Lommel function

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

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

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

${\displaystyle s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left(Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right),}$

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

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

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

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

where ${\displaystyle {}_{1}F_{2}}$ is a generalized hypergeometric function.

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

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