Lommel function

thumb thumb The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

z2d2ydz2+zdydz+(z2−ν2)y=zμ+1.

Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),

sμ,ν(z)=π2[Yν(z)∫0zxμJν(x)dx−Jν(z)∫0zxμYν(x)dx],
Sμ,ν(z)=sμ,ν(z)+2μ−1Γ(μ+ν+12)Γ(μ−ν+12)(sin⁡[(μ−ν)π2]Jν(z)−cos⁡[(μ−ν)π2]Yν(z)),

where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.

The s function can also be written as[1]

sμ,ν(z)=zμ+1(μ−ν+1)(μ+ν+1)1F2(1;μ2−ν2+32,μ2+ν2+32;−z24),

where pFq is a generalized hypergeometric function.

## See also

* Anger function
* Lommel polynomial
* Struve function
* Weber function

## References

1. ↑ Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)

* Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II, McGraw-Hill Book Company, Inc., New York-Toronto-London, http://apps.nrbook.com/bateman/Vol2.pdf
* Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann. 9 (3): 425–444, doi:10.1007/BF01443342, https://zenodo.org/record/1568162
* Lommel, E. (1880), "Zur Theorie der Bessel'schen Funktionen IV", Math. Ann. 16 (2): 183–208, doi:10.1007/BF01446386
* Paris, R. B. (2010), "Lommel function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/11.9
* Hazewinkel, Michiel, ed. (2001), "l/l060800", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=l/l060800

## External links

* Weisstein, Eric W. "Lommel Differential Equation." From MathWorld—A Wolfram Web Resource.
* Weisstein, Eric W. "Lommel Function." From MathWorld—A Wolfram Web Resource.

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