Editing Hankel transform (section)

==Definition==
The '''Hankel transform''' of order  <math>\nu</math> of a function ''f''(''r'') is given by

: <math>F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrm{d}r,</math>

where <math>J_\nu</math> is the [[Bessel function]] of the first kind of order <math>\nu</math> with <math>\nu \geq -1/2</math>. The inverse Hankel transform of {{math|''F<sub>ν</sub>''(''k'')}} is defined as

: <math>f(r) = \int_0^\infty F_\nu(k) J_\nu(kr) \,k\,\mathrm{d}k,</math>

which can be readily verified using the orthogonality relationship described below.

===Domain of definition===
Inverting a Hankel transform of a function ''f''(''r'') is valid at every point at which ''f''(''r'') is continuous, provided that the function is defined in (0,&nbsp;∞), is piecewise continuous and of [[bounded variation]] in every finite subinterval in (0,&nbsp;∞), and

: <math>\int_0^\infty |f(r)|\,r^{\frac{1}{2}} \,\mathrm{d}r < \infty.</math>

However, like the Fourier transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example <math>f(r) = (1 + r)^{-3/2}</math>.

===Alternative definition===
An alternative definition says that the Hankel transform of ''g''(''r'') is<ref>{{cite book |title=Hilbert spaces of entire functions |url=https://archive.org/details/hilbertspacesofe0000debr |url-access=registration |year=1968 |publisher=Prentice-Hall |location=London |isbn=978-0133889000 |author=Louis de Branges |authorlink=Louis de Branges de Bourcia |page=[https://archive.org/details/hilbertspacesofe0000debr/page/189 189]}}</ref>

: <math>h_\nu(k) = \int_0^\infty g(r) J_\nu(kr) \,\sqrt{kr}\,\mathrm{d}r.</math>

The two definitions are related:
: If <math>g(r) = f(r) \sqrt r</math>, then <math>h_\nu(k) = F_\nu(k) \sqrt k.</math>
This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse:
: <math>g(r) = \int_0^\infty h_\nu(k) J_\nu(kr) \,\sqrt{kr}\,\mathrm{d}k.</math>
The obvious domain now has the condition
: <math>\int_0^\infty |g(r)| \,\mathrm{d}r < \infty,</math>
but this can be extended. According to the reference given above, we can take the integral as the limit as the upper limit goes to infinity (an [[improper integral]] rather than a [[Lebesgue integral]]), and in this way the Hankel transform and its inverse work for all functions in [[L2-space|L<sup>2</sup>]](0, ∞).