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