<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Octonions Transcendentals</title> <link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.75.2"> <link rel="home" href="../../index.html" title="Boost.Octonions"> <link rel="up" href="../octonions.html" title="Octonions"> <link rel="prev" href="oct_create.html" title="Octonion Creation Functions"> <link rel="next" href="test_program.html" title="Test Program"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="oct_create.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="test_program.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <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> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="oct_create.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../octonions.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="test_program.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>