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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.bessel.bessel_over"></a><a class="link" href="bessel_over.html" title="Bessel Function Overview"> Bessel Function |
| Overview</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.bessel.bessel_over.ordinary_bessel_functions"></a><h5> |
| <a name="id1122507"></a> |
| <a class="link" href="bessel_over.html#math_toolkit.special.bessel.bessel_over.ordinary_bessel_functions">Ordinary |
| Bessel Functions</a> |
| </h5> |
| <p> |
| Bessel Functions are solutions to Bessel's ordinary differential equation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel1.png"></span> |
| </p> |
| <p> |
| where ν ​ is the <span class="emphasis"><em>order</em></span> of the equation, and may be an |
| arbitrary real or complex number, although integer orders are the most |
| common occurrence. |
| </p> |
| <p> |
| This library supports either integer or real orders. |
| </p> |
| <p> |
| Since this is a second order differential equation, there must be two linearly |
| independent solutions, the first of these is denoted J<sub>v</sub> ​ |
| and known as a Bessel |
| function of the first kind: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel2.png"></span> |
| </p> |
| <p> |
| This function is implemented in this library as <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>. |
| </p> |
| <p> |
| The second solution is denoted either Y<sub>v</sub> ​ or N<sub>v</sub> ​ |
| and is known as either a Bessel |
| Function of the second kind, or as a Neumann function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel3.png"></span> |
| </p> |
| <p> |
| This function is implemented in this library as <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a>. |
| </p> |
| <p> |
| The Bessel functions satisfy the recurrence relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel4.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel5.png"></span> |
| </p> |
| <p> |
| Have the derivatives: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel6.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel7.png"></span> |
| </p> |
| <p> |
| Have the Wronskian relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel8.png"></span> |
| </p> |
| <p> |
| and the reflection formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel9.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/bessel10.png"></span> |
| </p> |
| <a name="math_toolkit.special.bessel.bessel_over.modified_bessel_functions"></a><h5> |
| <a name="id1122827"></a> |
| <a class="link" href="bessel_over.html#math_toolkit.special.bessel.bessel_over.modified_bessel_functions">Modified |
| Bessel Functions</a> |
| </h5> |
| <p> |
| The Bessel functions are valid for complex argument <span class="emphasis"><em>x</em></span>, |
| and an important special case is the situation where <span class="emphasis"><em>x</em></span> |
| is purely imaginary: giving a real valued result. In this case the functions |
| are the two linearly independent solutions to the modified Bessel equation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel1.png"></span> |
| </p> |
| <p> |
| The solutions are known as the modified Bessel functions of the first and |
| second kind (or occasionally as the hyperbolic Bessel functions of the |
| first and second kind). They are denoted I<sub>v</sub> ​ and K<sub>v</sub> ​ |
| respectively: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel2.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel3.png"></span> |
| </p> |
| <p> |
| 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> |
| and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> |
| respectively. |
| </p> |
| <p> |
| The modified Bessel functions satisfy the recurrence relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel4.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel5.png"></span> |
| </p> |
| <p> |
| Have the derivatives: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel6.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel7.png"></span> |
| </p> |
| <p> |
| Have the Wronskian relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel8.png"></span> |
| </p> |
| <p> |
| and the reflection formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel9.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel10.png"></span> |
| </p> |
| <a name="math_toolkit.special.bessel.bessel_over.spherical_bessel_functions"></a><h5> |
| <a name="id1123132"></a> |
| <a class="link" href="bessel_over.html#math_toolkit.special.bessel.bessel_over.spherical_bessel_functions">Spherical |
| Bessel Functions</a> |
| </h5> |
| <p> |
| When solving the Helmholtz equation in spherical coordinates by separation |
| of variables, the radial equation has the form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/sbessel1.png"></span> |
| </p> |
| <p> |
| The two linearly independent solutions to this equation are called the |
| spherical Bessel functions j<sub>n</sub> ​ and y<sub>n</sub> ​, and are related to the ordinary Bessel |
| functions J<sub>n</sub> ​ and Y<sub>n</sub> ​ by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/sbessel2.png"></span> |
| </p> |
| <p> |
| The spherical Bessel function of the second kind y<sub>n</sub> ​ |
| is also known as the |
| spherical Neumann function n<sub>n</sub>. |
| </p> |
| <p> |
| 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> |
| and <a class="link" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">sph_neumann</a>. |
| </p> |
| </div> |
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| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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>) |
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