Abel Transform

Author

Eric W. Weisstein
June 2, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Code

AbelTransform[f_,r_,x_,opts___]:=2Integrate[f r/Sqrt[r^2-x^2],{r,x,Infinity},opts]

Π[x_,a_]:=UnitStep[x]-UnitStep[x-a]

Table

𝒜[1]==2 (Sqrt[(a^2)-(x^2)])

V5

AbelTransform[Π[r,a],r,x,Assumptions->0<x<a]

V5.1 Direct

2Integrate[(r/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]

V5.1

AbelTransform[Π[r,a],r,x]

Simplify[%,0<x<a]

...wrong...

V5.1 Assumptions

AbelTransform[Π[r,a],r,x,Assumptions->0<x<a]

...wrong...

V6

2Integrate[(r/(Sqrt[r^2-x^2])),{r,x,a}]

Simplify[%,0<x<a]

2Integrate[(r/(Sqrt[r^2-x^2])),{r,x,a},Assumptions->0<x<a]

AbelTransform[Π[r,a],r,x]

Simplify[%,0<x<a]

AbelTransform[Π[r,a],r,x,Assumptions->0<x<a]//Timing

2Integrate[(r/(Sqrt[(r^2)-(x^2)])),{r,x,a}]

Simplify[%,0<x<a]

2Integrate[(r/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]

V10

$Version

2Integrate[(r/(Sqrt[r^2-x^2])),{r,x,a}]//Timing

Simplify[Last[%],0<x<a]

2Integrate[(r/(Sqrt[r^2-x^2])),{r,x,a},Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a],r,x]//Timing

Simplify[Last[%],0<x<a]

AbelTransform[Π[r,a],r,x,Assumptions->0<x<a]//Timing

𝒜[(((Subscript[Π, a])(r))/(Sqrt[(a^2)-(r^2)]))]==π

V5.0

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x,Assumptions->0<x<a]//Timing

...crash...

V5.0 No Assumptions

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x]//Timing

V5.1 Direct

2Integrate[(r/(Sqrt[((r^2)-(x^2))((a^2)-(r^2))])),{r,x,a},Assumptions->0<x<a]

V5.1 Assumptions

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x,Assumptions->0<x<a]

...hangs...

V5.1

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x]

FullSimplify[%,0<x<a]

V6

2Integrate[(r/((Sqrt[(a^2)-(r^2)])(Sqrt[r^2-x^2]))),{r,x,a},Assumptions->0<x<a]//Timing

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x,Assumptions->0<x<a]//Timing

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x]//Timing

FullSimplify[Last[%],0<x<a]

V10

$Version

2Integrate[(r/((Sqrt[(a^2)-(r^2)])(Sqrt[r^2-x^2]))),{r,x,a},Assumptions->0<x<a]//Timing

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x,Assumptions->0<x<a]//Timing

bug(274306)

AbelTransform[((Π[r,a])/(Sqrt[(a^2)-(r^2)])),r,x]//Timing

bug(274308)

𝒜[(Sqrt[(a^2)-(r^2)])(Subscript[Π, a])(r)]==(1/2)π((a^2)-(x^2))

V5 Assumptions

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x]

V5.1 Direct

2Integrate[r(Sqrt[(((a^2)-(r^2))/((r^2)-(x^2)))]),{r,x,a},Assumptions->0<x<a]

Expand[%]//Simplify

V5.1 Assumptions

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x,Assumptions->0<x<a]

...wrong..

V5.1 No Assumptions

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x]//Timing

FullSimplify[%,0<x<a]

...wrong...

V6

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x]//Timing

FullSimplify[Last[%],0<x<a]//Timing

...hangs...

V6.1

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x]//Timing

FullSimplify[Last[%],0<x<a]//Timing

V10

$Version

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a](Sqrt[(a^2)-(r^2)]),r,x]//Timing

(* chooses x<a<0 branch *)

𝒜[((a^2)-(r^2))(Subscript[Π, a])(r)]==(4/3)((((a^2)-(x^2)))^(3/2))

V5 Assumptions

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x]

...wrong...

