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| 4 | <title>Quaternion Transcendentals</title> | |
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| 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 independant 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 © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 131 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 132 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 133 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 134 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 135 | 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>) | |
| 136 | </p> | |
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