Elliptic Integral of the Second Kind

Author

Eric W. Weisstein
August 15, 2005

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

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

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

Definition

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

v5.2

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

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

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

v6

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

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

Identities

(cscz) E((sin^2)(z))-(cosz) (cotz) K((sin^2)(z))==E(z|(csc^2)(z))

FunctionExpand[Csc[x] EllipticE[((Sin[x])^2)]-Cos[x] Cot[x] EllipticK[((Sin[x])^2)],-Pi/2<Re[x]<Pi/2]

Subtract@@@Table[{Csc[x] EllipticE[((Sin[x])^2)]-Cos[x] Cot[x] EllipticK[((Sin[x])^2)], EllipticE[x,((Csc[x])^2)]}/.{x->Random[Real,{-Pi/2,Pi/2}]},{100}]//Chop//Union

Plot3D[Evaluate[Re[Subtract@@{Csc[x] EllipticE[((Sin[x])^2)]-Cos[x] Cot[x] EllipticK[((Sin[x])^2)], EllipticE[x,((Csc[x])^2)]}/.x->x+I y]],{x,-10,10},{y,-10,10},PlotPoints->70,Mesh->False]

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

ComplexPlot3D[Subtract@@{Csc[z] EllipticE[((Sin[z])^2)]-Cos[z] Cot[z] EllipticK[((Sin[z])^2)], EllipticE[z,((Csc[z])^2)]},{z,-10,10},ComplexPlotLabel->blah,PlotPoints->150]//Timing

ComplexContourPlot[Subtract@@{Csc[z] EllipticE[((Sin[z])^2)]-Cos[z] Cot[z] EllipticK[((Sin[z])^2)], EllipticE[z,((Csc[z])^2)]},{z,-10,10},ComplexPlotLabel->blah,PlotPoints->150]//Timing