Lommel function
title: "Lommel function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["special-functions", "ordinary-differential-equations"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Lommel_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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The Lommel differential equation, named after Eugen von Lommel, 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}.
Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by ,
: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], :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 Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.
The s function can also be written as : s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_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 pFq is a generalized hypergeometric function.
References
References
- Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)
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