In mathematics, the transforms and transforms are complementary pairs of

integral transform In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily charac ...

s involving, respectively, the Neumann function (

Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

of the second kind) of order and the Struve function of the same order. For a given function , the -transform of order is given by :

F(k) = \int_0^\infty f(r) Y_(kr) \sqrt \, dr

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

f(r) = \int_0^\infty F(k) \mathbf_(kr) \sqrt \, 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, , 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 , 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 (-transforms) and Chapter XI (-transforms). * {{Cite journal , doi = 10.4153/CJM-1980-079-4, title = On the {{math, {{mathcal, ''Y''_''ν'' and {{math, {{mathcal, ''H''_''ν'' transformations, journal = Canadian Journal of Mathematics, volume = 32, issue = 5, pages = 1021, year = 1980, last1 = Rooney , first1 = P. G., doi-access = free Integral transforms