Namespaces
Variants
Actions

Difference between revisions of "Mathieu functions"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627601.png" />-periodic solutions of the [[Mathieu equation|Mathieu equation]]
+
<!--
 +
m0627601.png
 +
$#A+1 = 15 n = 0
 +
$#C+1 = 15 : ~/encyclopedia/old_files/data/M062/M.0602760 Mathieu functions
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627602.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
which exist only when the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627603.png" /> in the parameter plane lies on the boundary of the stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627604.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627605.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627606.png" />. The even Mathieu functions are the eigen functions of the integral equation
+
The  $  2 \pi $-
 +
periodic solutions of the [[Mathieu equation|Mathieu equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627607.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{d  ^ {2} u }{dz  ^ {2} }
 +
+ ( a + 16q  \cos  2z) u  = 0,\ \
 +
z \in \mathbf R ,
 +
$$
 +
 
 +
which exist only when the point  $  ( a, q) $
 +
in the parameter plane lies on the boundary of the stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in  $  z $
 +
for  $  | z | \rightarrow \infty $,
 +
provided  $  q \neq 0 $.
 +
The even Mathieu functions are the eigen functions of the integral equation
 +
 
 +
$$
 +
G( z)  = \lambda \int\limits _ {- \pi } ^  \pi  e ^ {k  \cos  z  \cos  t } G( t)  dt,\ \
 +
= \sqrt 32q .
 +
$$
  
 
An analogous equation is satisfied by the odd Mathieu functions. The notation for Mathieu functions is:
 
An analogous equation is satisfied by the odd Mathieu functions. The notation for Mathieu functions is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627608.png" /></td> </tr></table>
+
$$
 +
ce _ {0} ( z, q), ce _ {1} ( z, q) , .  .  . ; \  se _ {1} ( z, q) , se _ {2} ( z, q) , .  . . .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m0627609.png" /> these functions reduce to the [[Trigonometric system|trigonometric system]]
+
For $  q \rightarrow 0 $
 +
these functions reduce to the [[Trigonometric system|trigonometric system]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276010.png" /></td> </tr></table>
+
$$
 +
1 , \cos  z , . . . ; \ \
 +
\sin  z, \sin  2z \dots
 +
$$
  
and they possess the same orthogonality properties on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276011.png" />. The Mathieu functions admit Fourier-series expansions which converge for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276012.png" />; the coefficients of these series are convergent power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276013.png" />, for example,
+
and they possess the same orthogonality properties on the interval $  (- \pi , \pi ) $.  
 +
The Mathieu functions admit Fourier-series expansions which converge for small $  | q | \leq  r _ {n} $;  
 +
the coefficients of these series are convergent power series in $  q $,  
 +
for example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276014.png" /></td> </tr></table>
+
$$
 +
ce _ {0} ( z, q)  = \
 +
1 +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062760/m06276015.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ { n= } 1 ^  \infty  \left [ 2  ^ {n+} 1
 +
\frac{q  ^ {n} }{( n!)
 +
^ {2} }
 +
-  
 +
\frac{n( 3n+ 4) 2  ^ {n+} 3 q  ^ {n+} 2 }{(( n+ 1)!)
 +
^ {2} }
 +
+ O( q  ^ {n+} 4 ) \right ]  \cos  2nz.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,  "Equazioni differenziali nel campo reale" , '''1''' , Zanichelli  (1948)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.J.O. Strett,  "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer  (1932)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.W. Mac-Lachlan,  "Theory and application of Mathieu functions" , Clarendon Press  (1947)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,  "Equazioni differenziali nel campo reale" , '''1''' , Zanichelli  (1948)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.J.O. Strett,  "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer  (1932)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.W. Mac-Lachlan,  "Theory and application of Mathieu functions" , Clarendon Press  (1947)</TD></TR></table>

Latest revision as of 07:59, 6 June 2020


The $ 2 \pi $- periodic solutions of the Mathieu equation

$$ \frac{d ^ {2} u }{dz ^ {2} } + ( a + 16q \cos 2z) u = 0,\ \ z \in \mathbf R , $$

which exist only when the point $ ( a, q) $ in the parameter plane lies on the boundary of the stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in $ z $ for $ | z | \rightarrow \infty $, provided $ q \neq 0 $. The even Mathieu functions are the eigen functions of the integral equation

$$ G( z) = \lambda \int\limits _ {- \pi } ^ \pi e ^ {k \cos z \cos t } G( t) dt,\ \ k = \sqrt 32q . $$

An analogous equation is satisfied by the odd Mathieu functions. The notation for Mathieu functions is:

$$ ce _ {0} ( z, q), ce _ {1} ( z, q) , . . . ; \ se _ {1} ( z, q) , se _ {2} ( z, q) , . . . . $$

For $ q \rightarrow 0 $ these functions reduce to the trigonometric system

$$ 1 , \cos z , . . . ; \ \ \sin z, \sin 2z \dots $$

and they possess the same orthogonality properties on the interval $ (- \pi , \pi ) $. The Mathieu functions admit Fourier-series expansions which converge for small $ | q | \leq r _ {n} $; the coefficients of these series are convergent power series in $ q $, for example,

$$ ce _ {0} ( z, q) = \ 1 + $$

$$ + \sum _ { n= } 1 ^ \infty \left [ 2 ^ {n+} 1 \frac{q ^ {n} }{( n!) ^ {2} } - \frac{n( 3n+ 4) 2 ^ {n+} 3 q ^ {n+} 2 }{(( n+ 1)!) ^ {2} } + O( q ^ {n+} 4 ) \right ] \cos 2nz. $$

References

[1] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2
[2] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)
[3] G. Sansone, "Equazioni differenziali nel campo reale" , 1 , Zanichelli (1948)
[4] M.J.O. Strett, "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer (1932)
[5] N.W. Mac-Lachlan, "Theory and application of Mathieu functions" , Clarendon Press (1947)
How to Cite This Entry:
Mathieu functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathieu_functions&oldid=47791
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article