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Re: FoourierTransform of a function defined in sections


Hello.

On version 5.1 and newer you can work as follows:

Clear[hh]

hh[x_] := Piecewise[{{Sin[x]^2, Abs[x] <= Pi}, {0, x > Pi || x < Pi}}]

Here is its plot

Plot[hh[x], {x, -4, 4}]

Here is the FT

FourierTransform[hh[x], x, s]
-((2*Sqrt[2/Pi]*Sin[Pi*s])/(-4*s + s^3))

Here is the numrical value for s=1/2
N[% /. s -> 1/2]
0.8510768648563898

Here is the check

Chop[NIntegrate[(1/Sqrt[2*Pi])*hh[x]*Exp[I*s*x] /. s -> 1/2, {x,
-Infinity, Infinity}]]
0.8510768648563898


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