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| <div class="section"> |
| <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="id799499"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the exponential of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.cos"></a><h5> |
| <a name="id799620"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the cosine of the octonion |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.sin"></a><h5> |
| <a name="id799742"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the sine of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.tan"></a><h5> |
| <a name="id799864"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the tangent of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.cosh"></a><h5> |
| <a name="id799985"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the hyperbolic cosine of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.sinh"></a><h5> |
| <a name="id800107"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the hyperbolic sine of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.tanh"></a><h5> |
| <a name="id800229"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the hyperbolic tangent of the octonion. |
| </p> |
| <a name="boost_octonions.octonions.octonions_transcendentals.pow"></a><h5> |
| <a name="id800352"></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"><</span><span class="keyword">typename</span> <span class="identifier">T</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="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> |
| </pre> |
| <p> |
| Computes the n-th power of the octonion q. |
| </p> |
| </div> |
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| <td align="left"></td> |
| <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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