Y and H transforms

In mathematics, the Template:Math transforms and Template:Math transforms are complementary pairs of integral transforms involving, respectively, the Neumann function (Bessel function of the second kind) Template:Math of order Template:Math and the Struve function Template:Math of the same order.

For a given function Template:Math, the Template:Math-transform of order Template:Math is given by

$F(k)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(r)Y_{\nu }(kr)\sqrt{kr}dr$

The inverse of above is the Template:Math-transform of the same order; for a given function Template:Math, the Template:Math-transform of order Template:Math is given by

$f(r)=\underset{0}{\overset{\mathrm{\infty }}{\int }}F(k){𝐇}_{\nu }(kr)\sqrt{kr}dk$

These transforms are closely related to the Hankel transform, as both involve Bessel functions. In problems of mathematical physics and applied mathematics, the Hankel, Template:Math, Template:Math 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 Template:Math, Template:Math 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 (Template:Math-transforms) and Chapter XI (Template:Math-transforms).
• Template:Cite journal