User:Pershyn/sandbox

Weber's equation
The following equation is called Weber's equation:
 * $$\frac{d^2f}{dz^2} + \left(\nu +\frac12-\frac{z^2}{4}\right)f=0 $$


 * $$ ~D_\nu (z)= \dfrac{\Gamma(\nu+1)}{\sqrt{2 \pi}} \left( e^{\frac{1}{2}\nu \pi i} D_{-\nu - 1}(i z) + e^{-\frac{1}{2}\nu \pi i} D_{-\nu - 1}(-i z) \right) $$


 * $$ ~D_{\nu+1} (z) - z D_{\nu}(z) + \nu D_{\nu-1}(z) = 0 $$


 * $$\frac{d}{dz}~D_\nu (z) + \dfrac{1}{2} z D_\nu(z) - \nu D_{\nu-1}(z) = 0 $$