1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
|
<html lang="en">
<head>
<title>Mathieu Functions - GNU Scientific Library -- Reference Manual</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="GNU Scientific Library -- Reference Manual">
<meta name="generator" content="makeinfo 4.8">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Special-Functions.html" title="Special Functions">
<link rel="prev" href="Logarithm-and-Related-Functions.html" title="Logarithm and Related Functions">
<link rel="next" href="Power-Function.html" title="Power Function">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with the
Invariant Sections being ``GNU General Public License'' and ``Free Software
Needs Free Documentation'', the Front-Cover text being ``A GNU Manual'',
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled ``GNU Free Documentation
License''.
(a) The Back-Cover Text is: ``You have freedom to copy and modify this
GNU Manual, like GNU software.''-->
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
pre.display { font-family:inherit }
pre.format { font-family:inherit }
pre.smalldisplay { font-family:inherit; font-size:smaller }
pre.smallformat { font-family:inherit; font-size:smaller }
pre.smallexample { font-size:smaller }
pre.smalllisp { font-size:smaller }
span.sc { font-variant:small-caps }
span.roman { font-family:serif; font-weight:normal; }
span.sansserif { font-family:sans-serif; font-weight:normal; }
--></style>
</head>
<body>
<div class="node">
<p>
<a name="Mathieu-Functions"></a>
Next: <a rel="next" accesskey="n" href="Power-Function.html">Power Function</a>,
Previous: <a rel="previous" accesskey="p" href="Logarithm-and-Related-Functions.html">Logarithm and Related Functions</a>,
Up: <a rel="up" accesskey="u" href="Special-Functions.html">Special Functions</a>
<hr>
</div>
<h3 class="section">7.26 Mathieu Functions</h3>
<p><a name="index-Mathieu-functions-736"></a>
The routines described in this section compute the angular and radial
Mathieu functions, and their characteristic values. Mathieu
functions are the solutions of the following two differential
equations:
<pre class="example"> d^2y/dv^2 + (a - 2q\cos 2v)y = 0
d^2f/du^2 - (a - 2q\cosh 2u)f = 0
</pre>
<p class="noindent">The angular Mathieu functions ce_r(x,q), se_r(x,q) are
the even and odd periodic solutions of the first equation, which is known as Mathieu's equation. These exist
only for the discrete sequence of characteristic values a=a_r(q)
(even-periodic) and a=b_r(q) (odd-periodic).
<p>The radial Mathieu functions <!-- {$Mc^{(j)}_{r}(z,q)$} -->
Mc^{(j)}_{r}(z,q), <!-- {$Ms^{(j)}_@{r@}(z,q)$} -->
Ms^{(j)}_{r}(z,q) are the solutions of the second equation,
which is referred to as Mathieu's modified equation. The
radial Mathieu functions of the first, second, third and fourth kind
are denoted by the parameter j, which takes the value 1, 2, 3
or 4.
<!-- The angular Mathieu functions can be divided into four types as -->
<!-- @tex -->
<!-- \beforedisplay -->
<!-- $$ -->
<!-- \eqalign{ -->
<!-- x & = \sum_{m=0}^\infty A_{2m+p} \cos(2m+p)\phi, \quad p = 0, 1, \cr -->
<!-- x & = \sum_{m=0}^\infty B_{2m+p} \sin(2m+p)\phi, \quad p = 0, 1. -->
<!-- } -->
<!-- $$ -->
<!-- \afterdisplay -->
<!-- @end tex -->
<!-- @ifinfo -->
<!-- @example -->
<!-- x = \sum_(m=0)^\infty A_(2m+p) \cos(2m+p)\phi, p = 0, 1, -->
<!-- x = \sum_(m=0)^\infty B_(2m+p) \sin(2m+p)\phi, p = 0, 1. -->
<!-- @end example -->
<!-- @end ifinfo -->
<!-- @noindent -->
<!-- The nomenclature used for the angular Mathieu functions is @math{ce_n} -->
<!-- for the first solution and @math{se_n} for the second. -->
<!-- Similar solutions exist for the radial Mathieu functions by replacing -->
<!-- the trigonometric functions with their corresponding hyperbolic -->
<!-- functions as shown below. -->
<!-- @tex -->
<!-- \beforedisplay -->
<!-- $$ -->
<!-- \eqalign{ -->
<!-- x & = \sum_{m=0}^\infty A_{2m+p} \cosh(2m+p)u, \quad p = 0, 1, \cr -->
<!-- x & = \sum_{m=0}^\infty B_{2m+p} \sinh(2m+p)u, \quad p = 0, 1. -->
