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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.bessel.bessel_over"></a><a class="link" href="bessel_over.html" title="Bessel Function Overview">Bessel Function Overview</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.bessel.bessel_over.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.bessel.bessel_over.ordinary_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.ordinary_bessel_functions">Ordinary
  32         Bessel Functions</a>
  33       </h5>
  34 <p>
  35         Bessel Functions are solutions to Bessel's ordinary differential equation:
  36       </p>
  37 <div class="blockquote"><blockquote class="blockquote"><p>
  38           <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span>
  39
  40         </p></blockquote></div>
  41 <p>
  42         where &#957; is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary
  43         real or complex number, although integer orders are the most common occurrence.
  44       </p>
  45 <p>
  46         This library supports either integer or real orders.
  47       </p>
  48 <p>
  49         Since this is a second order differential equation, there must be two linearly
  50         independent solutions, the first of these is denoted J<sub>v</sub>
  51 and known as a Bessel
  52         function of the first kind:
  53       </p>
  54 <div class="blockquote"><blockquote class="blockquote"><p>
  55           <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
  56
  57         </p></blockquote></div>
  58 <p>
  59         This function is implemented in this library as <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>.
  60       </p>
  61 <p>
  62         The second solution is denoted either Y<sub>v</sub> or N<sub>v</sub>
  63 and is known as either a Bessel
  64         Function of the second kind, or as a Neumann function:
  65       </p>
  66 <div class="blockquote"><blockquote class="blockquote"><p>
  67           <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
  68
  69         </p></blockquote></div>
  70 <p>
  71         This function is implemented in this library as <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>.
  72       </p>
  73 <p>
  74         The Bessel functions satisfy the recurrence relations:
  75       </p>
  76 <div class="blockquote"><blockquote class="blockquote"><p>
  77           <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span>
  78
  79         </p></blockquote></div>
  80 <div class="blockquote"><blockquote class="blockquote"><p>
  81           <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span>
  82
  83         </p></blockquote></div>
  84 <p>
  85         Have the derivatives:
  86       </p>
  87 <div class="blockquote"><blockquote class="blockquote"><p>
  88           <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span>
  89
  90         </p></blockquote></div>
  91 <div class="blockquote"><blockquote class="blockquote"><p>
  92           <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span>
  93
  94         </p></blockquote></div>
  95 <p>
  96         Have the Wronskian relation:
  97       </p>
  98 <div class="blockquote"><blockquote class="blockquote"><p>
  99           <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
 100
 101         </p></blockquote></div>
 102 <p>
 103         and the reflection formulae:
 104       </p>
 105 <div class="blockquote"><blockquote class="blockquote"><p>
 106           <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
 107
 108         </p></blockquote></div>
 109 <div class="blockquote"><blockquote class="blockquote"><p>
 110           <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
 111
 112         </p></blockquote></div>
 113 <h5>
 114 <a name="math_toolkit.bessel.bessel_over.h1"></a>
 115         <span class="phrase"><a name="math_toolkit.bessel.bessel_over.modified_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.modified_bessel_functions">Modified
 116         Bessel Functions</a>
 117       </h5>
 118 <p>
 119         The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>,
 120         and an important special case is the situation where <span class="emphasis"><em>x</em></span>
 121         is purely imaginary: giving a real valued result. In this case the functions
 122         are the two linearly independent solutions to the modified Bessel equation:
 123       </p>
 124 <div class="blockquote"><blockquote class="blockquote"><p>
 125           <span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span>
 126
 127         </p></blockquote></div>
 128 <p>
 129         The solutions are known as the modified Bessel functions of the first and
 130         second kind (or occasionally as the hyperbolic Bessel functions of the first
 131         and second kind). They are denoted I<sub>v</sub> and K<sub>v</sub>
 132 respectively:
 133       </p>
 134 <div class="blockquote"><blockquote class="blockquote"><p>
 135           <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
 136
 137         </p></blockquote></div>
 138 <div class="blockquote"><blockquote class="blockquote"><p>
 139           <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
 140
 141         </p></blockquote></div>
 142 <p>
 143         These functions are implemented in this library as <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>
 144         and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> respectively.
 145       </p>
 146 <p>
 147         The modified Bessel functions satisfy the recurrence relations:
 148       </p>
 149 <div class="blockquote"><blockquote class="blockquote"><p>
 150           <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span>
 151
 152         </p></blockquote></div>
 153 <div class="blockquote"><blockquote class="blockquote"><p>
 154           <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
 155
 156         </p></blockquote></div>
 157 <p>
 158         Have the derivatives:
 159       </p>
 160 <div class="blockquote"><blockquote class="blockquote"><p>
 161           <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span>
 162
 163         </p></blockquote></div>
 164 <div class="blockquote"><blockquote class="blockquote"><p>
 165           <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span>
 166
 167         </p></blockquote></div>
 168 <p>
 169         Have the Wronskian relation:
 170       </p>
 171 <div class="blockquote"><blockquote class="blockquote"><p>
 172           <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
 173
 174         </p></blockquote></div>
 175 <p>
 176         and the reflection formulae:
 177       </p>
 178 <div class="blockquote"><blockquote class="blockquote"><p>
 179           <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
 180
 181         </p></blockquote></div>
 182 <div class="blockquote"><blockquote class="blockquote"><p>
 183           <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
 184
 185         </p></blockquote></div>
 186 <h5>
 187 <a name="math_toolkit.bessel.bessel_over.h2"></a>
 188         <span class="phrase"><a name="math_toolkit.bessel.bessel_over.spherical_bessel_functions"></a></span><a class="link" href="bessel_over.html#math_toolkit.bessel.bessel_over.spherical_bessel_functions">Spherical
 189         Bessel Functions</a>
 190       </h5>
 191 <p>
 192         When solving the Helmholtz equation in spherical coordinates by separation
 193         of variables, the radial equation has the form:
 194       </p>
 195 <div class="blockquote"><blockquote class="blockquote"><p>
 196           <span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span>
 197
 198         </p></blockquote></div>
 199 <p>
 200         The two linearly independent solutions to this equation are called the spherical
 201         Bessel functions j<sub>n</sub> and y<sub>n</sub> and are related to the ordinary Bessel functions
 202         J<sub>n</sub> and Y<sub>n</sub> by:
 203       </p>
 204 <div class="blockquote"><blockquote class="blockquote"><p>
 205           <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
 206
 207         </p></blockquote></div>
 208 <p>
 209         The spherical Bessel function of the second kind y<sub>n</sub>
 210 is also known as the spherical
 211         Neumann function n<sub>n</sub>.
 212       </p>
 213 <p>
 214         These functions are implemented in this library as <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_bessel</a>
 215         and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>.
 216       </p>
 217 </div>
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