Lommel function

The Lommel differential equation is an inhomogeneous form of the Bessel differential equation:

z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.

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

s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left[Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,\mathrm {d} x-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,\mathrm {d} x\right]

\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma {\bigg (}{\frac {\mu +\nu +1}{2}}{\bigg )}\Gamma {\bigg (}{\frac {\mu -\nu +1}{2}}{\bigg )}{\bigg (}\sin {\bigg [}(\mu -\nu ){\frac {\pi }{2}}{\bigg ]}J_{\nu }(z)-\cos {\bigg [}(\mu -\nu ){\frac {\pi }{2}}{\bigg ]}Y_{\nu }(z){\bigg )}

where J_ν(z) is a Bessel function of the first kind, and Y_ν(z) a Bessel function of the second kind.

References

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann., 9 (3): 425–444, doi:10.1007/BF01443342
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.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Solomentsev, E.D. (2001) [1994], "l/l060800", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.