ceph/src/boost/libs/math/doc/sf/inv_hyper.qbk

   1 [/ math.qbk
   2   Copyright 2006 Hubert Holin and John Maddock.
   3   Distributed under the Boost Software License, Version 1.0.
   4   (See accompanying file LICENSE_1_0.txt or copy at
   5   http://www.boost.org/LICENSE_1_0.txt).
   6 ]
   7
   8 [def __form1 [^\[0;+'''&#x221E;'''\[]]
   9 [def __form2 [^\]-'''&#x221E;''';+1\[]]
  10 [def __form3 [^\]-'''&#x221E;''';-1\[]]
  11 [def __form4 [^\]+1;+'''&#x221E;'''\[]]
  12 [def __form5 [^\[-1;-1+'''&#x03B5;'''\[]]
  13 [def __form6 '''&#x03B5;''']
  14 [def __form7 [^\]+1-'''&#x03B5;''';+1\]]]
  15
  16 [def __effects [*Effects: ]]
  17 [def __formula [*Formula: ]]
  18 [def __exm1 '''<code>e<superscript>x</superscript> - 1</code>'''[space]]
  19 [def __ex '''<code>e<superscript>x</superscript></code>''']
  20 [def __te '''2&#x03B5;''']
  21
  22 [section:inv_hyper Inverse Hyperbolic Functions]
  23
  24 [section:inv_hyper_over Inverse Hyperbolic Functions Overview]
  25
  26 The exponential funtion is defined, for all objects for which this makes sense,
  27 as the power series
  28 [equation special_functions_blurb1],
  29 with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]].
  30 In particular, the exponential function is well defined for real numbers,
  31 complex number, quaternions, octonions, and matrices of complex numbers,
  32 among others.
  33
  34 [: ['[*Graph of exp on R]] ]
  35
  36 [: [$../graphs/exp_on_r.png] ]
  37
  38 [: ['[*Real and Imaginary parts of exp on C]]]
  39 [: [$../graphs/im_exp_on_c.png]]
  40
  41 The hyperbolic functions are defined as power series which
  42 can be computed (for reals, complex, quaternions and octonions) as:
  43
  44 Hyperbolic cosine: [equation special_functions_blurb5]
  45
  46 Hyperbolic sine: [equation special_functions_blurb6]
  47
  48 Hyperbolic tangent: [equation special_functions_blurb7]
  49
  50 [: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]]
  51 [: [$../graphs/trigonometric.png]]
  52
  53 [: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]]
  54 [: [$../graphs/hyperbolic.png]]
  55
  56 The hyperbolic sine is one to one on the set of real numbers,
  57 with range the full set of reals, while the hyperbolic tangent is
  58 also one to one on the set of real numbers but with range __form1, and
  59 therefore both have inverses. The hyperbolic cosine is one to one from __form2
  60 onto __form3 (and from __form4 onto __form3); the inverse function we use
  61 here is defined on __form3 with range __form2.
  62
  63 The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
  64 and can be computed as [equation special_functions_blurb15].
  65
  66 The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
  67 and can be computed (for __form5) as [equation special_functions_blurb17].
  68
  69 The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
  70 and can be computed as [equation special_functions_blurb18].
  71
  72 [endsect]
  73
  74 [section:acosh acosh]
  75
  76 ``
  77 #include <boost/math/special_functions/acosh.hpp>
  78 ``
  79
  80    template<class T>
  81    ``__sf_result`` acosh(const T x);
  82
  83    template<class T, class ``__Policy``>
  84    ``__sf_result`` acosh(const T x, const ``__Policy``&);
  85
  86 Computes the reciprocal of (the restriction to the range of __form1)
  87 [link math_toolkit.inv_hyper.inv_hyper_over
  88 the hyperbolic cosine function], at x. Values returned are positive.
  89
  90 If x is in the range __form2 then returns the result of __domain_error.
  91
  92 The return type of this function is computed using the __arg_promotion_rules:
  93 the return type is `double` when T is an integer type, and T otherwise.
  94
  95 [optional_policy]
  96
  97 [graph acosh]
  98
  99 [h4 Accuracy]
 100
 101 Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
 102
 103 [h4 Testing]
 104
 105 This function is tested using a combination of random test values designed to give
 106 full function coverage computed at high precision using the "naive" formula:
 107
 108 [equation acosh1]
 109
 110 along with a selection of sanity check values
 111 computed using functions.wolfram.com to at least 50 decimal digits.
