Lommel function

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

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

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

$s_{\mu,\nu}(z) = \frac{1}{2} \pi \left[ Y_\nu (z) \int_0^z z^\mu J_\nu (z)\, dz - J_\nu (z) \int_0^z z^\mu Y_\nu (z)\, dz\right]$

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

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

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 (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), "l/l060800", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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.