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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.inv_hyper.inv_hyper_over"></a><a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview"> Inverse |
| Hyperbolic Functions Overview</a> |
| </h4></div></div></div> |
| <p> |
| The exponential funtion is defined, for all objects for which this makes |
| sense, as the power series <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb1.png"></span>, |
| with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and |
| <span class="emphasis"><em><code class="literal">0! = 1</code></em></span> by definition) being the |
| factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>. In particular, |
| the exponential function is well defined for real numbers, complex number, |
| quaternions, octonions, and matrices of complex numbers, among others. |
| </p> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/exp_on_r.png" alt="exp_on_r"></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/im_exp_on_c.png" alt="im_exp_on_c"></span> |
| </p></blockquote></div> |
| <p> |
| The hyperbolic functions are defined as power series which can be computed |
| (for reals, complex, quaternions and octonions) as: |
| </p> |
| <p> |
| Hyperbolic cosine: <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb5.png"></span> |
| </p> |
| <p> |
| Hyperbolic sine: <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb6.png"></span> |
| </p> |
| <p> |
| Hyperbolic tangent: <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb7.png"></span> |
| </p> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple; |
| sin: red; tan: blue)</strong></span></em></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/trigonometric.png" alt="trigonometric"></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple; |
| sinh: red; tanh: blue)</strong></span></em></span> |
| </p></blockquote></div> |
| <div class="blockquote"><blockquote class="blockquote"><p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/hyperbolic.png" alt="hyperbolic"></span> |
| </p></blockquote></div> |
| <p> |
| The hyperbolic sine is one to one on the set of real numbers, with range |
| the full set of reals, while the hyperbolic tangent is also one to one |
| on the set of real numbers but with range <code class="literal">[0;+∞[</code>, and |
| therefore both have inverses. The hyperbolic cosine is one to one from |
| <code class="literal">]-∞;+1[</code> onto <code class="literal">]-∞;-1[</code> (and from <code class="literal">]+1;+∞[</code> |
| onto <code class="literal">]-∞;-1[</code>); the inverse function we use here is defined |
| on <code class="literal">]-∞;-1[</code> with range <code class="literal">]-∞;+1[</code>. |
| </p> |
| <p> |
| The inverse of the hyperbolic tangent is called the Argument hyperbolic |
| tangent, and can be computed as <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb15.png"></span>. |
| </p> |
| <p> |
| The inverse of the hyperbolic sine is called the Argument hyperbolic sine, |
| and can be computed (for <code class="literal">[-1;-1+ε[</code>) as <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb17.png"></span>. |
| </p> |
| <p> |
| The inverse of the hyperbolic cosine is called the Argument hyperbolic |
| cosine, and can be computed as <span class="inlinemediaobject"><img src="../../../../equations/special_functions_blurb18.png"></span>. |
| </p> |
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| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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