Y and H transforms

In mathematics, the $Y$ transforms and $H$ transforms are complementary pairs of integral transforms involving, respectively, the Neumann function (Bessel function of the second kind) $Y$ of order $H$ and the Struve function $Y_{ν}$ of the same order.

For a given function $ν$, the $H_{ν}$-transform of order $f(r)$ is given by


 * $$F(k) = \int_0^\infty f(r) Y_{\nu}(kr) \sqrt{kr} \, dr $$

The inverse of above is the $Y$-transform of the same order; for a given function $ν$, the $H$-transform of order $F(k)$ is given by
 * $$f(r) = \int_0^\infty F(k) \mathbf{H}_{\nu}(kr) \sqrt{kr} \, dk $$

These transforms are closely related to the Hankel transform, as both involve Bessel functions. In problems of mathematical physics and applied mathematics, the Hankel, $H$, $ν$ transforms all may appear in problems having axial symmetry. Hankel transforms are however much more commonly seen due to their connection with the 2-dimensional Fourier transform. The $Y$, $H$ transforms appear in situations with singular behaviour on the axis of symmetry (Rooney).