Fourier Transform Pairs

f1[yy_,aa_]:=GraphicsRow[{ Plot[1/Sqrt[2 aa] Piecewise[{{0,yy<-aa},{1,yy>-aa&&yy<aa},{0,yy>aa}}],{yy,-5,5},
AxesLabel->{Style["x",24,Italic],Style["f(x)",24,Italic]},PlotStyle->{Thickness[.008],Black},PlotRange->{{-2.2,2.2},{-.1,1.1}},Ticks->{{-2,-1,0,1,2},{0,0.5,1}},TicksStyle->Directive[Gray,20]],
Plot[-((Sin[aa w] (-1+UnitStep[-aa]))/((Sqrt[aa]) (Sqrt[π]) w)),{w,-15,15},AxesLabel->{Style["k",24,Red,Italic],Style["(OverTilde[f])(k)",24,Italic,Red]},PlotStyle->{Thickness[.008],Red},PlotRange->{-.5,1},Ticks->{{-10,0,10},{-0.5,0,0.5,1}},TicksStyle->Directive[Gray,20]]}];
f2[yy_,aa_]:=GraphicsRow[{Plot[(2 aa/Pi)^(1/4) Exp[-aa yy^2],{yy,-3.75,3.75},AxesLabel->{Style["x",24,Italic],Style["f(x)",24,Italic]},PlotStyle->{Thickness[.008],Black},PlotRange->{0,1.1},Ticks->{{-3,0,3},{0,0.5,1}},TicksStyle->Directive[Gray,20]],
Plot[((E^(-((w^2)/(4 aa))))/((aa^(1/4)) (((2 π))^(1/4)))),{w,-7.2,7.2},AxesLabel->{Style["k",24,Red,Italic],Style["(OverTilde[f])(k)",24,Italic,Red]},PlotStyle->{Thickness[.008],Red},PlotRange->{0,1.1},Ticks->{{-6,0,6},{0,0.5,1}},TicksStyle->Directive[Gray,20]]}];
 f3[yy_,aa_]:=GraphicsRow[{Plot[Sqrt[((16 (aa^(5/2)))/(3 Sqrt[Pi/2]))  ] yy^2 Exp[-aa yy^2],{yy,-5,5},AxesLabel->{Style["x",24,Italic],Style["f(x)",24,Italic]},PlotStyle->{Thickness[.008],Black},PlotRange->{-.1,1},Ticks->{{-4,0,4},{0,0.5,1}},TicksStyle->Directive[Gray,20]],
Plot[(((Sqrt[(aa^(5/2))]) (E^(-((w^2)/(4 aa)))) (2 aa-(w^2)))/((Sqrt[3]) (aa^(5/2)) (((2 π))^(1/4)))),{w,-10,10},AxesLabel->{Style["k",24,Red,Italic],Style["(OverTilde[f])(k)",24,Italic,Red]},PlotStyle->{Thickness[.008],Red},PlotRange->{-.5,1},Ticks->{{-8,0,8},{-0.5,0,0.5,1}},TicksStyle->Directive[Gray,20]]}];
f4[yy_,aa_]:=GraphicsRow[{

Plot[(Sqrt[aa]) (E^(-aa yy)) ((E^(2 aa yy)) HeavisideTheta[-yy]+HeavisideTheta[yy]),{yy,-10,10},AxesLabel->{Style["x",24,Black,Italic],Row[{Style["f(x)",24,Italic,Black]}]},PlotStyle->{Thickness[.008],Black},PlotRange->{0,1.5},Ticks->{{-8,0,8},{0,0.5,1}},TicksStyle->Directive[Gray,20]],

Plot[Sqrt[aa/(2 Pi)] 2 aa/(w^2+aa^2),{w,-10,10},AxesLabel->{Style["k",24,Red,Italic],Style["(OverTilde[f])(k)",24,Italic,Red]},PlotStyle->{Thickness[.008],Red},PlotRange->{{-8.2,8.2},{0,1.2}},Ticks->{{-8,0,8},{0,1,2}},TicksStyle->Directive[Gray,20]]
}];

Manipulate[
Show[ffun[y,a1],ImageSize->{575,400}],

{{ffun,f1,Row[{Style["f",Italic],"(",Style["x",Italic],")"}]},
{f1->"top hat",
f2->Row[{  "exp(","-",Style["a",Italic]," ",((Style["x",Italic])^2),")"}],
f3-> Row[{((Style[ "x",Italic])^2), "exp(","-",Style["a",Italic]," "," ",((Style["x",Italic])^2),")"}],
f4-> Row[{ "exp(","-",Style["a",Italic]," \[LeftBracketingBar]",Style["x",Italic],"\[LeftBracketingBar])"}]}},

{{a1,.5,Style["a",Italic]},.5,2,Appearance->"Labeled"}
,SaveDefinitions->True,TrackedSymbols->True]

CAPTION

Here are four examples of Fourier transform pairs related by (OverTilde[f])(k)=(1/(Sqrt[2 π]))(∫_(-∞)^∞)f(x)(e^(i k x))dx, where the slider a is a width parameter.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Fourier Transform Pairs" from the Wolfram Demonstrations Project
 https://demonstrations.wolfram.com/FourierTransformPairs/

Contributed by: Porscha McRobbie and Eitan Geva

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