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| 4 | <title>Inverse Hyperbolic Functions Overview</title> | |
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| 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 funtion is defined, for all objects for which this makes | |
| 32 | sense, as the power series <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb1.svg"></span>, | |
| 33 | with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and <span class="emphasis"><em><code class="literal">0! | |
| 34 | = 1</code></em></span> by definition) being the factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>. | |
| 35 | In particular, the exponential function is well defined for real numbers, | |
| 36 | complex number, quaternions, octonions, and matrices of complex numbers, | |
| 37 | among others. | |
| 38 | </p> | |
| 39 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 40 | <span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span> | |
| 41 | </p></blockquote></div> | |
| 42 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 43 | <span class="inlinemediaobject"><img src="../../../graphs/exp_on_r.png"></span> | |
| 44 | </p></blockquote></div> | |
| 45 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 46 | <span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span> | |
| 47 | </p></blockquote></div> | |
| 48 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 49 | <span class="inlinemediaobject"><img src="../../../graphs/im_exp_on_c.png"></span> | |
| 50 | </p></blockquote></div> | |
| 51 | <p> | |
| 52 | The hyperbolic functions are defined as power series which can be computed | |
| 53 | (for reals, complex, quaternions and octonions) as: | |
| 54 | </p> | |
| 55 | <p> | |
| 56 | Hyperbolic cosine: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb5.svg"></span> | |
| 57 | </p> | |
| 58 | <p> | |
| 59 | Hyperbolic sine: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb6.svg"></span> | |
| 60 | </p> | |
| 61 | <p> | |
| 62 | Hyperbolic tangent: <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb7.svg"></span> | |
| 63 | </p> | |
| 64 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 65 | <span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple; | |
| 66 | sin: red; tan: blue)</strong></span></em></span> | |
| 67 | </p></blockquote></div> | |
| 68 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 69 | <span class="inlinemediaobject"><img src="../../../graphs/trigonometric.png"></span> | |
| 70 | </p></blockquote></div> | |
| 71 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 72 | <span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple; | |
| 73 | sinh: red; tanh: blue)</strong></span></em></span> | |
| 74 | </p></blockquote></div> | |
| 75 | <div class="blockquote"><blockquote class="blockquote"><p> | |
| 76 | <span class="inlinemediaobject"><img src="../../../graphs/hyperbolic.png"></span> | |
| 77 | </p></blockquote></div> | |
| 78 | <p> | |
| 79 | The hyperbolic sine is one to one on the set of real numbers, with range | |
| 80 | the full set of reals, while the hyperbolic tangent is also one to one on | |
| 81 | the set of real numbers but with range <code class="literal">[0;+∞[</code>, and therefore | |
| 82 | both have inverses. The hyperbolic cosine is one to one from <code class="literal">]-∞;+1[</code> | |
| 83 | onto <code class="literal">]-∞;-1[</code> (and from <code class="literal">]+1;+∞[</code> onto | |
| 84 | <code class="literal">]-∞;-1[</code>); the inverse function we use here is defined on | |
| 85 | <code class="literal">]-∞;-1[</code> with range <code class="literal">]-∞;+1[</code>. | |
| 86 | </p> | |
| 87 | <p> | |
| 88 | The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent, | |
| 89 | and can be computed as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb15.svg"></span>. | |
| 90 | </p> | |
| 91 | <p> | |
| 92 | The inverse of the hyperbolic sine is called the Argument hyperbolic sine, | |
| 93 | and can be computed (for <code class="literal">[-1;-1+ε[</code>) as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb17.svg"></span>. | |
| 94 | </p> | |
| 95 | <p> | |
| 96 | The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine, | |
| 97 | and can be computed as <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb18.svg"></span>. | |
| 98 | </p> | |
| 99 | </div> | |
| 100 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 101 | <td align="left"></td> | |
| 102 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 103 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 104 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 105 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 106 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 107 | 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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