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

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

Out= 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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