1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Quaternion Transcendentals</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="../quaternions.html" title="Chapter 16. Quaternions"> 9<link rel="prev" href="create.html" title="Quaternion Creation Functions"> 10<link rel="next" href="quat_tests.html" title="Test Program"> 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="create.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quaternions.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="quat_tests.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.trans"></a><a class="link" href="trans.html" title="Quaternion Transcendentals">Quaternion Transcendentals</a> 28</h2></div></div></div> 29<p> 30 There is no <code class="computeroutput"><span class="identifier">log</span></code> or <code class="computeroutput"><span class="identifier">sqrt</span></code> provided for quaternions in this implementation, 31 and <code class="computeroutput"><span class="identifier">pow</span></code> is likewise restricted 32 to integral powers of the exponent. There are several reasons to this: on the 33 one hand, the equivalent of analytic continuation for quaternions ("branch 34 cuts") remains to be investigated thoroughly (by me, at any rate...), 35 and we wish to avoid the nonsense introduced in the standard by exponentiations 36 of complexes by complexes (which is well defined, but not in the standard...). 37 Talking of nonsense, saying that <code class="computeroutput"><span class="identifier">pow</span><span class="special">(</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">)</span></code> is "implementation 38 defined" is just plain brain-dead... 39 </p> 40<p> 41 We do, however provide several transcendentals, chief among which is the exponential. 42 This author claims the complete proof of the "closed formula" as 43 his own, as well as its independent invention (there are claims to prior invention 44 of the formula, such as one by Professor Shoemake, and it is possible that 45 the formula had been known a couple of centuries back, but in absence of bibliographical 46 reference, the matter is pending, awaiting further investigation; on the other 47 hand, the definition and existence of the exponential on the quaternions, is 48 of course a fact known for a very long time). Basically, any converging power 49 series with real coefficients which allows for a closed formula in <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span> can be transposed to <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span>. More transcendentals of this type could 50 be added in a further revision upon request. It should be noted that it is 51 these functions which force the dependency upon the <a href="../../../../../boost/math/special_functions/sinc.hpp" target="_top">boost/math/special_functions/sinc.hpp</a> 52 and the <a href="../../../../../boost/math/special_functions/sinhc.hpp" target="_top">boost/math/special_functions/sinhc.hpp</a> 53 headers. 54 </p> 55<h5> 56<a name="math_toolkit.trans.h0"></a> 57 <span class="phrase"><a name="math_toolkit.trans.exp"></a></span><a class="link" href="trans.html#math_toolkit.trans.exp">exp</a> 58 </h5> 59<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 60</pre> 61<p> 62 Computes the exponential of the quaternion. 63 </p> 64<h5> 65<a name="math_toolkit.trans.h1"></a> 66 <span class="phrase"><a name="math_toolkit.trans.cos"></a></span><a class="link" href="trans.html#math_toolkit.trans.cos">cos</a> 67 </h5> 68<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 69</pre> 70<p> 71 Computes the cosine of the quaternion 72 </p> 73<h5> 74<a name="math_toolkit.trans.h2"></a> 75 <span class="phrase"><a name="math_toolkit.trans.sin"></a></span><a class="link" href="trans.html#math_toolkit.trans.sin">sin</a> 76 </h5> 77<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 78</pre> 79<p> 80 Computes the sine of the quaternion. 81 </p> 82<h5> 83<a name="math_toolkit.trans.h3"></a> 84 <span class="phrase"><a name="math_toolkit.trans.tan"></a></span><a class="link" href="trans.html#math_toolkit.trans.tan">tan</a> 85 </h5> 86<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 87</pre> 88<p> 89 Computes the tangent of the quaternion. 90 </p> 91<h5> 92<a name="math_toolkit.trans.h4"></a> 93 <span class="phrase"><a name="math_toolkit.trans.cosh"></a></span><a class="link" href="trans.html#math_toolkit.trans.cosh">cosh</a> 94 </h5> 95<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">cosh</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 96</pre> 97<p> 98 Computes the hyperbolic cosine of the quaternion. 99 </p> 100<h5> 101<a name="math_toolkit.trans.h5"></a> 102 <span class="phrase"><a name="math_toolkit.trans.sinh"></a></span><a class="link" href="trans.html#math_toolkit.trans.sinh">sinh</a> 103 </h5> 104<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">sinh</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 105</pre> 106<p> 107 Computes the hyperbolic sine of the quaternion. 108 </p> 109<h5> 110<a name="math_toolkit.trans.h6"></a> 111 <span class="phrase"><a name="math_toolkit.trans.tanh"></a></span><a class="link" href="trans.html#math_toolkit.trans.tanh">tanh</a> 112 </h5> 113<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">tanh</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">);</span> 114</pre> 115<p> 116 Computes the hyperbolic tangent of the quaternion. 117 </p> 118<h5> 119<a name="math_toolkit.trans.h7"></a> 120 <span class="phrase"><a name="math_toolkit.trans.pow"></a></span><a class="link" href="trans.html#math_toolkit.trans.pow">pow</a> 121 </h5> 122<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">quaternion</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n</span><span class="special">);</span> 123</pre> 124<p> 125 Computes the n-th power of the quaternion q. 126 </p> 127</div> 128<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 129<td align="left"></td> 130<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 131 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 132 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 133 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 134 Daryle Walker and Xiaogang Zhang<p> 135 Distributed under the Boost Software License, Version 1.0. (See accompanying 136 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>) 137 </p> 138</div></td> 139</tr></table> 140<hr> 141<div class="spirit-nav"> 142<a accesskey="p" href="create.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quaternions.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="quat_tests.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 143</div> 144</body> 145</html> 146
