Talk:Hankel transform

Radially symmetric functions

The Hankel transform of order $n-1$ arises when you wish to perform the Fourier transform of a radially symmetric function (i.e. $f(r)$ where $r=|\mathbf {r} |$) in $n$ dimensions. I don't have good references for this, but it is referred to in the Integral transform section of The Encyclopedic Dictionary of Mathematics --Farmhouse121 03:46, 30 June 2006 (UTC)

in Mathematica

If you'd like to evaluate a Hankel Transform in Mathematica, it's relatively straightforward. Here's an example. FIrst we define the function to be transformed; in this case, let's use f(r) = r .

f[r_] := r;    

and now call Mathematica's numerical integrator:

NIntegrate[ f[r]*r*BesselJ[0, k*r], {r,0,Infinity}, Method -> Oscillatory ]

which will give the output -1/k^3, although it will likely give you a warning that it's unhappy. Because it's numerical integration, you need to actually specify the value of k before the computation, not after. If you don't specify

Method -> Oscillatory

then it won't converge, and you'll probably get a meaningless answer below machine precision.

Note that this integral converges only because the Bessel Function is oscillatory. If we require that $\int _{0}^{\infty }|f(r)|r^{1/2}dr<\infty$, then f(r) = r violates this condition since it increases without bound. Lavaka 17:25, 11 April 2007 (UTC)

Numerical evaluation of Hankel integrals: a fast and accurate algorithm (the QFHT)

If one's objective is to evaluate Hankel integrals of zero and higher orders for the purpose of doing optical beam propagation calculations in cylindrical coordinates, I truly believe that the fastest, most efficient, most accurate, and most compact algorithm for evaluating Hankel transforms of all orders digitally remains a "quasi Fast Hankel Transform" (QFHT) algorithm that I hacked together in the late 1970s, based on ideas derived from the early Cooley-Tukey Fast Fourier Transform (FFT) developments.

A Phys Rev paper summarizing this approach and giving earlier references is:

    S.-C. Sheng and A. E. Siegman, "Nonlinear optical calculations 
    using fast transform methods: second harmonic generation with 
    depletion and diffraction," Phys. Rev. A, vol. 21, pp. 599--606  
    (February 1980)/

The one unusual and potentially annoying aspect of this approach is that it (inherently and necessarily) uses exponentially rather than uniformly spaced sampling points along the radial coordinates. The vast improvements in computer technology since the 1970s may mean that other algorithms using equally spaced radial points and more conventional integration methods will do everything one needs with adequate accuracy.

If anyone actually tries this early approach and has good results with it, I'd be glad to hear about this, just to know that it still works!

Permission granted to anyone who wants to use this algorithm or distribute this information further, including inserting it into Wikipedia proper.

--A. E. Siegman, Stanford University <siegman@stanford.edu> —Preceding unsigned comment added by Siegman (talk • contribs) 21:09, 2 February 2009 (UTC)

Holy mathspeak, Batman

Can someone with the right decoder ring take a stab at explaining what the Hankel transform is in English instead of math language? Our intro paragraph currently relies entirely on an equation involving integral symbols and Bessel functions. --Doradus (talk) 20:23, 3 December 2010 (UTC)

I would say...

The Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind J_ν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis. The necessary coefficient F_ν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function.

However I suspect that the mathematicians would slap me down for such imprecise and inaccurate talk. --catslash (talk) 22:29, 3 December 2010 (UTC)

Thanks! I find that quite readable. I'd say be bold and stick it in there, and see what other editors say. --Doradus (talk) 02:44, 4 December 2010 (UTC)

Ok I've added it. --Doradus (talk) 18:41, 14 December 2010 (UTC)

Three questions

• Why must ν ≥ 1/2 ? Does orthogonality depend on this? or completeness?
• Can all functions of be transformed (for positive r)? What about functions with discontinuities?
• What applications are there? Solution of wave equations in cylindrical coordinates perhaps? or spherical coordinates?

Thanks --catslash (talk) 21:46, 14 December 2010 (UTC)

Another question

Shouldn't it be e^{-k.r} for the forward hankel transform (note the negative) — Preceding unsigned comment added by 101.171.42.170 (talk) 11:49, 17 May 2012 (UTC)

The article uses conventions for the Fourier transform that are strange (to me at least), so many of the signs appear to be the "wrong" way around. Sławomir Biały (talk) 11:58, 17 May 2012 (UTC)

Definition of Fourier transform incorrect

The definition of the Fourier transform is incorrect, since it has the wrong sign in the exponent. No convention, as far as I know, uses the possitive sign when calculating the Fourier transform and the negative sign when calculating the reverse Fourier transform. It should be the other way around. I started to change this once before, but it got reverted, which is understandable since I didn't go all the way and make the change on all places that needed it. However, it still needs to be changed, but I'm not sure that I have time right now to figure out what the exponents should look like in all expressions. So my contribution is: To create a to do list :) —Kri (talk) 08:36, 7 June 2012 (UTC)

citation to a proof of the orthogonality relation

Could somebody please provide a citation to a proof of the orthogonality relation?

Son of eugene (talk) 23:41, 7 June 2015 (UTC)

Why do we have it ?

The German Wikipedia at least mentions that the transformation is used in image processing for instance in handling image defects. But that's all I could find. It would be great if the article could be extended by examples of usefulness ... within mathematics and in practical applications. What are its advantages, in which situations do they become significant, are there weaknesses or drawbacks ... the whole nine yards. JB. --92.193.152.16 (talk) 16:44, 1 April 2020 (UTC)