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Mathematica
8: New and Improved Core Algorithms
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Core Algorithms
Perform Numerical Hankel Transform
Hankel transform
is given by
. Hankel transform is computed numerically thanks to
Mathematica
8's efficient algorithms for integration of oscillatory functions.
In[1]:=
X
Hankel1[k_?NumberQ] := Quiet[NIntegrate[ Cos[Pi r]/(1 + r^2) BesselJ[1, k r] r, {r, 0, Infinity}]]; fplot = Plot[Cos[Pi r]/(1 + r^2), {r, 0, 10}, PlotRange -> All, PlotStyle -> Darker[Red], Epilog -> Inset[TraditionalForm[ Style[HoldForm[f[r] == Cos[Pi r]/(1 + r^2)], 16]], Scaled[{0.3, 0.65}], {Left, Bottom}]]; Plot[Hankel1[k], {k, 0, 12}, PlotRange -> All, ImageSize -> 500, PlotLabel -> Style[Row[{"Hankel Transform", " ", TraditionalForm[ HoldForm[Subscript[\[ScriptCapitalH], 1][f][k]]]}], 16, FontFamily -> "Helvetica"], Epilog -> Inset[Framed[Show[fplot, ImageSize -> 280], RoundingRadius -> 10, FrameStyle -> LightGray], Scaled[{1, 0}], {Right, Bottom}]]
Out[1]=