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  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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
  56 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
  66 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
  76 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
  86 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
  96 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">cosh</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
 106 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">sinh</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
 116 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">tanh</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
 126 <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span>  <span class="identifier">pow</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</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>
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 134 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 135       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 136       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 137       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 138       Daryle Walker and Xiaogang Zhang<p>
 139         Distributed under the Boost Software License, Version 1.0. (See accompanying
 140         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>)
 141       </p>
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