<title>Bessel Function Overview</title>
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<p>
Bessel Functions are solutions to Bessel's ordinary differential equation:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel1.svg"></span>
- </p>
-<p>
- where ν   is the <span class="emphasis"><em>order</em></span> of the equation, and may be an arbitrary
+ 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>
</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>  
+ 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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
+
+ </p></blockquote></div>
<p>
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>.
</p>
<p>
- The second solution is denoted either Y<sub>v</sub>   or N<sub>v</sub>  
+ 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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
+
+ </p></blockquote></div>
<p>
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>.
</p>
<p>
The Bessel functions satisfy the recurrence relations:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel4.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel5.svg"></span>
+
+ </p></blockquote></div>
<p>
Have the derivatives:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel6.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel7.svg"></span>
+
+ </p></blockquote></div>
<p>
Have the Wronskian relation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
+
+ </p></blockquote></div>
<p>
and the reflection formulae:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_over.h1"></a>
<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
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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel1.svg"></span>
+
+ </p></blockquote></div>
<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>  
+ 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.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
+
+ </p></blockquote></div>
<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>
The modified Bessel functions satisfy the recurrence relations:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel4.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
+
+ </p></blockquote></div>
<p>
Have the derivatives:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel6.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel7.svg"></span>
+
+ </p></blockquote></div>
<p>
Have the Wronskian relation:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
+
+ </p></blockquote></div>
<p>
and the reflection formulae:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_over.h2"></a>
<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
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.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/sbessel1.svg"></span>
+
+ </p></blockquote></div>
<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.svg"></span>
+ 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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/sbessel2.svg"></span>
+
+ </p></blockquote></div>
<p>
- The spherical Bessel function of the second kind y<sub>n</sub>  
+ 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>
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