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MathGroup Archive 2007

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Hankel transformation // Fourier transformation for a circular function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76300] Hankel transformation // Fourier transformation for a circular function
  • From: Mathieu G <ellocomateo at free.fr>
  • Date: Sat, 19 May 2007 04:39:56 -0400 (EDT)

Hello,
I would like to compute the Hankel transformation (taken from Mathworld) 
of a 2D Gaussian beam & a circular aperture.
Then multiplying the two and doing the Hankel transformation again 
should give me the convolution of the 2D Gaussian beam with the 
aperture... But I must do something wrong?!
Also it seems there is some problem with my function DHole which I would 
like to use instead of Function[f, 1] in my definition of TFHole.

Any idea, experts?

Here is what I get:
What is that kind of output with $??



In[12]:= HankelTransform[f_, q_, assump_List: {}] := Module[{x},
   FullSimplify[
    Integrate[2 Pi f[x] BesselJ[0, 2 Pi q x] x, {x, 0, Infinity},
     Assumptions -> {x > 0}], assump]
   ]

HankelTransform[f_, q_, a_, assump_List: {}] := Module[{x},
   FullSimplify[Integrate[2 Pi f[x] BesselJ[0, 2 Pi q x] x, {x, 0, a},
     Assumptions -> {x > 0}], assump]
   ]

Gaussian2D[r_, Radius_: 1, Amplitude_: 1] :=
  Amplitude Exp[-1/2 (r/Radius)^2]
DHole[r_, HoleSize_: 1] := Boole[r <= HoleSize];

TFBeam[q_] := HankelTransform[Gaussian2D[#] &, q, {q > 0}]
TFBeam[q]
TFHole[q_] :=
  HankelTransform[Function[f, 1], q, a, {q > 0, a > 0}] /. a -> 1
TFHole[q]
TFConvolution[q_] := TFBeam[q]*TFHole[q]
Convolution[q_] := HankelTransform[TFConvolution[#] &, q, {q > 0}]
Convolution[q]

Out[17]= 2 \[ExponentialE]^(-2 \[Pi]^2 q^2) \[Pi]

Out[19]= BesselJ[1, 2 \[Pi] q]/q

Out[22]= Integrate[
  4 \[ExponentialE]^(-2 \[Pi]^2 x$5948^2) \[Pi]^2 BesselJ[0,
    2 \[Pi] q x$5948] BesselJ[1, 2 \[Pi] x$5948], {x$5948,
   0, \[Infinity]}, Assumptions -> {x$5948 > 0}]


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