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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>acosh</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
<link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions">
<link rel="prev" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">
<link rel="next" href="asinh.html" title="asinh">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
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<div class="spirit-nav">
<a accesskey="p" href="inv_hyper_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="asinh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">acosh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Computes the reciprocal of (the restriction to the range of <code class="literal">[0;+&#8734;[</code>)
<a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic cosine
function</a>, at x. Values returned are positive.
</p>
<p>
If x is in the range <code class="literal">]-&#8734;;+1[</code> then returns the result of
<a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/acosh.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h0"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.accuracy">Accuracy</a>
</h5>
<p>
Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
</p>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h1"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.testing">Testing</a>
</h5>
<p>
This function is tested using a combination of random test values designed
to give full function coverage computed at high precision using the "naive"
formula:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>
</p>
<p>
along with a selection of sanity check values computed using functions.wolfram.com
to at least 50 decimal digits.
</p>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h2"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.implementation">Implementation</a>
</h5>
<p>
For sufficiently large x, we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/acosh2.svg"></span>
</p>
<p>
For x sufficiently close to 1 we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/acosh4.svg"></span>
</p>
<p>
Otherwise for x close to 1 we can use the following rearrangement of the
primary definition to preserve accuracy:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/acosh3.svg"></span>
</p>
<p>
Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary
definition</a> is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="inv_hyper_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="asinh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
libs/math/doc/html/math_toolkit/inv_hyper/asinh.html
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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>asinh</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
<link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions">
<link rel="prev" href="acosh.html" title="acosh">
<link rel="next" href="atanh.html" title="atanh">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="acosh.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="atanh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.asinh"></a><a class="link" href="asinh.html" title="asinh">asinh</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">asinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">asinh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">asinh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the
hyperbolic sine function</a>.
</p>
<p>
The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/asinh.svg" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
<a name="math_toolkit.inv_hyper.asinh.h0"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.asinh.accuracy"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.accuracy">Accuracy</a>
</h5>
<p>
Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
</p>
<h5>
<a name="math_toolkit.inv_hyper.asinh.h1"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.asinh.testing"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.testing">Testing</a>
</h5>
<p>
This function is tested using a combination of random test values designed
to give full function coverage computed at high precision using the "naive"
formula:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/asinh1.svg"></span>
</p>
<p>
along with a selection of sanity check values computed using functions.wolfram.com
to at least 50 decimal digits.
</p>
<h5>
<a name="math_toolkit.inv_hyper.asinh.h2"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.asinh.implementation"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.implementation">Implementation</a>
</h5>
<p>
For sufficiently large x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/06/01/0001/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/asinh2.svg"></span>
</p>
<p>
While for very small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/0001/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/asinh3.svg"></span>
</p>
<p>
For 0.5 &gt; x &gt; &#949; the following rearrangement of the primary definition
is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span>
</p>
<p>
Otherwise evalution is via the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/" target="_top">primary
definition</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span>
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
</tr></table>
<hr>
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<a accesskey="p" href="acosh.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="atanh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
libs/math/doc/html/math_toolkit/inv_hyper/atanh.html
+134
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<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>atanh</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
<link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions">
<link rel="prev" href="asinh.html" title="asinh">
<link rel="next" href="../owens_t.html" title="Owen's T function">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
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</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">atanh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">atanh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">atanh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the
hyperbolic tangent function</a>, at x.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
If x is in the range <code class="literal">]-&#8734;;-1[</code> or in the range <code class="literal">]+1;+&#8734;[</code>
then returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
If x is in the range <code class="literal">[-1;-1+&#949;[</code>, then the result of -<a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
is returned, with &#949; &#160;
denoting numeric_limits&lt;T&gt;::epsilon().
</p>
<p>
If x is in the range <code class="literal">]+1-&#949;;+1]</code>, then the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
is returned, with &#949; &#160;
denoting numeric_limits&lt;T&gt;::epsilon().
</p>
<p>
The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/atanh.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.inv_hyper.atanh.h0"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.accuracy"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.accuracy">Accuracy</a>
</h5>
<p>
Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
</p>
<h5>
<a name="math_toolkit.inv_hyper.atanh.h1"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.testing"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.testing">Testing</a>
</h5>
<p>
This function is tested using a combination of random test values designed
to give full function coverage computed at high precision using the "naive"
formula:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span>
</p>
<p>
along with a selection of sanity check values computed using functions.wolfram.com
to at least 50 decimal digits.
</p>
<h5>
<a name="math_toolkit.inv_hyper.atanh.h2"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.implementation"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.implementation">Implementation</a>
</h5>
<p>
For sufficiently small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh2.svg"></span>
</p>
<p>
Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/" target="_top">primary
definition</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span>
</p>
<p>
or its equivalent form:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh3.svg"></span>
</p>
<p>
is used.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.inv_hyper_over"></a><a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">Inverse Hyperbolic
Functions Overview</a>
</h3></div></div></div>
<p>
The exponential funtion is defined, for all objects for which this makes
sense, as the power series <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb1.svg"></span>,
with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and <span class="emphasis"><em><code class="literal">0!
= 1</code></em></span> by definition) being the factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>.
In particular, the exponential function is well defined for real numbers,
complex number, quaternions, octonions, and matrices of complex numbers,
among others.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/exp_on_r.png"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/im_exp_on_c.png"></span>
</p></blockquote></div>
<p>
The hyperbolic functions are defined as power series which can be computed
(for reals, complex, quaternions and octonions) as:
</p>
<p>
Hyperbolic cosine: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb5.svg"></span>
</p>
<p>
Hyperbolic sine: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb6.svg"></span>
</p>
<p>
Hyperbolic tangent: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb7.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple;
sin: red; tan: blue)</strong></span></em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/trigonometric.png"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple;
sinh: red; tanh: blue)</strong></span></em></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/hyperbolic.png"></span>
</p></blockquote></div>
<p>
The hyperbolic sine is one to one on the set of real numbers, with range
the full set of reals, while the hyperbolic tangent is also one to one on
the set of real numbers but with range <code class="literal">[0;+&#8734;[</code>, and therefore
both have inverses. The hyperbolic cosine is one to one from <code class="literal">]-&#8734;;+1[</code>
onto <code class="literal">]-&#8734;;-1[</code> (and from <code class="literal">]+1;+&#8734;[</code> onto
<code class="literal">]-&#8734;;-1[</code>); the inverse function we use here is defined on
<code class="literal">]-&#8734;;-1[</code> with range <code class="literal">]-&#8734;;+1[</code>.
</p>
<p>
The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
and can be computed as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb15.svg"></span>.
</p>
<p>
The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
and can be computed (for <code class="literal">[-1;-1+&#949;[</code>) as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb17.svg"></span>.
</p>
<p>
The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
and can be computed as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb18.svg"></span>.
</p>
</div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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