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280 lines
7.5 KiB
Plaintext
280 lines
7.5 KiB
Plaintext
[/ math.qbk
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Copyright 2006 Hubert Holin and John Maddock.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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[def __form1 [^\[0;+'''∞'''\[]]
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[def __form2 [^\]-'''∞''';+1\[]]
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[def __form3 [^\]-'''∞''';-1\[]]
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[def __form4 [^\]+1;+'''∞'''\[]]
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[def __form5 [^\[-1;-1+'''ε'''\[]]
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[def __form6 '''ε''']
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[def __form7 [^\]+1-'''ε''';+1\]]]
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[def __effects [*Effects: ]]
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[def __formula [*Formula: ]]
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[def __exm1 '''<code>e<superscript>x</superscript> - 1</code>'''[space]]
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[def __ex '''<code>e<superscript>x</superscript></code>''']
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[def __te '''2ε''']
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[section:inv_hyper Inverse Hyperbolic Functions]
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[section:inv_hyper_over Inverse Hyperbolic Functions Overview]
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The exponential funtion is defined, for all objects for which this makes sense,
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as the power series
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[equation special_functions_blurb1],
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with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]].
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In particular, the exponential function is well defined for real numbers,
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complex number, quaternions, octonions, and matrices of complex numbers,
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among others.
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[: ['[*Graph of exp on R]] ]
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[: [$../graphs/exp_on_r.png] ]
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[: ['[*Real and Imaginary parts of exp on C]]]
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[: [$../graphs/im_exp_on_c.png]]
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The hyperbolic functions are defined as power series which
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can be computed (for reals, complex, quaternions and octonions) as:
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Hyperbolic cosine: [equation special_functions_blurb5]
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Hyperbolic sine: [equation special_functions_blurb6]
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Hyperbolic tangent: [equation special_functions_blurb7]
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[: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]]
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[: [$../graphs/trigonometric.png]]
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[: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]]
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[: [$../graphs/hyperbolic.png]]
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The hyperbolic sine is one to one on the set of real numbers,
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with range the full set of reals, while the hyperbolic tangent is
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also one to one on the set of real numbers but with range __form1, and
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therefore both have inverses. The hyperbolic cosine is one to one from __form2
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onto __form3 (and from __form4 onto __form3); the inverse function we use
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here is defined on __form3 with range __form2.
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The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
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and can be computed as [equation special_functions_blurb15].
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The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
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and can be computed (for __form5) as [equation special_functions_blurb17].
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The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
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and can be computed as [equation special_functions_blurb18].
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[endsect]
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[section:acosh acosh]
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``
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#include <boost/math/special_functions/acosh.hpp>
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``
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template<class T>
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``__sf_result`` acosh(const T x);
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template<class T, class ``__Policy``>
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``__sf_result`` acosh(const T x, const ``__Policy``&);
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Computes the reciprocal of (the restriction to the range of __form1)
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[link math_toolkit.inv_hyper.inv_hyper_over
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the hyperbolic cosine function], at x. Values returned are positive.
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If x is in the range __form2 then returns the result of __domain_error.
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The return type of this function is computed using the __arg_promotion_rules:
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the return type is `double` when T is an integer type, and T otherwise.
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[optional_policy]
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[graph acosh]
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[h4 Accuracy]
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Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
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[h4 Testing]
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This function is tested using a combination of random test values designed to give
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full function coverage computed at high precision using the "naive" formula:
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[equation acosh1]
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along with a selection of sanity check values
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computed using functions.wolfram.com to at least 50 decimal digits.
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[h4 Implementation]
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For sufficiently large x, we can use the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/
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approximation]:
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[equation acosh2]
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For x sufficiently close to 1 we can use the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/
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approximation]:
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[equation acosh4]
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Otherwise for x close to 1 we can use the following rearrangement of the
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primary definition to preserve accuracy:
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[equation acosh3]
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Otherwise the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/
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primary definition] is used:
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[equation acosh1]
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[endsect]
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[section:asinh asinh]
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``
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#include <boost/math/special_functions/asinh.hpp>
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``
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template<class T>
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``__sf_result`` asinh(const T x);
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template<class T, class ``__Policy``>
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``__sf_result`` asinh(const T x, const ``__Policy``&);
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Computes the reciprocal of
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[link math_toolkit.inv_hyper.inv_hyper_over
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the hyperbolic sine function].
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The return type of this function is computed using the __arg_promotion_rules:
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the return type is `double` when T is an integer type, and T otherwise.
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[graph asinh]
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[optional_policy]
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[h4 Accuracy]
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Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
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[h4 Testing]
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This function is tested using a combination of random test values designed to give
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full function coverage computed at high precision using the "naive" formula:
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[equation asinh1]
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along with a selection of sanity check values
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computed using functions.wolfram.com to at least 50 decimal digits.
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[h4 Implementation]
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For sufficiently large x we can use the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/06/01/0001/
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approximation]:
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[equation asinh2]
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While for very small x we can use the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/0001/
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approximation]:
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[equation asinh3]
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For 0.5 > x > [epsilon] the following rearrangement of the primary definition is used:
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[equation asinh4]
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Otherwise evalution is via the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/
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primary definition]:
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[equation asinh4]
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[endsect]
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[section:atanh atanh]
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``
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#include <boost/math/special_functions/atanh.hpp>
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``
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template<class T>
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``__sf_result`` atanh(const T x);
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template<class T, class ``__Policy``>
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``__sf_result`` atanh(const T x, const ``__Policy``&);
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Computes the reciprocal of
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[link math_toolkit.inv_hyper.inv_hyper_over
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the hyperbolic tangent function], at x.
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[optional_policy]
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If x is in the range
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__form3
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or in the range
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__form4
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then returns the result of __domain_error.
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If x is in the range
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__form5,
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then the result of -__overflow_error is returned, with
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__form6[space]
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denoting numeric_limits<T>::epsilon().
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If x is in the range
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__form7,
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then the result of __overflow_error is returned, with
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__form6[space]
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denoting
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numeric_limits<T>::epsilon().
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The return type of this function is computed using the __arg_promotion_rules:
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the return type is `double` when T is an integer type, and T otherwise.
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[graph atanh]
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[h4 Accuracy]
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Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
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[h4 Testing]
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This function is tested using a combination of random test values designed to give
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full function coverage computed at high precision using the "naive" formula:
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[equation atanh1]
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along with a selection of sanity check values
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computed using functions.wolfram.com to at least 50 decimal digits.
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[h4 Implementation]
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For sufficiently small x we can use the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/ approximation]:
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[equation atanh2]
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Otherwise the
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[@http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/ primary definition]:
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[equation atanh1]
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or its equivalent form:
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[equation atanh3]
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is used.
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[endsect]
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[endsect]
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