Complete Elliptic Integral of the Second Kind

Author

Eric W. Weisstein
October 10, 2005

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

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

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

Plots

<<Utilities`Plot`

Real Axis

Plot[EllipticE[k^2],{k,0,1},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{k,EllipticE[k]}]

Complex Plots

ComplexPlot3D[EllipticE[k^2],{k,-10,10},
PlotPoints->100,
ComplexPlotLabel->EllipticE[k],
AxesLabel->TraditionalForm/@{(Subscript[k, x]),(Subscript[k, y]),""}]//Timing

ComplexContourPlot[EllipticE[k^2],{k,-10,10},
ComplexPlotLabel->EllipticE[k],
FrameLabel->TraditionalForm/@{(Subscript[k, x]),(Subscript[k, y])}]//Timing

Definitions

E((π/2)|(k^2))

EllipticE[π/2,k^2]

(1/2) (Subscript[, 2])(Subscript[F, 1])(-(1/2),(1/2);1;(k^2)) π

Hypergeometric2F1[-1/2,1/2,1,k^2]π/2

(∫_0^(K((k^2))))((dn(u|(k^2)))^2)du

Integrate[JacobiDN[u,k^2]^2,{u,0,EllipticK[k^2]}]/.JacobiAmplitude[EllipticK[m_],m_]:>Pi/2

FunctionExpand[JacobiAmplitude[EllipticK[(k^2)],(k^2)]]

N[Table[JacobiAmplitude[EllipticK[x+I y],x+I y],{x,0,1,1/10},{y,0,1,1/10}],17]//Chop

Sum

(1/2) π (1-(∑_(n=1)^∞)((((((((2 n-1)!!)/((2 n)!!))))^2) (k^(2 n)))/(2 n-1)))

V6

π/2(1-FullSimplify[Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^2)]((k^(2n))/(2n-1)),{n,∞}]])

V7

π/2(1-Sum[((((((2n-1)!!)/((2n)!!))))^2)((k^(2n))/(2n-1)),{n,∞}])

FullSimplify[%]

ODE

(soln=DSolve[ode=(1-x^2)D[x y'[x],x]+x y[x]==0,y[x],x])//Timing

soln[[1,1,2]]//TraditionalForm

FunctionExpand[MeijerG[{{},{(1/2),(3/2)}},{{0,0},{}},(x^2)]]

FullSimplify[FunctionExpand[(1-(x^2)) HypergeometricPFQ[{(1/2),(3/2)},{2},1-(x^2)]]]

Plot[{EllipticE[k^2],(EllipticE[1-(k^2)]-EllipticK[1-(k^2)])},{k,-1,1},PlotStyle->{Red,Blue},AxesLabel->TraditionalForm/@{x,y[x]}]

Special Values

{#,EllipticE[Sqrt[#]]}&/@{-I ∞,-∞,-1,0,1,I,∞,I ∞}//TableForm

Negative Parameter

{EllipticE[-1.],EllipticE[1/2.](Sqrt[2])}

Sort this mess out...

With[{m=.2`30,ϕ=π/2},
{EllipticE[u,-m],(Sqrt[1+m])(EllipticE[JacobiAmplitude[u(Sqrt[m+1]),m],(m/(m+1))]-(m/(Sqrt[m+1]))JacobiSN[u(Sqrt[m+1]),(m/(m+1))]JacobiCD[JacobiAmplitude[u(Sqrt[m+1]),m],(m/(m+1))])}
]

From Abramowitz and Stegun p. 593, but doesn't seem to work