computed using
	56	a simplified version of this implementation
	57	(with all the special case handling removed).
	58
	59	[h4 Accuracy]
	60
	61	[table_sph_bessel]
	62
	63	[table_sph_neumann]
	64
65	[h4 Implementation]
66
67	Other than error handling and a couple of special cases these functions
68	are implemented directly in terms of their definitions:
69
70	[equation sbessel2]
71
72	The special cases occur for:
73
74	j[sub 0][space]= __sinc_pi(x) = sin(x) / x
75
76	and for small ['x < 1], we can use the series:
77
78	[equation sbessel5]
79
80	which neatly avoids the problem of calculating 0/0 that can occur with the
81	main definition as x [rarr] 0.
82
83	[endsect]
84
85	[/
86	Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
87	Distributed under the Boost Software License, Version 1.0.
88	(See accompanying file LICENSE_1_0.txt or copy at
89	http://www.boost.org/LICENSE_1_0.txt).
90	]