Cylinder Function

Author

Eric W. Weisstein
June 7, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/CylinderFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CylinderFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plot

CylinderFunction[x_,y_]:=If[x^2+y^2<=1,1,0]

Plot3D[CylinderFunction[x,y],{x,-2,2},{y,-2,2},PlotPoints->50]

Plot3D[1-UnitStep[x^2+y^2-1],{x,-2,2},{y,-2,2},PlotPoints->50]

LiveGraphics3D

WriteLiveForm["cylfun.m",
	Plot3D[CylinderFunction[x,y],{x,-2,2},{y,-2,2},PlotPoints->30]]

Fourier Transform

v5

FourierTransform[1-UnitStep[x^2+y^2-1],{x,y},{kx,ky},FourierParameters->{0,-2π}]//Timing

...wrong...

FullSimplify[Integrate[Exp[I k r Cos[θ-α]]r,{r,0,a}],0<θ<2π&&k>0&&α>0]

Integrate[%,{θ,0,2π}]//Timing

Integrate[Exp[I k r Cos[b]]r,{b,0,2π},{r,0,a},Assumptions->a>0&&k>0&&r>0]//Timing

Integrate[Exp[I k r Cos[b]]r,{r,0,a},{b,0,2π},Assumptions->a>0&&k>0&&r>0]//Timing

v6

FourierTransform[1-UnitStep[x^2+y^2-1],{x,y},{kx,ky},FourierParameters->{0,-2π}]//Timing

Integrate[Exp[I k r Cos[θ-α]],{θ,0,2π}]

FullSimplify[Integrate[Exp[I k r Cos[θ-α]]r,{r,0,a}],0<θ<2π&&k>0&&α>0]

Integrate[%,{θ,0,2π}]

Integrate[Exp[I k r Cos[b]]r,{b,0,2π},{r,0,a},Assumptions->k>0&&r>0]//Timing

Integrate[Exp[I k r Cos[b]]r,{r,0,a},{b,0,2π},Assumptions->r>0&&a>0&&k>0]//Timing