1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Inverse Hyperbolic Functions Overview</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions"> 9<link rel="prev" href="../inv_hyper.html" title="Inverse Hyperbolic Functions"> 10<link rel="next" href="acosh.html" title="acosh"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../inv_hyper.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="acosh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<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 28 Functions Overview</a> 29</h3></div></div></div> 30<p> 31 The exponential function is defined, for all objects for which this makes 32 sense, as the power series 33 </p> 34<div class="blockquote"><blockquote class="blockquote"><p> 35 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb1.svg"></span> 36 37 </p></blockquote></div> 38<p> 39 with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and <span class="emphasis"><em><code class="literal">0! 40 = 1</code></em></span> by definition) being the factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>. 41 In particular, the exponential function is well defined for real numbers, 42 complex number, quaternions, octonions, and matrices of complex numbers, 43 among others. 44 </p> 45<div class="blockquote"><blockquote class="blockquote"><p> 46 <span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span> 47 </p></blockquote></div> 48<div class="blockquote"><blockquote class="blockquote"><p> 49 <span class="inlinemediaobject"><img src="../../../graphs/exp_on_r.png"></span> 50 </p></blockquote></div> 51<div class="blockquote"><blockquote class="blockquote"><p> 52 <span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span> 53 </p></blockquote></div> 54<div class="blockquote"><blockquote class="blockquote"><p> 55 <span class="inlinemediaobject"><img src="../../../graphs/im_exp_on_c.png"></span> 56 </p></blockquote></div> 57<p> 58 The hyperbolic functions are defined as power series which can be computed 59 (for reals, complex, quaternions and octonions) as: 60 </p> 61<p> 62 Hyperbolic cosine: 63 </p> 64<div class="blockquote"><blockquote class="blockquote"><p> 65 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb5.svg"></span> 66 67 </p></blockquote></div> 68<p> 69 Hyperbolic sine: 70 </p> 71<div class="blockquote"><blockquote class="blockquote"><p> 72 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb6.svg"></span> 73 74 </p></blockquote></div> 75<p> 76 Hyperbolic tangent: 77 </p> 78<div class="blockquote"><blockquote class="blockquote"><p> 79 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb7.svg"></span> 80 81 </p></blockquote></div> 82<div class="blockquote"><blockquote class="blockquote"><p> 83 <span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple; 84 sin: red; tan: blue)</strong></span></em></span> 85 </p></blockquote></div> 86<div class="blockquote"><blockquote class="blockquote"><p> 87 <span class="inlinemediaobject"><img src="../../../graphs/trigonometric.png"></span> 88 </p></blockquote></div> 89<div class="blockquote"><blockquote class="blockquote"><p> 90 <span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple; 91 sinh: red; tanh: blue)</strong></span></em></span> 92 </p></blockquote></div> 93<div class="blockquote"><blockquote class="blockquote"><p> 94 <span class="inlinemediaobject"><img src="../../../graphs/hyperbolic.png"></span> 95 </p></blockquote></div> 96<p> 97 The hyperbolic sine is one to one on the set of real numbers, with range 98 the full set of reals, while the hyperbolic tangent is also one to one on 99 the set of real numbers but with range <code class="literal">[0;+∞[</code>, and therefore 100 both have inverses. 101 </p> 102<p> 103 The hyperbolic cosine is one to one from <code class="literal">]-∞;+1[</code> onto 104 <code class="literal">]-∞;-1[</code> (and from <code class="literal">]+1;+∞[</code> onto <code class="literal">]-∞;-1[</code>). 105 </p> 106<p> 107 The inverse function we use here is defined on <code class="literal">]-∞;-1[</code> 108 with range <code class="literal">]-∞;+1[</code>. 109 </p> 110<p> 111 The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent, 112 and can be computed as 113 </p> 114<div class="blockquote"><blockquote class="blockquote"><p> 115 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb15.svg"></span> 116 117 </p></blockquote></div> 118<p> 119 The inverse of the hyperbolic sine is called the Argument hyperbolic sine, 120 and can be computed (for <code class="literal">[-1;-1+ε[</code>) as 121 </p> 122<div class="blockquote"><blockquote class="blockquote"><p> 123 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb17.svg"></span> 124 125 </p></blockquote></div> 126<p> 127 The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine, 128 and can be computed as 129 </p> 130<div class="blockquote"><blockquote class="blockquote"><p> 131 <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb18.svg"></span> 132 133 </p></blockquote></div> 134</div> 135<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 136<td align="left"></td> 137<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 138 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 139 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 140 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 141 Daryle Walker and Xiaogang Zhang<p> 142 Distributed under the Boost Software License, Version 1.0. (See accompanying 143 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>) 144 </p> 145</div></td> 146</tr></table> 147<hr> 148<div class="spirit-nav"> 149<a accesskey="p" href="../inv_hyper.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="acosh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 150</div> 151</body> 152</html> 153
