Hankel Transform

Author

Eric W. Weisstein
March 16, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegralTransforms/HankelTransform.nb.

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Initialization

HankelTransform1[f_,q_,assump_List:{}]:=Module[{x},
FullSimplify[Integrate[2Pi f[x]BesselJ[0,2Pi q x]x,{x,0,Infinity},
Assumptions->{x>0}],assump]
]

HankelTransform1[f_,q_,a_,assump_List:{}]:=Module[{x},
FullSimplify[Integrate[2Pi f[x]BesselJ[0,2Pi q x]x,{x,0,a},
Assumptions->{x>0}],assump]
]

Options

?HankelTransform

Options[HankelTransform]

Options[FourierTransform]

(Subscript[Π, a])(r)

Built-in

HankelTransform[a UnitStep[r],r,s]

HankelTransform1

HankelTransform[Function[f,1],q,a,{q>0,a>0}]

((sin (2 π a r))/r)

Built-in

HankelTransform[(1/(2π))((Sin[a (r/(2π))])/(r/(2π))),r,s,Assumptions->a>0]

FullSimplify[%]

HankelTransform1

HankelTransform[Sin[2Pi a#]/#&,q]

HankelTransform[Sin[2Pi a#]/#&,q,a]

(1/2) δ(r-a)

Built-in

HankelTransform[(1/(2π))DiracDelta[(r/(2π))- a]/2,r,q]

HankelTransform1

Plot[UnitStep[x],{x,-1,1},PlotStyle->{Red,Thickness[.007]}]

HankelTransform[DiracDelta[#-a]/2&,q]

M (a r)

m[x_]:=FullSimplify[(x^(-3)Integrate[BesselJ[0,xx],{xx,0,x}]-BesselJ[0,x]/x^2)](2Pi)

m[x]

Built-in

HankelTransform1

HankelTransform[m[#  a]&,q,{a>0}]

(e^(-π (r^2)))

Built-in

HankelTransform[(1/(2π)) Exp[-Pi ((((r/(2π ))))^2)],r,q]

HankelTransform1

HankelTransform[Exp[-Pi #^2]&,q,{q>0}]

(1/(Sqrt[(a^2)+(r^2)]))

Built-in

HankelTransform[(1/(2π)) (1/(Sqrt[(a^2)+((((r/(2π))))^2)])),r,q,Assumptions->a>0]

HankelTransform1

HankelTransform[1/Sqrt[a^2+#^2]&,q,{q>0,a>0}]

(1/((((a^2)+(r^2)))^(3/2)))

Built-in

HankelTransform[(1/(2π)) (1/((((a^2)+((((r/(2π))))^2)))^(3/2))),r,q,Assumptions->a>0]

HankelTransform1

HankelTransform[(a^2+#^2)^(-3/2)&,q,{q>0,a>0}]

(1/((a^2)+(r^2)))

Built-in

HankelTransform[(1/(2π)) (1/((a^2)+((((r/(2π))))^2))),r,q,Assumptions->a>0]

FullSimplify[%]

HankelTransform1

HankelTransform[1/(a^2+#^2)&,q,{q>0,a>0}]

((2 (a^2))/((((a^2)+(r^2)))^2))

Built-in

HankelTransform[(1/(2π)) ((2(a^2))/((((a^2)+((((r/(2π))))^2)))^2)),r,q,Assumptions->a>0]

FullSimplify[%,a>0]

HankelTransform1

HankelTransform[2a^2/(a^2+#^2)^2&,q,{q>0,a>0}]

((4 (a^4))/((((a^2)+(r^2)))^3))

Built-in

HankelTransform[(1/(2π)) ((4(a^4))/((((a^2)+((((r/(2π))))^2)))^3)),r,q,Assumptions->a>0]

FullSimplify[%,a>0]

HankelTransform1

HankelTransform[4a^4/(a^2+#^2)^3&,q,{q>0,a>0}]

FullSimplify[4 (a^2) (π^3) (q^2) BesselK[2,2 a π q]-(4Pi^2 a q BesselK[1,2Pi a q]+4 Pi^3 a^2 q^2 BesselK[0,2Pi a q])]

(a^2)-(r^2)

Built-in

HankelTransform[(1/(2π))((a^2)-(r^2)),r,q]

HankelTransform[(1/(2π))((a^2)-((((r/(2π))))^2)),r,q,Assumptions->a>0]

FullSimplify[%,a>0]

HankelTransform1

HankelTransform[a^2-#^2&,q,a,{q>0,a>0}]

(1/r)

Built-in

HankelTransform[(1/(2π))(1/(r/(2π))),r,q]

HankelTransform1

HankelTransform[1/#&,q,{q>0}]

(e^(-a r))

Built-in

FullSimplify[HankelTransform[(1/(2π))Exp[-a (r/(2π))],r,q],a>0]

HankelTransform1

HankelTransform[Exp[-a #]&,q,{q>0,a>0}]

((e^(-a r))/r)

Built-in

FullSimplify[HankelTransform[(1/(2π))((Exp[-a (r/(2π))])/(r/(2π))),r,q],a>0]

HankelTransform1

HankelTransform[Exp[-a #]/#&,q,{q>0,a>0}]

((δ(r))/(2 π r))

Built-in

HankelTransform[(1/(2π))((DiracDelta[(r/(2π))])/(2π(r/(2π)))),r,q]

HankelTransform1

HankelTransform[DiracDelta[#]/2/Pi/#&,q]

(1/(2 (a^2)))

Built-in

HankelTransform[(1/(2π))(1/(2(a^2))),r,q]

HankelTransform1

HankelTransform[1/(2a^2)&,q,a,{a>0,q>0}]

(r^2) (e^(-π (r^2)))

Built-in

HankelTransform[(1/(2π))((((r/(2π))))^2)Exp[-π ((((r/(2π))))^2)],r,q]//Simplify

HankelTransform1

HankelTransform[#^2 Exp[-Pi #^2]&,q,{q>0}]