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26 | <div class="titlepage"><div><div><h2 class="title" style="clear: both"> | |
27 | <a name="math_toolkit.oct_trans"></a><a class="link" href="oct_trans.html" title="Octonions Transcendentals">Octonions 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 octonions 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 octonions ("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 | That it allows for a "closed formula" is a result of the author (the | |
43 | existence and definition of the exponential, on the octonions among others, | |
44 | on the other hand, is a few centuries old). Basically, any converging power | |
45 | 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>O</strong></span></em></span>. More transcendentals of this type could | |
46 | be added in a further revision upon request. It should be noted that it is | |
47 | 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> | |
48 | and the <a href="../../../../../boost/math/special_functions/sinhc.hpp" target="_top">boost/math/special_functions/sinhc.hpp</a> | |
49 | headers. | |
50 | </p> | |
51 | <h5> | |
52 | <a name="math_toolkit.oct_trans.h0"></a> | |
53 | <span class="phrase"><a name="math_toolkit.oct_trans.exp"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.exp">exp</a> | |
54 | </h5> | |
55 | <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> | |
56 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
57 | </pre> | |
58 | <p> | |
59 | Computes the exponential of the octonion. | |
60 | </p> | |
61 | <h5> | |
62 | <a name="math_toolkit.oct_trans.h1"></a> | |
63 | <span class="phrase"><a name="math_toolkit.oct_trans.cos"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.cos">cos</a> | |
64 | </h5> | |
65 | <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> | |
66 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
67 | </pre> | |
68 | <p> | |
69 | Computes the cosine of the octonion | |
70 | </p> | |
71 | <h5> | |
72 | <a name="math_toolkit.oct_trans.h2"></a> | |
73 | <span class="phrase"><a name="math_toolkit.oct_trans.sin"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.sin">sin</a> | |
74 | </h5> | |
75 | <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> | |
76 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
77 | </pre> | |
78 | <p> | |
79 | Computes the sine of the octonion. | |
80 | </p> | |
81 | <h5> | |
82 | <a name="math_toolkit.oct_trans.h3"></a> | |
83 | <span class="phrase"><a name="math_toolkit.oct_trans.tan"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.tan">tan</a> | |
84 | </h5> | |
85 | <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> | |
86 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
87 | </pre> | |
88 | <p> | |
89 | Computes the tangent of the octonion. | |
90 | </p> | |
91 | <h5> | |
92 | <a name="math_toolkit.oct_trans.h4"></a> | |
93 | <span class="phrase"><a name="math_toolkit.oct_trans.cosh"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_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> | |
96 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
97 | </pre> | |
98 | <p> | |
99 | Computes the hyperbolic cosine of the octonion. | |
100 | </p> | |
101 | <h5> | |
102 | <a name="math_toolkit.oct_trans.h5"></a> | |
103 | <span class="phrase"><a name="math_toolkit.oct_trans.sinh"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.sinh">sinh</a> | |
104 | </h5> | |
105 | <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> | |
106 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
107 | </pre> | |
108 | <p> | |
109 | Computes the hyperbolic sine of the octonion. | |
110 | </p> | |
111 | <h5> | |
112 | <a name="math_toolkit.oct_trans.h6"></a> | |
113 | <span class="phrase"><a name="math_toolkit.oct_trans.tanh"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.tanh">tanh</a> | |
114 | </h5> | |
115 | <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> | |
116 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">);</span> | |
117 | </pre> | |
118 | <p> | |
119 | Computes the hyperbolic tangent of the octonion. | |
120 | </p> | |
121 | <h5> | |
122 | <a name="math_toolkit.oct_trans.h7"></a> | |
123 | <span class="phrase"><a name="math_toolkit.oct_trans.pow"></a></span><a class="link" href="oct_trans.html#math_toolkit.oct_trans.pow">pow</a> | |
124 | </h5> | |
125 | <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> | |
126 | <span class="identifier">octonion</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">octonion</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">o</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n</span><span class="special">);</span> | |
127 | </pre> | |
128 | <p> | |
129 | Computes the n-th power of the octonion q. | |
130 | </p> | |
131 | </div> | |
132 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
133 | <td align="left"></td> | |
134 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
135 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
136 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
137 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
138 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
139 | 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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