View source for Lommel function

{{MSC|34}}
{{TEX|done}}

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

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

(see 10.71 in {{Cite|Wa}}). If the numbers $\rho+\nu\ge 0$ and $\rho-\nu\ge 0$ are not integers, then

$$y=2^{\rho -2}\G(\frac{\rho+\nu}{2})\G(\frac{\rho-\nu}{2})\sum_{k=0}^\infty\frac{(-1)^k(x/2)^{\rho+2k}}{\G(k+1+(\rho+\nu)/2)\G(k+1+(\rho-\nu)/2)}$$
(see 10.7 in {{Cite|Wa}}).

If $\rho = \nu-2n$, where $n\ge 0$ 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|Bessel functions]]). Lommel functions in two variables are also known.

See also
[[Anger function|Anger function]];
[[Weber function|Weber function]];
[[Struve function|Struve function]].

Lommel functions were studied by E. Lommel
{{Cite|Lo}}.

====References====
{|
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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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|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}} 

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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}} 

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