Lommel function

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

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

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

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

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

The s function can also be written as^[1]

${\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),}${\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),}

where _pF_q is a generalized hypergeometric function.

• Anger function
• Lommel polynomial
• Struve function
• Weber function

• 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. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248 .
• Solomentsev, E.D. (2001) [1994], "Lommel function", Encyclopedia of Mathematics , EMS Press

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

• Special functions
• Ordinary differential equations

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