Complete Elliptic Integral of the First Kind

Author

Eric W. Weisstein
January 16, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Along Real Axis

V5

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

V6

Plot[EllipticK[k^2],{k,0,1},AxesOrigin->{0,Pi/2},PlotStyle->Red,AxesLabel->{k,EllipticK[k]}]

Plot[EllipticK[k^2],{k,0,1},AxesOrigin->{0,Pi/2},PlotStyle->Red,AxesLabel->{k,EllipticK[k]},PlotRange->{1.5,5.5}]

Complex Plots

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

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

Integral Definition

(∫_0^(π/2))(1/(Sqrt[1-(k^2) (sin^2)(θ)]))dθ

v4.2

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

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

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

v5

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

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

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

v5.1

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

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

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

Subtract@@@Table[{EllipticK[(k^2)],((EllipticK[((k^2)/(-1+(k^2)))])/(Sqrt[1-(k^2)]))}/.k->Random[],{50}]//Chop//Union

V7

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

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

Definitions

F((π/2)|(k^2))

EllipticF[Pi/2,k^2]

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

Hypergeometric2F1[1/2,1/2,1,k^2]Pi/2

((π (Subscript[P, -1/2])(((1+(k^2))/(1-(k^2)))))/(2 (Sqrt[1-(k^2)])))

FullSimplify[(π/(2 (Sqrt[1-(k^2)])))LegendreP[-1/2,((1+(k^2))/(1-(k^2)))]]

Equal@@@Table[
{
(π/(2Sqrt[1-(k^2)]))LegendreP[-(1/2),((1+(k^2))/(1-(k^2)))],
EllipticK[k^2]
}/.k->Random[Complex,5(I+1){-1,1}],{1000}]//Union

Equal@@@Table[
{
(π/(2Sqrt[1-(k^2)]))LegendreP[-(1/2),((1+(k^2))/(1-(k^2)))],
((EllipticK[((k^2)/(-1+(k^2)))])/(Sqrt[1-(k^2)]))
}/.k->Random[Complex,5(I+1){-1,1}],{1000}]//Union

Equal@@@Table[
{
(π/(2Sqrt[1-(k^2)]))LegendreP[-1/2,(((1+(k^2))/(1-(k^2))))],
((EllipticK[((k^2)/(-1+(k^2)))])/(Sqrt[1-(k^2)])),
EllipticK[ k^2]
}/.k->Random[Real,{-1,1}],{100}]//Union

Table[
{
(π/(2Sqrt[1-(k^2)]))LegendreP[-1/2,(((1+(k^2))/(1-(k^2))))],
((EllipticK[((k^2)/(-1+(k^2)))])/(Sqrt[1-(k^2)])),
EllipticK[ k^2]
}/.k->Random[Real,{1,2}],{10}]

Table[
{
(π/(2Sqrt[1-(k^2)]))LegendreP[-1/2,(((1+(k^2))/(1-(k^2))))],
((EllipticK[((k^2)/(-1+(k^2)))])/(Sqrt[1-(k^2)])),
EllipticK[ k^2]
}/.k->Random[Real,{-2,-1}],{10}]

Sums

(π/2)  (∑_(n=0)^∞)((((((2 n-1)!!)/((2 n)!!))))^2) (k^(2 n))

V5

(π/2) Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^2)] (k^(2n)),{n,0,∞}]

V7

(π/2) Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^2)] (k^(2n)),{n,0,∞}]

((π^2)/4)  (∑_(n=0)^∞)((((((2 n-1)!!)/((2 n)!!))))^3) (((2 k (Sqrt[1-(k^2)])))^(2 n))

V5

((π^2)/4)Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^3)(((2k Sqrt[1-k^2]))^(2n))],{n,0,∞}]//FullSimplify

FullSimplify[%,k∈Reals]

FullSimplify[%,0<k<1/Sqrt[2]]

V7

((π^2)/4)Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^3)(((2k Sqrt[1-k^2]))^(2n))],{n,0,∞}]

FullSimplify[%,k∈Reals]

FullSimplify[%,0<k<1/Sqrt[2]]

ODE

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

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

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

Plot[{EllipticK[k^2],EllipticK[1-(k^2)]},{k,-1,1},PlotStyle->{Red,Blue},AxesLabel->{k,{Style[EllipticK[k],Red],Style[EllipticK[k'],Blue]}},ImageSize->250,PlotRange->{1,8}]

Identities

K(-m)==(1/(Sqrt[m+1]))K((m/(m+1)))

Subtract@@@Table[{EllipticK[-m],(1/(Sqrt[m+1]))EllipticK[(m/(m+1))]}/.m->Random[Complex],{100}]//Chop//Union

Special Values

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

Integrals

v5.0

Integrate[EllipticK[k^2],{k,0,1}]

Integrate[EllipticK[k],{k,0,1}]

Integrate[EllipticK[Sqrt[k]],{k,0,1}]

Integrate[EllipticK[Sqrt[k]]/Sqrt[k],{k,0,1}]