<!-- } -->
<!-- $$ -->
<!-- \afterdisplay -->
<!-- @end tex -->
<!-- @ifinfo -->
<!-- @example -->
<!-- x = \sum_(m=0)^\infty A_(2m+p) \cosh(2m+p)u, p = 0, 1, -->
<!-- x = \sum_(m=0)^\infty B_(2m+p) \sinh(2m+p)u, p = 0, 1. -->
<!-- @end example -->
<!-- @end ifinfo -->
<!-- @noindent -->
<!-- The nomenclature used for the radial Mathieu functions is @math{Mc_n} -->
<!-- for the first solution and @math{Ms_n} for the second. The hyperbolic -->
<!-- series do not always converge at an acceptable rate. Therefore most -->
<!-- texts on the subject suggest using the following equivalent equations -->
<!-- that are expanded in series of Bessel and Hankel functions. -->
<!-- @tex -->
<!-- \beforedisplay -->
<!-- $$ -->
<!-- \eqalign{ -->
<!-- Mc_{2n}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k} -->
<!-- A_{2m}^{2n}(q)\left[J_m(u_1)Z_m^{(j)}(u_2) + -->
<!-- J_m(u_1)Z_m^{(j)}(u_2)\right]/A_2^{2n} \cr -->
<!-- Mc_{2n+1}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k} -->
<!-- A_{2m+1}^{2n+1}(q)\left[J_m(u_1)Z_{m+1}^{(j)}(u_2) + -->
<!-- J_{m+1}(u_1)Z_m^{(j)}(u_2)\right]/A_1^{2n+1} \cr -->
<!-- Ms_{2n}^{(j)}(x,q) & = \sum_{m=1}^\infty (-1)^{r+k} -->
<!-- B_{2m}^{2n}(q)\left[J_{m-1}(u_1)Z_{m+1}^{(j)}(u_2) + -->
<!-- J_{m+1}(u_1)Z_{m-1}^{(j)}(u_2)\right]/B_2^{2n} \cr -->
<!-- Ms_{2n+1}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k} -->
<!-- B_{2m+1}^{2n+1}(q)\left[J_m(u_1)Z_{m+1}^{(j)}(u_2) + -->
<!-- J_{m+1}(u_1)Z_m^{(j)}(u_2)\right]/B_1^{2n+1} -->
<!-- } -->
<!-- $$ -->
<!-- \afterdisplay -->
<!-- @end tex -->
<!-- @ifinfo -->
<!-- @example -->
<!-- Mc_(2n)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) A_(2m)^(2n)(q) -->
<!-- [J_m(u_1)Z_m^(j)(u_2) + J_m(u_1)Z_m^(j)(u_2)]/A_2^(2n) -->
<!-- Mc_(2n+1)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) A_(2m+1)^(2n+1)(q) -->
<!-- [J_m(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_m^(j)(u_2)]/A_1^(2n+1) -->
<!-- Ms_(2n)^(j)(x,q) = \sum_(m=1)^\infty (-1)^(r+k) B_(2m)^(2n)(q) -->
<!-- [J_(m-1)(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_(m-1)^(j)(u_2)]/B_2^(2n) -->
<!-- Ms_(2n+1)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) B_(2m+1)^(2n+1)(q) -->
<!-- [J_m(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_m^(j)(u_2)]/B_1^(2n+1) -->
<!-- @end example -->
<!-- @end ifinfo -->
<!-- @noindent -->
<!-- where @c{$u_1 = \sqrt{q} \exp(-x)$} -->
<!-- @math{u_1 = \sqrt@{q@} \exp(-x)} and @c{$u_2 = \sqrt@{q@} \exp(x)$} -->
<!-- @math{u_2 = \sqrt@{q@} \exp(x)} and -->
<!-- @tex -->
<!-- \beforedisplay -->
<!-- $$ -->
<!-- \eqalign{ -->
<!-- Z_m^{(1)}(u) & = J_m(u) \cr -->
<!-- Z_m^{(2)}(u) & = Y_m(u) \cr -->
<!-- Z_m^{(3)}(u) & = H_m^{(1)}(u) \cr -->
<!-- Z_m^{(4)}(u) & = H_m^{(2)}(u) -->
<!-- } -->
<!-- $$ -->
<!-- \afterdisplay -->
<!-- @end tex -->
<!-- @ifinfo -->
<!-- @example -->
<!-- Z_m^(1)(u) = J_m(u) -->
<!-- Z_m^(2)(u) = Y_m(u) -->
<!-- Z_m^(3)(u) = H_m^(1)(u) -->
<!-- Z_m^(4)(u) = H_m^(2)(u) -->
<!-- @end example -->
<!-- @end ifinfo -->
<!-- @noindent -->
<!-- where @math{J_m(u)}, @math{Y_m(u)}, @math{H_m^{(1)}(u)}, and -->
<!-- @math{H_m^{(2)}(u)} are the regular and irregular Bessel functions and -->
<!-- the Hankel functions, respectively. -->
<p>For more information on the Mathieu functions, see Abramowitz and
Stegun, Chapter 20. These functions are defined in the header file
<samp><span class="file">gsl_sf_mathieu.h</span></samp>.
<ul class="menu">
<li><a accesskey="1" href="Mathieu-Function-Workspace.html">Mathieu Function Workspace</a>
<li><a accesskey="2" href="Mathieu-Function-Characteristic-Values.html">Mathieu Function Characteristic Values</a>
<li><a accesskey="3" href="Angular-Mathieu-Functions.html">Angular Mathieu Functions</a>
<li><a accesskey="4" href="Radial-Mathieu-Functions.html">Radial Mathieu Functions</a>
</ul>
<hr>The GNU Scientific Library - a free numerical library licensed under the GNU GPL<br>Back to the <a href="/software/gsl/">GNU Scientific Library Homepage</a></body></html>
|
