$Y$ 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) $Y ν$ of order $ν$ and the Struve function $H ν$ of the same order.

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

F(k)=\int _{0}^{\infty }f(r)Y_{\nu }(kr){\sqrt {kr}}\,dr

The inverse of above is the $H$ -transform of the same order; for a given function $F (k)$ , the $H$ -transform of order $ν$ 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, $Y$ , $H$ 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).

References[edit]

Bateman Manuscript Project: Tables of Integral Transforms Vol. II. Contains extensive tables of transforms: Chapter IX ( $Y$ -transforms) and Chapter XI ( $H$ -transforms).
Rooney, P. G. (1980). "On the $Y ν$ and $H ν$ transformations". Canadian Journal of Mathematics. 32 (5): 1021. doi:10.4153/CJM-1980-079-4.