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Brian Silverman	32ed54e	2018-08-04 23:37:28 -0700	[diff] [blame^]	1	[def __effects [*Effects: ]]
				2	[def __formula [*Formula: ]]
				3	[def __exm1 '''<code>e<superscript>x</superscript> - 1</code>''']
				4	[def __ex '''<code>e<superscript>x</superscript></code>''']
				5	[def __te '''2ε''']
				6
				7	[mathpart inverse_complex..Complex Number Functions]
				8
				9	The following complex number algorithms are the inverses of trigonometric functions currently
				10	present in the C++ standard. Equivalents to these functions are part of the C99 standard, and
				11	are part of the [tr1].
				12
				13	[section:complex_implementation Implementation and Accuracy]
				14
				15	Although there are deceptively simple formulae available for all of these functions, a naive
				16	implementation that used these formulae would fail catastrophically for some input
				17	values. The Boost versions of these functions have been implemented using the methodology
				18	described in "Implementing the Complex Arcsine and Arccosine Functions Using Exception Handling"
				19	by T. E. Hull Thomas F. Fairgrieve and Ping Tak Peter Tang, ACM Transactions on Mathematical Software,
				20	Vol. 23, No. 3, September 1997. This means that the functions are well defined over the entire
				21	complex number range, and produce accurate values even at the extremes of that range, where as a naive
				22	formula would cause overflow or underflow to occur during the calculation, even though the result is
				23	actually a representable value. The maximum theoretical relative error for all of these functions
				24	is less than 9.5[epsilon] for every machine-representable point in the complex plane. Please refer to
				25	comments in the header files themselves and to the above mentioned paper for more information
				26	on the implementation methodology.
				27
				28	[endsect]
				29
				30	[section:asin asin]
				31
				32	[h4 Header:]
				33
				34	#include <boost/math/complex/asin.hpp>
				35
				36	[h4 Synopsis:]
				37
				38	template<class T>
				39	std::complex<T> asin(const std::complex<T>& z);
				40
				41	__effects returns the inverse sine of the complex number z.
				42
				43	__formula [$../images/asin.png]
				44
				45	[endsect]
				46
				47	[section:acos acos]
				48
				49	[h4 Header:]
				50
				51	#include <boost/math/complex/acos.hpp>
				52
				53	[h4 Synopsis:]
				54
				55	template<class T>
				56	std::complex<T> acos(const std::complex<T>& z);
				57
				58	__effects returns the inverse cosine of the complex number z.
				59
				60	__formula [$../images/acos.png]
				61
				62	[endsect]
				63
				64	[section:atan atan]
				65
				66	[h4 Header:]
				67
				68	#include <boost/math/complex/atan.hpp>
				69
				70	[h4 Synopsis:]
				71
				72	template<class T>
				73	std::complex<T> atan(const std::complex<T>& z);
				74
				75	__effects returns the inverse tangent of the complex number z.
				76
				77	__formula [$../images/atan.png]
				78
				79	[endsect]
				80
				81	[section:asinh asinh]
				82
				83	[h4 Header:]
				84
				85	#include <boost/math/complex/asinh.hpp>
				86
				87	[h4 Synopsis:]
				88
				89	template<class T>
				90	std::complex<T> asinh(const std::complex<T>& z);
				91
				92	__effects returns the inverse hyperbolic sine of the complex number z.
				93
				94	__formula [$../images/asinh.png]
				95
				96	[endsect]
				97
				98	[section:acosh acosh]
				99
				100	[h4 Header:]
				101
				102	#include <boost/math/complex/acosh.hpp>
				103
				104	[h4 Synopsis:]
				105
				106	template<class T>
				107	std::complex<T> acosh(const std::complex<T>& z);
				108
				109	__effects returns the inverse hyperbolic cosine of the complex number z.
				110
				111	__formula [$../images/acosh.png]
				112
				113	[endsect]
				114
				115	[section:atanh atanh]
				116
				117	[h4 Header:]
				118
				119	#include <boost/math/complex/atanh.hpp>
				120
				121	[h4 Synopsis:]
				122
				123	template<class T>
				124	std::complex<T> atanh(const std::complex<T>& z);
				125
				126	__effects returns the inverse hyperbolic tangent of the complex number z.
				127
				128	__formula [$../images/atanh.png]
				129
				130	[endsect]
				131
				132	[section:complex_history History]
				133
				134	* 2005/12/17: Added support for platforms with no meaningful numeric_limits<>::infinity().
				135	* 2005/12/01: Initial version, added as part of the TR1 library.
				136
				137
				138	[endsect]
				139
				140	[endmathpart]
				141
				142	[/
				143	Copyright 2008, 2009 John Maddock and Paul A. Bristow.
				144	Distributed under the Boost Software License, Version 1.0.
				145	(See accompanying file LICENSE_1_0.txt or copy at
				146	http://www.boost.org/LICENSE_1_0.txt).
				147	]
				148
				149