Elliptic Alpha Function

Author

Eric W. Weisstein
January 25, 2009

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

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

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

Initialization

EllipticLambda1[r_]:=((EllipticTheta[2,0,Exp[-π(Sqrt[r])]])^2)/((EllipticTheta[3,0,Exp[-π(Sqrt[r])]])^2)

EllipticLambda2[r_]:=(Sqrt[ModularLambda[I (Sqrt[r])]])

MakeBoxes[EllipticLambda1[r_],TraditionalForm]:=RowBox[{SubsuperscriptBox["λ",1,"*"],"(",MakeBoxes[r,TraditionalForm],")"}]

MakeBoxes[EllipticLambda2[r_],TraditionalForm]:=RowBox[{SubsuperscriptBox["λ",2,"*"],"(",MakeBoxes[r,TraditionalForm],")"}]

Definitions

α(r)==(π/(4 ((EllipticK[(((Subscript[λ, 1]^( * ))(r))^2)])^2)))-(((Sqrt[r]) (EllipticE[(((Subscript[λ, 1]^( * ))(r))^2)]))/(EllipticK[(((Subscript[λ, 1]^( * ))(r))^2)]))+(Sqrt[r])

EllipticAlpha1[r_]:=(π/(4 ((EllipticK[((EllipticLambda1[r])^2)])^2)))+(Sqrt[r])-((EllipticE[((EllipticLambda1[r])^2)](Sqrt[r]))/(EllipticK[((EllipticLambda1[r])^2)]))

EllipticAlpha1[1]

FullSimplify[FunctionExpand[%]]

N[%]

1/2//N

EllipticAlpha1[2]

FullSimplify[FunctionExpand[%]]

N[%]

(Sqrt[2])-1//N

α(r)==(π/(4 ((EllipticK[(((Subscript[λ, 2]^( * ))(r))^2)])^2)))-(((Sqrt[r]) (EllipticE[(((Subscript[λ, 2]^( * ))(r))^2)]))/(EllipticK[(((Subscript[λ, 2]^( * ))(r))^2)]))+(Sqrt[r])

EllipticAlpha2[r_]:=(π/(4 ((EllipticK[((EllipticLambda2[r])^2)])^2)))+(Sqrt[r])-((EllipticE[((EllipticLambda2[r])^2)](Sqrt[r]))/(EllipticK[((EllipticLambda2[r])^2)]))

EllipticAlpha2[1]

FullSimplify[FunctionExpand[%]]

N[%]

1/2//N

EllipticAlpha2[2]

FullSimplify[FunctionExpand[%]]

N[%]

(Sqrt[2])-1//N

α(r)==((EllipticE[1-(((Subscript[λ, 1]^( * ))(r))^2)])/(EllipticK[(((Subscript[λ, 1]^( * ))(r))^2)]))-(π/(4 ((EllipticK[(((Subscript[λ, 1]^( * ))(r))^2)])^2)))

EllipticAlpha3[r_]:=((EllipticE[1-((EllipticLambda1[r])^2)])/(EllipticK[((EllipticLambda1[r])^2)]))-(π/(4((EllipticK[((EllipticLambda1[r])^2)])^2)))

EllipticAlpha3[1]

FullSimplify[FunctionExpand[%]]

N[%]

EllipticAlpha3[2]

FullSimplify[FunctionExpand[%]]

N[%]

α(r)==((EllipticE[1-(((Subscript[λ, 2]^( * ))(r))^2)])/(EllipticK[(((Subscript[λ, 2]^( * ))(r))^2)]))-(π/(4 ((EllipticK[(((Subscript[λ, 2]^( * ))(r))^2)])^2)))

EllipticAlpha4[r_]:=((EllipticE[1-((EllipticLambda2[r])^2)])/(EllipticK[((EllipticLambda2[r])^2)]))-(π/(4((EllipticK[((EllipticLambda2[r])^2)])^2)))

EllipticAlpha4[1]

FullSimplify[FunctionExpand[%]]

N[%]

EllipticAlpha4[2]

FullSimplify[FunctionExpand[%]]

N[%]

Identities

α(r)==α(r) ((((Subscript[λ, 2]^( * ))(4 r)+1))^2)-2 (Sqrt[r]) (Subscript[λ, 2]^( * ))(4 r)

(((1+EllipticLambda2[4r]))^2)EllipticAlpha2[r]-2(Sqrt[r])EllipticLambda2[4r]/.r->1

N[%]

EllipticAlpha2[4]

N[%]

Checks

{(π/(4 ((EllipticK[(ModularLambda[I (Sqrt[r])])])^2)))-((2 (Sqrt[r]) (ModularLambda[I (Sqrt[r])]) EllipticE'((ModularLambda[I (Sqrt[r])])))/(EllipticK[(ModularLambda[I (Sqrt[r])])])),(π/(4 ((EllipticK[(ModularLambda[I (Sqrt[r])])])^2)))-(((Sqrt[r]) (EllipticE[(ModularLambda[I (Sqrt[r])])]))/(EllipticK[(ModularLambda[I (Sqrt[r])])]))+(Sqrt[r]),(((1/π)-((4 (Sqrt[r]) Derivative[0,0,1][EllipticTheta](4,0,(E^(-π (Sqrt[r])))))/((E^(π (Sqrt[r]))) (EllipticTheta[4, 0, (E^(-π (Sqrt[r])))]))))/((EllipticTheta[3, 0, (E^(-π (Sqrt[r])))])^4))}

N[With[{r=Random[Complex]},{Pi/(4*EllipticK[ModularLambda[I*Sqrt[r]]]^2)-(2*Sqrt[r]*ModularLambda[I*Sqrt[r]]*Derivative[1][EllipticE][ModularLambda[I*Sqrt[r]]])/EllipticK[ModularLambda[I*Sqrt[r]]],Pi/(4*EllipticK[ModularLambda[I*Sqrt[r]]]^2)+Sqrt[r]-Sqrt[r]*(EllipticE[ModularLambda[I*Sqrt[r]]]/EllipticK[ModularLambda[I*Sqrt[r]]]),(Pi^(-1)-4*Sqrt[r]*(1/(E^(Pi*Sqrt[r])*EllipticTheta[4,0,E^((-Pi)*Sqrt[r])]))*Derivative[0,0,1][EllipticTheta][4,0,E^((-Pi)*Sqrt[r])])/EllipticTheta[3,0,E^((-Pi)*Sqrt[r])]^4}]]