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<title>Quaternion Transcendentals</title>
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.trans"></a><a class="link" href="trans.html" title="Quaternion Transcendentals">Quaternion Transcendentals</a>
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</h2></div></div></div>
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<p>
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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,
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and <code class="computeroutput"><span class="identifier">pow</span></code> is likewise restricted
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to integral powers of the exponent. There are several reasons to this: on the
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one hand, the equivalent of analytic continuation for quaternions ("branch
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cuts") remains to be investigated thoroughly (by me, at any rate...),
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and we wish to avoid the nonsense introduced in the standard by exponentiations
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of complexes by complexes (which is well defined, but not in the standard...).
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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
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defined" is just plain brain-dead...
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</p>
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<p>
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We do, however provide several transcendentals, chief among which is the exponential.
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This author claims the complete proof of the "closed formula" as
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his own, as well as its independant invention (there are claims to prior invention
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of the formula, such as one by Professor Shoemake, and it is possible that
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the formula had been known a couple of centuries back, but in absence of bibliographical
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reference, the matter is pending, awaiting further investigation; on the other
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hand, the definition and existence of the exponential on the quaternions, is
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of course a fact known for a very long time). Basically, any converging power
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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
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be added in a further revision upon request. It should be noted that it is
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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>
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and the <a href="../../../../../boost/math/special_functions/sinhc.hpp" target="_top">boost/math/special_functions/sinhc.hpp</a>
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headers.
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</p>
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<h5>
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<a name="math_toolkit.trans.h0"></a>
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<span class="phrase"><a name="math_toolkit.trans.exp"></a></span><a class="link" href="trans.html#math_toolkit.trans.exp">exp</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the exponential of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h1"></a>
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<span class="phrase"><a name="math_toolkit.trans.cos"></a></span><a class="link" href="trans.html#math_toolkit.trans.cos">cos</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the cosine of the quaternion
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</p>
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<h5>
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<a name="math_toolkit.trans.h2"></a>
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<span class="phrase"><a name="math_toolkit.trans.sin"></a></span><a class="link" href="trans.html#math_toolkit.trans.sin">sin</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the sine of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h3"></a>
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<span class="phrase"><a name="math_toolkit.trans.tan"></a></span><a class="link" href="trans.html#math_toolkit.trans.tan">tan</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the tangent of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h4"></a>
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<span class="phrase"><a name="math_toolkit.trans.cosh"></a></span><a class="link" href="trans.html#math_toolkit.trans.cosh">cosh</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the hyperbolic cosine of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h5"></a>
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<span class="phrase"><a name="math_toolkit.trans.sinh"></a></span><a class="link" href="trans.html#math_toolkit.trans.sinh">sinh</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the hyperbolic sine of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h6"></a>
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<span class="phrase"><a name="math_toolkit.trans.tanh"></a></span><a class="link" href="trans.html#math_toolkit.trans.tanh">tanh</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the hyperbolic tangent of the quaternion.
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</p>
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<h5>
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<a name="math_toolkit.trans.h7"></a>
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<span class="phrase"><a name="math_toolkit.trans.pow"></a></span><a class="link" href="trans.html#math_toolkit.trans.pow">pow</a>
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</h5>
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<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>
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</pre>
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<p>
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Computes the n-th power of the quaternion q.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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