Lommel function

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

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

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

${\displaystyle s_{\mu ,\nu }(z)=\frac{\pi }{2}[Y_{\nu }(z){\displaystyle \underset{0}{\overset{z}{\int }}}{x}^{\mu }{J}_{\nu }(x)dx-J_{\nu }(z){\displaystyle \underset{0}{\overset{z}{\int }}}{x}^{\mu }{Y}_{\nu }(x)dx],}$

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

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]

${\displaystyle s_{\mu ,\nu }(z)=\frac{z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}{}_1{F}_2(1;\frac{\mu }{2}-\frac{\nu }{2}+\frac{3}{2},\frac{\mu }{2}+\frac{\nu }{2}+\frac{3}{2};-\frac{z^2}{4}),}$

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, 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 

• 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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