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 sμ,ν(z) and Sμ,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 Jν(z) is a Bessel function of the first kind, and Yν(z) a Bessel function of the second kind.