V5.1 Direct

2Integrate[((r((a^2)-(r^2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]

V5.1 Assumptions

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x,Assumptions->0<x<a]

...wrong...

V5.1 No Assumptions

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x]

FullSimplify[%,0<x<a]

...wrong...

V6

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x]//Timing

...wrong...

V10

$Version

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a]((a^2)-(r^2)),r,x]//Timing

(* chooses x<a<0 branch *)

2Integrate[(UnitStep[r]-UnitStep[r-a]) (r/(Sqrt[r^2-x^2]))((a^2)-(r^2)),{r,x,Infinity},Assumptions->0<x<a]//Timing

2Integrate[(UnitStep[r]-UnitStep[r-a]) (r/(Sqrt[r^2-x^2]))((a^2)-(r^2)),{r,x,Infinity}]//Timing

(* chooses x<a<0 branch *)

𝒜[((((a^2)-(r^2)))^(3/2))(Subscript[Π, a])(r)]==(3/8)((π((a^2)-(x^2)))^2)

V5 Assumptions

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x]

V5.1 Direct

2Integrate[((r((((a^2)-(r^2)))^(3/2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]

V5.1 Assumptions

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->0<x<a]

V5.1 No Assumptions

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x]

FullSimplify[%,0<x<a]

V6

2Integrate[((r((((a^2)-(r^2)))^(3/2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]//Timing

2Integrate[((r((((a^2)-(r^2)))^(3/2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a}]//Timing

FullSimplify[%,0<x<a]

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->0<x<a]//Timing

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->x>a]//Timing

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x]//Timing

V10

$Version

2Integrate[((r((((a^2)-(r^2)))^(3/2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]//Timing

2Integrate[((r((((a^2)-(r^2)))^(3/2)))/(Sqrt[(r^2)-(x^2)])),{r,x,a}]//Timing

FullSimplify[%,0<x<a]

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->0<x<a]//Timing

bug(274306)

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x,Assumptions->x>a]//Timing

FullSimplify[Last[%12],#]&/@{0<x<a,x<0&&a>0,x>a&&a>0}

...correct...

AbelTransform[Π[r,a]((((a^2)-(r^2)))^(3/2)),r,x]//Timing

FullSimplify[Last[%13],#]&/@{x<0&&a>0,x>a&&a>0}

𝒜((a-r) (Subscript[Π, a])(r))==a (Sqrt[(a^2)-(x^2)])-(x^2) (cosh^(-1))((a/x))

V5 Assumptions

AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[Π[r,a](a-r),r,x]

V5.1 Direct

2Integrate[((r(a-r))/(Sqrt[(r^2)-(x^2)])),{r,x,a},Assumptions->0<x<a]

FullSimplify[Subtract@@{a (Sqrt[(a^2)-(x^2)])+(x^2) Log[(x/(a+(Sqrt[(a^2)-(x^2)])))],a (Sqrt[(a^2)-(x^2)])-(x^2)ArcCosh[a/x]},0<x<a]

V5.1 Assumptions

AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a]

...incorrect divergence message...

FullSimplify[%,0<x<a]

...wrong...

V5.1 No Assumptions

AbelTransform[Π[r,a](a-r),r,x]

...incorrect divergence messages...

FullSimplify[%,0<x<a]

....wrong...

V6

AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a]//Timing

FullSimplify[Subtract@@{%,a(Sqrt[(a^2)-(x^2)])-(x^2)ArcCosh[(a/x)]},0<x<a]

AbelTransform[Π[r,a](a-r),r,x]//Timing

...hangs...

