Lommel function

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.