       Re: Fourier Trasform of a Bessel Function

• To: mathgroup at smc.vnet.net
• Subject: [mg87814] Re: Fourier Trasform of a Bessel Function
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Fri, 18 Apr 2008 02:36:53 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <fu6mij\$nno\$1@smc.vnet.net>

```Vladislav wrote:

> I try to make the following
>
>> FourierTransform[BesselJ[n, x], x, w];
>> % /. n -> 0
>
> I try
>> FourierTransform[BesselJ[0, x], x, w]
> I obtain another result. Why?

The issue is that, by default, FourierTransform[] does not
generate/check any conditions on the parameters and thus returns the
most general case.

So, your first expression returns zero for n == 0, which is true for any
w > 1 or w < -1. However, for -1 < w < 1 the true value is not zero (see
below). And of course the expression is not defined for w == 1 or w == -1.

You can force FourierTransform[] to generate conditions and also put
some constraints on the parameters thanks to the options
GenerateConditions -> True and Assumptions, perspectively.

In:= expr = FourierTransform[BesselJ[0, x], x, w]

Out= (Sqrt[2/\[Pi]] (-HeavisideTheta[-1 + w] +
HeavisideTheta[1 + w]))/Sqrt[1 - w^2]

In:= FullSimplify[expr, Assumptions -> w > 1]

Out= 0

In:= FullSimplify[expr, Assumptions -> w < -1]

Out= 0

In:= FullSimplify[expr, Assumptions -> -1 < w < 1]

Out= Sqrt/Sqrt[\[Pi] - \[Pi] w^2]

In:= Plot[(-HeavisideTheta[-1 + w] +
HeavisideTheta[1 + w]), {w, -5, 5}]
Plot[expr, {w, -5, 5}]

In:= FourierTransform[BesselJ[n, x], x, w,
GenerateConditions -> True]

Out= If[w^2 > 1 && Re[n] > -1, (
E^((I n \[Pi])/
2) (-w + Abs[w]) (Sqrt[-1 + w^2] + Abs[w])^-n Sin[n \[Pi]])/(
Sqrt[2 \[Pi]] w Sqrt[-1 + w^2]),
FourierTransform[BesselJ[n, x], x, w, GenerateConditions -> True]]

In:= % /. n -> 0

Out= If[w^2 > 1, (
E^((I 0 \[Pi])/
2) (-w + Abs[w]) (Sqrt[-1 + w^2] + Abs[w])^-0 Sin[0 \[Pi]])/(
Sqrt[2 \[Pi]] w Sqrt[-1 + w^2]),
FourierTransform[BesselJ[0, x], x, w, GenerateConditions -> True]]

In:= % // Simplify

Out= \[Piecewise] {
{((Sqrt[2/\[Pi]] (-HeavisideTheta[-1 + w] + HeavisideTheta[1 + w]))/
Sqrt[1 - w^2]), -1 <= w <= 1}
}

Regards,
-- Jean-Marc

```

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