Re: FourierTransform of a function defined in sections

• To: mathgroup at smc.vnet.net
• Subject: [mg70189] Re: FourierTransform of a function defined in sections
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 7 Oct 2006 07:06:59 -0400 (EDT)
• Organization: The University of Western Australia
• References: <eg4soa\$ffu\$1@smc.vnet.net>

```In article <eg4soa\$ffu\$1 at smc.vnet.net>,
"Eckhard Schlemm" <e.schlemm at hotmail.de> wrote:

> I want Mathematica to calculate the FourierTransform of a function which is
> defined by Sin[x]^2 for Abs[x]<PI and zero else.

What version of Mathematica are you using? For Piecewise functions, it
is preferable to use Piecewise.

> I tried and defined the function g as follows:
>
> g[x_]:=If[Abs[x]>PI,0,Sin[x]^2];

I assume that you have Pi instead of PI.

> That works fine.

Note that in version 5.2, PiecewiseExpand converts g[x] to a Piecewise
function -- and such operations are done automatically, if required. For
example,

Integrate[g[x], x]

yields

Piecewise[{{x/2 - Sin[2 x]/4, Abs[x] <= Pi}}, 0]

> But if I have mathematica try to determine the FourierTransform by
>
> FourierTransform[g[x],x,p]
>
> I always get the error that the recursion limit and the iteration limit were
> exceeded...
>
> what am I'm doing wrong?

In version 5.2 the following three expressions are identical:

FourierTransform[g[x], x, k]

FourierTransform[Piecewise[{{Sin[x]^2, -Pi <= x <= Pi}}], x, k]

1/Sqrt[2 Pi] Integrate[E^(I k x) Sin[x]^2, {x, -Pi, Pi}]

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)
AUSTRALIA                               http://physics.uwa.edu.au/~paul

```

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