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

[math]\displaystyle{ F(k) = \int_0^\infty f(r) Y_{\nu}(kr) \sqrt{kr} \, dr }[/math]

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

[math]\displaystyle{ f(r) = \int_0^\infty F(k) \mathbf{H}_{\nu}(kr) \sqrt{kr} \, dk }[/math]

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

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.''Y''_''ν''+and+''H''_''ν''+transformations&rft.jtitle=Canadian+Journal+of+Mathematics&rft.aulast=Rooney&rft.aufirst=P.+G.&rft.au=Rooney, P.+G.&rft.date=1980&rft.volume=32&rft.issue=5&rft.pages=1021&rft_id=info:doi/10.4153/CJM-1980-079-4&rfr_id=info:sid/en.wikibooks.org:Y_and_H_transforms">

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