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) Yν of order ν and the Struve function Hν of the same order.

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

F ( k ) = ∫ 0 ∞ f ( r ) Y ν ( k r ) k r d r {\displaystyle 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 ) = ∫ 0 ∞ F ( k ) H ν ( k r ) k r d k {\displaystyle 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

  • 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 Yν and Hν transformations". Canadian Journal of Mathematics. 32 (5): 1021. doi:10.4153/CJM-1980-079-4.