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Lommel function

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2010 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

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

(see 10.71 in [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 [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). 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://www.encyclopediaofmath.org/index.php?title=Lommel_function&oldid=31326
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article