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}.

Two 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.

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