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Difference between revisions of "Lommel function"

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A solution of the non-homogeneous [[Bessel equation|Bessel equation]]
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{{MSC|34}}
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{{TEX|done}}
  
<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/l/l060/l060800/l0608001.png" /></td> </tr></table>
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A solution of the non-homogeneous
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[[Bessel equation|Bessel equation]]
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$$x^2y''+xy'+(x^2-\nu^2)y = x^\rho.$$
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If $\rho = \nu+2n$, where $n$ is a natural number, then
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l0608002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l0608003.png" /> is a natural number, then
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$$y=(-1)^{n-1}(n-1)!\; 2^{\nu+2n-2} \sum_{k=0}^{n-1} (-1)^k\big(\frac{x}{2}\big)^{\nu+2k} \frac{\def\G{\Gamma}\G(\nu+n)}{k!\G(\nu+k+1)}.$$
  
<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/l/l060/l060800/l0608004.png" /></td> </tr></table>
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If the numbers $\rho+\nu\ge 0$ and $\rho-\nu\ge 0$ are not integers, then
  
<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/l/l060/l060800/l0608005.png" /></td> </tr></table>
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$$y=2^{\rho -2}\G(\frac{\rho+\nu}{2})\G(\frac{\rho-\nu}{2})\sum_{k=0}^\infty\frac{(-1)^k(x/2)^{\rho+2k}}{\G(k+1+(\rho+\nu)/2)\G(k+1+(\rho-\nu)/2)}.$$
  
If the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l0608006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l0608007.png" /> are not integers, then
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If $\rho = \nu-2n$, where $\nu\ge 0$ is an integer and $\nu$ is not an integer $\le n$, then
  
<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/l/l060/l060800/l0608008.png" /></td> </tr></table>
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$$y=\frac{\G(\nu-n)}{n!\; 2^{-\nu+2n+2}}\Big[2J_\nu(x)\ln\frac{x}{2}-\sum_{k=0}^{n-1}\frac{(n-k-1)!}{\G(\nu-n+k+1)}\big(\frac{x}{2}\big)^{\nu-2n+2k}-\sum_{k=0}^{\infty}\Big(\frac{(-1)^k(x/2)^{\nu+2k}}{k!\;\G(\nu+k+1)}-\frac{\G'(k+1)}{\G(k+1)}-\frac{\G'(\nu+k+1)}{\G(\nu+k+1)}\Big)\Big].$$
  
<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/l/l060/l060800/l0608009.png" /></td> </tr></table>
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Here, for $n=0$ the first sum is taken to be zero, and $J_\nu(x)$ is a Bessel function (cf.
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[[Bessel functions|Bessel functions]]). Lommel functions in two variables are also known.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080011.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080012.png" /> is not an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080013.png" />, then
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See also
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[[Anger function|Anger function]];
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[[Weber function|Weber function]];
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[[Struve function|Struve function]].
  
<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/l/l060/l060800/l06080014.png" /></td> </tr></table>
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Lommel functions were studied by E. Lommel
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{{Cite|Lo}}.
  
<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/l/l060/l060800/l06080015.png" /></td> </tr></table>
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====References====
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{|
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|-
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|valign="top"|{{Ref|Ka}}||valign="top"|  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint  (1947) 
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|-
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|valign="top"|{{Ref|Lo}}||valign="top"|  E. Lommel,  "Zur Theorie der Bessel'schen Funktionen IV" ''Math. Ann.'', '''16'''  (1880)  pp. 183–208  {{MR|1510022}}  JFM&nbsp;{{ZBL|12.0773.01}} JFM&nbsp;{{ZBL|12.0398.01}}
  
<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/l/l060/l060800/l06080016.png" /></td> </tr></table>
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|-
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|valign="top"|{{Ref|Wa}}||valign="top"|  G.N. Watson,  "A treatise on the theory of Bessel functions", '''1''', Cambridge Univ. Press  (1952)  {{MR|1349110}} {{MR|1570252}} {{MR|0010746}} {{MR|1520278}}  {{ZBL|0849.33001}} {{ZBL|0174.36202}} {{ZBL|0063.08184}} JFM&nbsp;{{ZBL|48.0412.02}} JFM&nbsp;{{ZBL|50.0264.01}}
  
Here, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080017.png" /> the first sum is taken to be zero, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060800/l06080018.png" /> is a Bessel function (cf. [[Bessel functions|Bessel functions]]). Lommel functions in two variables are also known.
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|-
 
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See also [[Anger function|Anger function]]; [[Weber function|Weber function]]; [[Struve function|Struve function]].
 
 
 
Lommel functions were studied by E. Lommel [[#References|[1]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lommel,  "Zur Theorie der Bessel'schen Funktionen IV"  ''Math. Ann.'' , '''16'''  (1880)  pp. 183–208</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1''' , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR></table>
 

Revision as of 11:43, 21 February 2014

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

A solution of the non-homogeneous Bessel equation $$x^2y''+xy'+(x^2-\nu^2)y = x^\rho.$$ If $\rho = \nu+2n$, where $n$ is a natural number, then

$$y=(-1)^{n-1}(n-1)!\; 2^{\nu+2n-2} \sum_{k=0}^{n-1} (-1)^k\big(\frac{x}{2}\big)^{\nu+2k} \frac{\def\G{\Gamma}\G(\nu+n)}{k!\G(\nu+k+1)}.$$

If the numbers $\rho+\nu\ge 0$ and $\rho-\nu\ge 0$ are not integers, then

$$y=2^{\rho -2}\G(\frac{\rho+\nu}{2})\G(\frac{\rho-\nu}{2})\sum_{k=0}^\infty\frac{(-1)^k(x/2)^{\rho+2k}}{\G(k+1+(\rho+\nu)/2)\G(k+1+(\rho-\nu)/2)}.$$

If $\rho = \nu-2n$, where $\nu\ge 0$ is an integer and $\nu$ is not an integer $\le n$, then

$$y=\frac{\G(\nu-n)}{n!\; 2^{-\nu+2n+2}}\Big[2J_\nu(x)\ln\frac{x}{2}-\sum_{k=0}^{n-1}\frac{(n-k-1)!}{\G(\nu-n+k+1)}\big(\frac{x}{2}\big)^{\nu-2n+2k}-\sum_{k=0}^{\infty}\Big(\frac{(-1)^k(x/2)^{\nu+2k}}{k!\;\G(\nu+k+1)}-\frac{\G'(k+1)}{\G(k+1)}-\frac{\G'(\nu+k+1)}{\G(\nu+k+1)}\Big)\Big].$$

Here, for $n=0$ the first sum is taken to be zero, and $J_\nu(x)$ is a Bessel function (cf. Bessel functions). Lommel functions in two variables are also known.

See also Anger function; Weber function; Struve function.

Lommel functions were studied by E. Lommel [Lo].

References

[Ka] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1947)
[Lo] E. Lommel, "Zur Theorie der Bessel'schen Funktionen IV" Math. Ann., 16 (1880) pp. 183–208 MR1510022 JFM Zbl 12.0773.01 JFM Zbl 12.0398.01
[Wa] G.N. Watson, "A treatise on the theory of Bessel functions", 1, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184 JFM Zbl 48.0412.02 JFM Zbl 50.0264.01
How to Cite This Entry:
Lommel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lommel_function&oldid=12539
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article