Editing Hankel transform (section)

== Some Hankel transform pairs ==
<ref>{{cite book |last=Papoulis |first=Athanasios |title=Systems and Transforms with Applications to Optics |year=1981 |publisher=Krieger Publishing Company |location=Florida USA |isbn=978-0898743586 |pages=140–175}}</ref>

{|class="wikitable"
! <math>f(r)</math>
! <math>F_0(k)</math>
|-
| <math>1</math>
| <math>\frac{\delta(k)}{k}</math>
|-
| <math>\frac{1}{r}</math>
| <math>\frac{1}{k}</math>
|-
| <math>r</math>
| <math>-\frac{1}{k^3}</math>
|-
| <math>r^3</math>
| <math>\frac{9}{k^5}</math>
|-
| <math>r^m</math>
| <math>\frac{\, 2^{m+1} \, \Gamma \left( \tfrac{m}{2} + 1 \right) \,}{k^{m+2} \, \Gamma\left( -\tfrac{m}{2} \right)}, \quad -2 < \mathcal{R_e} \{ m \} < -\tfrac{1}{2}</math>
|-
| <math>\frac{1}{\sqrt{r^2 + z^2\,}}</math>
| <math>\frac{\, e^{-k|z|} \,}{k}</math><!--  = \sqrt{ \frac{2|z| \,}{\pi k}}K_{-\tfrac{1}{2}}(k|z|) \, ref Smythe 1968  -->
|-
| <math>\frac{1}{\, z^2 + r^2 \,}</math>
| <math>K_0(kz), \quad z \in \mathbb{C}</math>
|-
|rowspan="2"| <math>\frac{e^{iar}}{r}</math>
| <math>\frac{i}{\, \sqrt{a^2 - k^2 \,} \,}, \quad a > 0, \; k < a</math>
|-
| <math>\frac{1}{\,\sqrt{k^2 - a^2\,}\,}, \quad a > 0, \; k > a</math>
|-
| <math>e^{-\frac{1}{2} a^2r^2}</math>
| <math>\frac{1}{\,a^2\,} \, e^{-\tfrac{k^2}{2\,a^2}}</math>
|-
| <math>\frac{1}{r} J_0(lr) \, e^{-sr}</math>
| <math>\frac{2}{\, \pi \sqrt{ (k + l)^2 + s^2 \,} \,} K\left( \sqrt{\frac{4kl}{(k + l)^2 + s^2} \,} \right)</math>
|-
| <math>-r^2 f(r)</math>
| <math>\frac{\, \mathrm{d}^2 F_0 \,}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_0 \,}{\mathrm{d}k}</math>
|}

{|class="wikitable"
! <math>f(r)</math>
! <math>F_\nu(k)</math>
|-
| <math>r^s</math>
| <math>\frac{2^{s+1}}{\, k^{s+2} \,} \, \frac{\Gamma\left(\tfrac{1}{2}(2 + \nu + s)\right)}{\Gamma(\tfrac{1}{2} (\nu - s))}</math>
|-
| <math>r^{\nu-2s} \Gamma(s, r^2 h)</math>
| <math>\tfrac{1}{2} \left(\tfrac k 2\right)^{2s-\nu-2} \gamma\left(1 - s + \nu, \tfrac{k^2}{4h} \right)</math>
|-
| <math>e^{-r^2} r^\nu \, U(a, b, r^2)</math>
| <math>\frac{\Gamma(2 + \nu - b)}{\, 2\, \Gamma(2 + \nu - b + a)} \left(\tfrac k 2\right)^\nu \, e^{-\frac{k^2}{4} \,} \, _1F_1\left( a, 2 + a - b + \nu, \tfrac{k^2}{4} \right)</math>
|-
| <math>r^n J_\mu(lr) \, e^{-sr}</math>
| Expressable in terms of [[elliptic integral]]s.<ref>{{cite journal | last1 = Kausel | first1 = E. | last2 = Irfan Baig | first2 = M.M. | year = 2012 | title = Laplace transform of products of Bessel functions: A visitation of earlier formulas | journal = Quarterly of Applied Mathematics | volume = 70 | pages = 77–97 | hdl = 1721.1/78923 | url = http://dspace.mit.edu/bitstream/1721.1/78923/1/Kausel_Baig.pdf | doi = 10.1090/s0033-569x-2011-01239-2 | doi-access = free}}</ref>
|-
| <math>-r^2 f(r)</math>
| <math>\frac{\mathrm{d}^2 F_\nu}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_\nu \,}{\mathrm{d}k} - \frac{\nu^2}{k^2} \, F_\nu</math>
|}

{{math|''K<sub>n</sub>''(''z'')}} is a [[modified Bessel function of the second kind]].
{{math|''K''(''z'')}} is the [[complete elliptic integral of the first kind]].

The expression
: <math>\frac{\, \mathrm{d}^2 F_0 \,}{\mathrm{d}k^2} + \frac{1}{k} \frac{\, \mathrm{d} F_0 \,}{\mathrm{d}k}</math>
coincides with the expression for the [[Laplace operator]] in [[polar coordinates]] {{math|( ''k'', ''θ'' )}} applied to a spherically symmetric function {{math| ''F''<sub>0</sub>(''k'') .}}

The Hankel transform of [[Zernike polynomial]]s are essentially Bessel Functions (Noll 1976):

: <math>R_n^m(r) = (-1)^{\frac{n-m}{2}} \int_0^\infty J_{n+1}(k) J_m(kr) \,\mathrm{d}k</math>

for even {{math|''n'' − ''m'' ≥ 0}}.