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<title>Octonions Transcendentals</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_octonions.octonions.octonions_transcendentals"></a><a class="link" href="octonions_transcendentals.html" title="Octonions Transcendentals">Octonions
      Transcendentals</a>
</h3></div></div></div>
<p>
        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,
        and <code class="computeroutput"><span class="identifier">pow</span></code> is likewise restricted
        to integral powers of the exponent. There are several reasons to this: on
        the one hand, the equivalent of analytic continuation for octonions ("branch
        cuts") remains to be investigated thoroughly (by me, at any rate...),
        and we wish to avoid the nonsense introduced in the standard by exponentiations
        of complexes by complexes (which is well defined, but not in the standard...).
        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
        defined" is just plain brain-dead...
      </p>
<p>
        We do, however provide several transcendentals, chief among which is the
        exponential. That it allows for a "closed formula" is a result
        of the author (the existence and definition of the exponential, on the octonions
        among others, on the other hand, is a few centuries old). Basically, any
        converging power 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 be added in a further revision upon request.
        It should be noted that it is 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>
        and the <a href="../../../../../../../boost/math/special_functions/sinhc.hpp" target="_top">boost/math/special_functions/sinhc.hpp</a>
        headers.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.exp"></a><h5>
<a name="id2675119"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.exp">exp</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the exponential of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.cos"></a><h5>
<a name="id2675245"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.cos">cos</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the cosine of the octonion
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.sin"></a><h5>
<a name="id2675371"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.sin">sin</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the sine of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.tan"></a><h5>
<a name="id2675497"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.tan">tan</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the tangent of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.cosh"></a><h5>
<a name="id2675623"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.cosh">cosh</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the hyperbolic cosine of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.sinh"></a><h5>
<a name="id2675749"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.sinh">sinh</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the hyperbolic sine of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.tanh"></a><h5>
<a name="id2675876"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.tanh">tanh</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the hyperbolic tangent of the octonion.
      </p>
<a name="boost_octonions.octonions.octonions_transcendentals.pow"></a><h5>
<a name="id2676002"></a>
        <a class="link" href="octonions_transcendentals.html#boost_octonions.octonions.octonions_transcendentals.pow">pow</a>
      </h5>
<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> 
<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>
</pre>
<p>
        Computes the n-th power of the octonion q.
      </p>
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<td align="right"><div class="copyright-footer">Copyright  2001 -2003 Hubert Holin<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        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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