Lommel function

In mathematics, the Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation

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

Its solutions are given by the Lommel functions $s_{\mu ,\nu }(z)$ and $S_{\mu ,\nu }(z)$ :^[1]

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 $J_{\nu }(z)$ is a Bessel function of the first kind and $Y_{\nu }(z)$ a Bessel function of the second kind.

The function $s_{\mu ,\nu }$ can also be written as^[2]

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

References

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953). Higher Transcendental Functions. Vol. I, II. New York: McGraw-Hill. MR 0058756.

Lommel, E. (1880). "Zur Theorie der Bessel'schen Functionen". Mathematische Annalen (in German). 16 (2): 183–208. doi:10.1007/BF01446386.

Watson, G. N. (1922). Treatise on the Theory of Bessel Functions. Cambridge University Press. JFM 48.0412.02.

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