Elliptic Integral of the First Kind

Author

Eric W. Weisstein
November 13, 2018

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/EllipticIntegraloftheFirstKind.nb.

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Integrals

EllipticF[ϕ,(k^2)]==(∫_0^ϕ)(1/(Sqrt[1-(k^2) sin(θ)]))dθ

v4.1.5

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ}]//Timing

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

v4.2.1

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ}]//Timing

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

v5.0.1

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ}]//Timing

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

v5.2

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ}]//Timing

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ},Assumptions->-1<k<1]//Timing

Integrate[(1/(Sqrt[1-(k^2)((Sin[θ])^2)])),{θ,0,ϕ},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

(∫_0^(sin(ϕ)))(1/((Sqrt[1-(k^2) (t^2)]) (Sqrt[1-(t^2)])))dt

V4.2.1

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

V5.0.1

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

V5.1.1

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

V5.2

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]}]//Timing

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]},Assumptions->-1<k<1]//Timing

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]}]//Timing

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

V12

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]}]//Timing

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]},Assumptions->-1<k<1]//Timing

Integrate[(1/(Sqrt[1-(k^2)(t^2)]))(1/(Sqrt[1-(t^2)])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]}]//Timing

Integrate[(1/(Sqrt[(1-(k^2)(t^2))(1-(t^2))])),{t,0,Sin[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

EllipticF[ϕ,(k^2)]==(∫_0^(tan(ϕ)))(1/(Sqrt[((v^2)+1) (-(k^2) (v^2)+(v^2)+1)]))dv

With[{k=.1,ϕ=.2},{
Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]}],
-I EllipticF[I ArcSinh[Tan[ϕ]],1-(k^2)],
EllipticF[ϕ,(k^2)]
}]

v4.1.5

Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

FullSimplify[Last[%5],-1<k<1&&0<ϕ<π/2]//Timing

v4.2.1

Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

v5.0.1

Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

v5.2

Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]},Assumptions->-1<k<1&&0<ϕ<π/2]//Timing

Integrate[(1/(Sqrt[(1+(v^2))(1+(v^2)-(k^2)(v^2))])),{v,0,Tan[ϕ]},GenerateConditions->False]//Timing

FullSimplify[Last[%],Assumptions->-1<k<1&&0<ϕ<(π/2)]

Subtract@@@Table[{-I EllipticF[I ArcSinh[Tan[ϕ]],1-(k^2)],EllipticF[ϕ,(k^2)]}/.{k->Random[Real,{-1,1},50],ϕ->Random[Real,{0,π/2},50]},{20}]//Chop//Union

Identities

csc(x) (EllipticF[x, (csc^2)(x)])==(EllipticK[(sin^2)(x)])

Table[Csc[x]EllipticF[x,Csc[x]^2]-EllipticK[Sin[x]^2]/.x->Random[Complex,(10I+Pi/2){-1,1},40],{1000}]//Chop//Union

Table[EllipticF[x,Csc[x]^2]-Sin[x]EllipticK[Sin[x]^2]/.x->Random[Complex,(10I+Pi/2){-1,1},50],{1000}]//Chop//Union

<<Utilities`Typesetting`
<<Utilities`Plot`

MakeBoxes[label1,TraditionalForm]:=RowBox[{RowBox[{RowBox[{"F","(",RowBox[{"z",",",RowBox[{"csc","z"}]}],")"}],"-",RowBox[{"sin"," ","z"," ",RowBox[{"K","(",RowBox[{"sin","z"}],")"}]}]}]}]

label1//TraditionalForm

ComplexPlot3D[EllipticF[z,Csc[z]^2]-Sin[z]EllipticK[Sin[z]^2],{z,-10,10},ComplexPlotLabel->label1,PlotPoints->150]//Timing

ComplexContourPlot[EllipticF[z,Csc[z]^2]-Sin[z]EllipticK[Sin[z]^2],{z,-10,10},ComplexPlotLabel->label1,PlotPoints->150]//Timing

MakeBoxes[label2,TraditionalForm]:=RowBox[{RowBox[{"csc","z"," ",RowBox[{"F","(",RowBox[{"z",",",RowBox[{"csc","z"}]}],")"}]}],"-",RowBox[{"K","(",RowBox[{"sin","z"}],")"}]}]

label2//TraditionalForm

ComplexPlot3D[Csc[z]EllipticF[z,Csc[z]^2]-EllipticK[Sin[z]^2],{z,-10,10},ComplexPlotLabel->label2,PlotPoints->150]//Timing

ComplexContourPlot[Csc[z]EllipticF[z,Csc[z]^2]-EllipticK[Sin[z]^2],{z,-10,10},ComplexPlotLabel->label2]//Timing

(EllipticF[(JacobiAmplitude[u, m]), m])==u

FullSimplify[EllipticF[JacobiAmplitude[u,m],m]]

FunctionExpand[%]

Table[EllipticF[JacobiAmplitude[u,m],m]-u/.{u->RandomComplex[],m->RandomComplexDisk[]},{100}]//Chop//Union

Differential identities

((d K)/(d k))== (E/(k ((k')^2)))-(K/k)

W&W §22.736, p. 521

((d K)/(d k))== (E/(k ((k')^2)))-(K/k)

FullSimplify[Subtract@@{D[EllipticK[k^2],k],((EllipticE[k^2])/(k(1-(k^2))))-((EllipticK[k^2])/k)}]

(d/(d k))(k(1-(k^2))((d K)/(d k)))==k K(k)

(d/(d k))(k(1-(k^2))((d K)/(d k)))==k K(k)

FullSimplify[Subtract@@{D[k(1-(k^2))D[EllipticK[k^2],k],k],k  EllipticK[(k^2)]}]

(EllipticE[(k^2)])==k (1-(k^2)) (((EllipticK[(k^2)])/k)+((∂(EllipticK[(k^2)]))/(∂k)))

FullSimplify[Subtract@@{EllipticE[(k^2)],k(1-(k^2))(D[EllipticK[(k^2)],k]+((EllipticK[(k^2)])/k))}]

(EllipticE[(k^2)])==(1-(k^2)) ((EllipticK[(k^2)])+k ((∂(EllipticK[(k^2)]))/(∂k)))

FullSimplify[Subtract@@{EllipticE[(k^2)],(1-(k^2))(k D[EllipticK[(k^2)],k]+EllipticK[(k^2)])}]