 112
 113 [h4 Implementation]
 114
 115 For sufficiently large x, we can use the
 116 [@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/
 117 approximation]:
 118
 119 [equation acosh2]
 120
 121 For x sufficiently close to 1 we can use the
 122 [@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/
 123 approximation]:
 124
 125 [equation acosh4]
 126
 127 Otherwise for x close to 1 we can use the following rearrangement of the
 128 primary definition to preserve accuracy:
 129
 130 [equation acosh3]
 131
 132 Otherwise the
 133 [@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/
 134 primary definition] is used:
 135
 136 [equation acosh1]
 137
 138 [endsect]
 139
 140 [section:asinh asinh]
 141
 142 ``
 143 #include <boost/math/special_functions/asinh.hpp>
 144 ``
 145
 146    template<class T>
 147    ``__sf_result`` asinh(const T x);
 148
 149    template<class T, class ``__Policy``>
 150    ``__sf_result`` asinh(const T x, const ``__Policy``&);
 151
 152 Computes the reciprocal of
 153 [link math_toolkit.inv_hyper.inv_hyper_over
 154 the hyperbolic sine function].
 155
 156 The return type of this function is computed using the __arg_promotion_rules:
 157 the return type is `double` when T is an integer type, and T otherwise.
 158
 159 [graph asinh]
 160
 161 [optional_policy]
 162
 163 [h4 Accuracy]
 164
 165 Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
 166
 167 [h4 Testing]
 168
 169 This function is tested using a combination of random test values designed to give
 170 full function coverage computed at high precision using the "naive" formula:
 171
 172 [equation asinh1]
 173
 174 along with a selection of sanity check values
 175 computed using functions.wolfram.com to at least 50 decimal digits.
 176
 177 [h4 Implementation]
 178
 179 For sufficiently large x we can use the
 180 [@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/06/01/0001/
 181 approximation]:
 182
 183 [equation asinh2]
 184
 185 While for very small x we can use the
 186 [@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/0001/
 187 approximation]:
 188
 189 [equation asinh3]
 190
 191 For 0.5 > x > [epsilon] the following rearrangement of the primary definition is used:
 192
 193 [equation asinh4]
 194
 195 Otherwise evalution is via the
 196 [@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/
 197 primary definition]:
 198
 199 [equation asinh4]
 200
 201 [endsect]
 202
 203 [section:atanh atanh]
 204
 205 ``
 206 #include <boost/math/special_functions/atanh.hpp>
 207 ``
 208
 209    template<class T>
 210    ``__sf_result`` atanh(const T x);
 211
 212    template<class T, class ``__Policy``>
 213    ``__sf_result`` atanh(const T x, const ``__Policy``&);
 214
 215 Computes the reciprocal of
 216 [link math_toolkit.inv_hyper.inv_hyper_over
 217 the hyperbolic tangent function], at x.
 218
 219 [optional_policy]
 220
 221 If x is in the range
 222 __form3
 223 or in the range
 224 __form4
 225 then returns the result of __domain_error.
 226
 227 If x is in the range
 228 __form5,
 229 then the result of -__overflow_error is returned, with
 230 __form6[space]
 231 denoting numeric_limits<T>::epsilon().
 232
 233 If x is in the range
 234 __form7,
 235 then the result of __overflow_error is returned, with
 236 __form6[space]
 237 denoting
 238 numeric_limits<T>::epsilon().
 239
 240 The return type of this function is computed using the __arg_promotion_rules:
 241 the return type is `double` when T is an integer type, and T otherwise.
 242
 243 [graph atanh]
 244
 245 [h4 Accuracy]
 246
 247 Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
 248
 249 [h4 Testing]
 250
 251 This function is tested using a combination of random test values designed to give
 252 full function coverage computed at high precision using the "naive" formula:
 253
 254 [equation atanh1]
 255
 256 along with a selection of sanity check values
 257 computed using functions.wolfram.com to at least 50 decimal digits.
 258
 259 [h4 Implementation]
 260
 261 For sufficiently small x we can use the
 262 [@http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/ approximation]:
 263
 264 [equation atanh2]
 265
 266 Otherwise the
 267 [@http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/ primary definition]:
 268
 269 [equation atanh1]
 270
 271 or its equivalent form:
 272
 273 [equation atanh3]
 274
 275 is used.
 276
 277 [endsect]
 278
 279 [endsect]