mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-18 10:01:57 -05:00
149 lines
13 KiB
HTML
149 lines
13 KiB
HTML
|
<html>
|
||
|
|
<head>
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||
|
|
<title>Octonions Transcendentals</title>
|
||
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
||
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.5.1">
|
||
|
|
<link rel="up" href="../octonions.html" title="Chapter 10. Octonions">
|
||
|
|
<link rel="prev" href="oct_create.html" title="Octonion Creation Functions">
|
||
|
|
<link rel="next" href="oct_tests.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="oct_tests.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
<div class="section">
|
||
|
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
|
|
<a name="math_toolkit.oct_trans"></a><a class="link" href="oct_trans.html" title="Octonions Transcendentals">Octonions Transcendentals</a>
|
||
|
|
</h2></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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h0"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h1"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h2"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h3"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h4"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h5"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h6"></a>
|
||
|
|
<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>
|
||
|
|
</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>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.oct_trans.h7"></a>
|
||
|
|
<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>
|
||
|
|
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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="oct_tests.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|