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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.inv_hyper.inv_hyper_over"></a><a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">Inverse Hyperbolic
  28       Functions Overview</a>
  29 </h3></div></div></div>
  30 <p>
  31         The exponential funtion is defined, for all objects for which this makes
  32         sense, as the power series
  33       </p>
  34 <div class="blockquote"><blockquote class="blockquote"><p>
  35           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb1.svg"></span>
  36
  37         </p></blockquote></div>
  38 <p>
  39         with <span class="emphasis"><em><code class="literal">n! = 1x2x3x4x5...xn</code></em></span> (and <span class="emphasis"><em><code class="literal">0!
  40         = 1</code></em></span> by definition) being the factorial of <span class="emphasis"><em><code class="literal">n</code></em></span>.
  41         In particular, the exponential function is well defined for real numbers,
  42         complex number, quaternions, octonions, and matrices of complex numbers,
  43         among others.
  44       </p>
  45 <div class="blockquote"><blockquote class="blockquote"><p>
  46           <span class="emphasis"><em><span class="bold"><strong>Graph of exp on R</strong></span></em></span>
  47         </p></blockquote></div>
  48 <div class="blockquote"><blockquote class="blockquote"><p>
  49           <span class="inlinemediaobject"><img src="../../../graphs/exp_on_r.png"></span>
  50         </p></blockquote></div>
  51 <div class="blockquote"><blockquote class="blockquote"><p>
  52           <span class="emphasis"><em><span class="bold"><strong>Real and Imaginary parts of exp on C</strong></span></em></span>
  53         </p></blockquote></div>
  54 <div class="blockquote"><blockquote class="blockquote"><p>
  55           <span class="inlinemediaobject"><img src="../../../graphs/im_exp_on_c.png"></span>
  56         </p></blockquote></div>
  57 <p>
  58         The hyperbolic functions are defined as power series which can be computed
  59         (for reals, complex, quaternions and octonions) as:
  60       </p>
  61 <p>
  62         Hyperbolic cosine:
  63       </p>
  64 <div class="blockquote"><blockquote class="blockquote"><p>
  65           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb5.svg"></span>
  66
  67         </p></blockquote></div>
  68 <p>
  69         Hyperbolic sine:
  70       </p>
  71 <div class="blockquote"><blockquote class="blockquote"><p>
  72           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb6.svg"></span>
  73
  74         </p></blockquote></div>
  75 <p>
  76         Hyperbolic tangent:
  77       </p>
  78 <div class="blockquote"><blockquote class="blockquote"><p>
  79           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb7.svg"></span>
  80
  81         </p></blockquote></div>
  82 <div class="blockquote"><blockquote class="blockquote"><p>
  83           <span class="emphasis"><em><span class="bold"><strong>Trigonometric functions on R (cos: purple;
  84           sin: red; tan: blue)</strong></span></em></span>
  85         </p></blockquote></div>
  86 <div class="blockquote"><blockquote class="blockquote"><p>
  87           <span class="inlinemediaobject"><img src="../../../graphs/trigonometric.png"></span>
  88         </p></blockquote></div>
  89 <div class="blockquote"><blockquote class="blockquote"><p>
  90           <span class="emphasis"><em><span class="bold"><strong>Hyperbolic functions on r (cosh: purple;
  91           sinh: red; tanh: blue)</strong></span></em></span>
  92         </p></blockquote></div>
  93 <div class="blockquote"><blockquote class="blockquote"><p>
  94           <span class="inlinemediaobject"><img src="../../../graphs/hyperbolic.png"></span>
  95         </p></blockquote></div>
  96 <p>
  97         The hyperbolic sine is one to one on the set of real numbers, with range
  98         the full set of reals, while the hyperbolic tangent is also one to one on
  99         the set of real numbers but with range <code class="literal">[0;+&#8734;[</code>, and therefore
 100         both have inverses.
 101       </p>
 102 <p>
 103         The hyperbolic cosine is one to one from <code class="literal">]-&#8734;;+1[</code> onto
 104         <code class="literal">]-&#8734;;-1[</code> (and from <code class="literal">]+1;+&#8734;[</code> onto <code class="literal">]-&#8734;;-1[</code>).
 105       </p>
 106 <p>
 107         The inverse function we use here is defined on <code class="literal">]-&#8734;;-1[</code>
 108         with range <code class="literal">]-&#8734;;+1[</code>.
 109       </p>
 110 <p>
 111         The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
 112         and can be computed as
 113       </p>
 114 <div class="blockquote"><blockquote class="blockquote"><p>
 115           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb15.svg"></span>
 116
 117         </p></blockquote></div>
 118 <p>
 119         The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
 120         and can be computed (for <code class="literal">[-1;-1+&#949;[</code>) as
 121       </p>
 122 <div class="blockquote"><blockquote class="blockquote"><p>
 123           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb17.svg"></span>
 124
 125         </p></blockquote></div>
 126 <p>
 127         The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
 128         and can be computed as
 129       </p>
 130 <div class="blockquote"><blockquote class="blockquote"><p>
 131           <span class="inlinemediaobject"><img src="../../../equations/special_functions_blurb18.svg"></span>
 132
 133         </p></blockquote></div>
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