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.