V6.1

(i=AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a])//Timing

FullSimplify[Subtract@@{i,a(Sqrt[(a^2)-(x^2)])-(x^2)ArcCosh[(a/x)]},0<x<a]

(i=AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a])//Timing

AbelTransform[Π[r,a](a-r),r,x]//Timing

V10

$Version

AbelTransform[Π[r,a](a-r),r,x,Assumptions->0<x<a]//Timing

FullSimplify[Subtract@@{i,a(Sqrt[(a^2)-(x^2)])-(x^2)ArcCosh[(a/x)]},0<x<a]

AbelTransform[Π[r,a](a-r),r,x]//Timing

(* chooses x<a<0 branch *)

𝒜[(1/π)(cosh^(-1))((a/r))]==a-x

V5 Assumptions

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x]

V5.1 Direct

(2/π)Integrate[(r/(Sqrt[(r^2)-(x^2)]))ArcCosh[(a/r)],{r,x,a},Assumptions->0<x<a]

V5.1 Assumptions

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]

V5.1 No Assumptions

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x]

...very slow, unevaluated, spurious messages...

V5.2

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]//Timing

V6

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]//Timing

V6.1

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]//Timing

(2/π)Integrate[(r/(Sqrt[(r^2)-(x^2)]))ArcCosh[(a/r)],{r,x,a},Assumptions->0<x<a]//Timing

With[{a=1},(2/π)Integrate[(r/(Sqrt[(r^2)-(x^2)]))ArcCosh[(a/r)],{r,x,a},Assumptions->0<x<a]]

With[{a=1},AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]]//Timing

V10

$Version

AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]//Timing

(2/π)Integrate[(r/(Sqrt[(r^2)-(x^2)]))ArcCosh[(a/r)],{r,x,a},Assumptions->0<x<a]//Timing

With[{a=1},(2/π)Integrate[(r/(Sqrt[(r^2)-(x^2)]))ArcCosh[(a/r)],{r,x,a},Assumptions->0<x<a]]

With[{a=1},AbelTransform[Π[r,a]ArcCosh[a/r]/Pi,r,x,Assumptions->0<x<a]]//Timing

𝒜[δ(r-a)]==((2 a)/(Sqrt[(a^2)-(x^2)]))(Subscript[Π, a])(x)

V5 Assumptions

AbelTransform[DiracDelta[r-a],r,x,Assumptions->0<x<a]

V5 No Assumptions

AbelTransform[DiracDelta[r-a],r,x]

V5.1 Direct

2Integrate[(r/(Sqrt[(r^2)-(x^2)]))DiracDelta[r-a],{r,x,a},Assumptions->0<x<a]

...wrong...

V5.1 Assumptions

AbelTransform[DiracDelta[r-a],r,x,Assumptions->0<x<a]

V5.1 No Assumptions

AbelTransform[DiracDelta[r-a],r,x]

FullSimplify[%,0<x<a]//Together

V6

AbelTransform[DiracDelta[r-a],r,x,Assumptions->0<x<a]//Timing

AbelTransform[DiracDelta[r-a],r,x]//Timing

FullSimplify[Last[%],0<x<a]

V10

$Version

AbelTransform[DiracDelta[r-a],r,x,Assumptions->0<x<a]//Timing

AbelTransform[DiracDelta[r-a],r,x]//Timing

FullSimplify[Last[%18],0<x<a]

𝒜[(e^(-(r^2)/(σ^2)))]==σ(Sqrt[π])(e^(-(x^2)/(σ^2)))

V5.0

AbelTransform[Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]

V5.1

AbelTransform[Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]

V6

AbelTransform[Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]//Timing

V10

$Version

AbelTransform[Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]//Timing

AbelTransform[Exp[-r^2/σ^2],r,x]//Timing

FullSimplify[Last[%],#]&/@{σ>0&&x∈Reals,σ>0&&x<0,σ>0&&x>0}

𝒜[(r^2) (e^(-((r^2)/(σ^2))))]==σ((x^2)+(1/2)σ)(Sqrt[π])(e^(-(x^2)/(σ^2)))

