Lommel function

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

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

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

[math]\displaystyle{ s_{\mu,\nu}(z) = \frac{\pi}{2} \left[ Y_{\nu} (z) \! \int_{0}^{z} \!\! x^{\mu} J_{\nu}(x) \, dx - J_\nu (z) \! \int_{0}^{z} \!\! x^{\mu} Y_{\nu}(x) \, dx \right], }[/math]

[math]\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), }[/math]

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

See also

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

References

• 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.