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| <div class="titlepage"><div><div><h2 class="title" style="clear: both"> |
| <a name="math_toolkit.trans"></a><a class="link" href="trans.html" title="Quaternion Transcendentals">Quaternion 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 quaternions 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 quaternions ("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. |
| This author claims the complete proof of the "closed formula" as |
| his own, as well as its independant invention (there are claims to prior invention |
| of the formula, such as one by Professor Shoemake, and it is possible that |
| the formula had been known a couple of centuries back, but in absence of bibliographical |
| reference, the matter is pending, awaiting further investigation; on the other |
| hand, the definition and existence of the exponential on the quaternions, is |
| of course a fact known for a very long time). 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>H</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.trans.h0"></a> |
| <span class="phrase"><a name="math_toolkit.trans.exp"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the exponential of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h1"></a> |
| <span class="phrase"><a name="math_toolkit.trans.cos"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the cosine of the quaternion |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h2"></a> |
| <span class="phrase"><a name="math_toolkit.trans.sin"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the sine of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h3"></a> |
| <span class="phrase"><a name="math_toolkit.trans.tan"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the tangent of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h4"></a> |
| <span class="phrase"><a name="math_toolkit.trans.cosh"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the hyperbolic cosine of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h5"></a> |
| <span class="phrase"><a name="math_toolkit.trans.sinh"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the hyperbolic sine of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h6"></a> |
| <span class="phrase"><a name="math_toolkit.trans.tanh"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the hyperbolic tangent of the quaternion. |
| </p> |
| <h5> |
| <a name="math_toolkit.trans.h7"></a> |
| <span class="phrase"><a name="math_toolkit.trans.pow"></a></span><a class="link" href="trans.html#math_toolkit.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">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> |
| </pre> |
| <p> |
| Computes the n-th power of the quaternion q. |
| </p> |
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| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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