V5.0

AbelTransform[r^2 Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]

V5.1

AbelTransform[r^2 Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]

V6

AbelTransform[(r^2)Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]//Timing

V10

AbelTransform[(r^2)Exp[-r^2/σ^2],r,x,Assumptions->σ>0&&x>0]//Timing

𝒜[((((r^2)-((σ^2)/2)) (e^(-((r^2)/(σ^2)))))/(σ (Sqrt[π])))]==(x^2)(e^(-(x^2)/(σ^2)))

V5.0

AbelTransform[(r^2 -σ^2/2)Exp[-r^2/σ^2]/(σ Sqrt[π]),r,x,Assumptions->σ>0&&x>0]

V5.1

AbelTransform[(r^2 -σ^2/2)Exp[-r^2/σ^2]/(σ Sqrt[π]),r,x,Assumptions->σ>0&&x>0]

V6

AbelTransform[(r^2 -σ^2/2)Exp[-r^2/σ^2]/(σ Sqrt[π]),r,x,Assumptions->σ>0&&x>0]

V10

$Version

AbelTransform[(r^2 -σ^2/2)Exp[-r^2/σ^2]/(σ Sqrt[π]),r,x,Assumptions->σ>0&&x>0]

𝒜[(1/((b^2)+(r^2)))]==(π/(Sqrt[(b^2)+(x^2)]))

V5.0

AbelTransform[1/(b^2+r^2),r,x,Assumptions->b>0&&x>0]

AbelTransform[1/(b^2+r^2),r,x]

V5.1

AbelTransform[1/(b^2+r^2),r,x,Assumptions->b>0&&x>0]

V6

AbelTransform[1/(b^2+r^2),r,x,Assumptions->b>0&&x>0]

V10

$Version

AbelTransform[1/(b^2+r^2),r,x,Assumptions->b>0&&x>0]

AbelTransform[1/(b^2+r^2),r,x,Assumptions->b>0]

𝒜((BesselJ[0, ω r]))==((2 cos(x ω))/ω)

V5.0

AbelTransform[BesselJ[0,ω r],r,x,Assumptions->ω>0]

AbelTransform[BesselJ[0,ω r],r,x]

V5.1

AbelTransform[BesselJ[0,ω r],r,x,Assumptions->ω>0&&x>0]

V6

AbelTransform[BesselJ[0,ω r],r,x,Assumptions->ω>0&&x>0]

V10

$Version

AbelTransform[BesselJ[0,ω r],r,x,Assumptions->ω>0&&x>0]

𝒜(M(r))==((8 (π^4))/((x^2) (ω^3)))sin(((x ω)/(2 π)))

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

M[x]

V5

AbelTransform[M[r  ω/Pi],r,x,Assumptions->ω>0]

V5.1

AbelTransform[M[r  ω/Pi],r,x,Assumptions->ω>0&&x>0]

Simplify[Subtract@@{-((4 (π^4) (-1+Cos[((x ω)/π)]))/((x^2) (ω^3))),((8 (π^4) ((Sin[((x ω)/(2 π))])^2))/((x^2) (ω^3)))}]

AbelTransform[M[r  ω/Pi],r,x]

V6

AbelTransform[M[x],r,x,Assumptions->ω>0]//Timing

V10

$Version

AbelTransform[M[x],r,x,Assumptions->ω>0]//Timing

With[{ω=5,x=2/10},{
2 NIntegrate[(((π^2) r (BesselJ[1,x] StruveH[0,x]-BesselJ[0,x] StruveH[1,x]))/((x^2) (Sqrt[(r^2)-(x^2)]))),{r,x,10^5},WorkingPrecision->20],
(((8 (π^4)) Sin[((x ω)/(2 π))])/((x^2) (ω^3)))
}]//N//Chop

...something’s wrong here...