Elliptic Lambda Function

Author

Eric W. Weisstein
January 25, 2009

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

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

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

Initialization

<<Utilities`Typesetting`

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])]])

Notation

ModularLambda[τ]//TraditionalForm

Check

ModularLambda[I]//N

Divide@@(EllipticTheta[#,0,Exp[I Pi I]]^4&/@{2,3})//N

Identities

(ModularLambda[τ+2])==(ModularLambda[τ])

FullSimplify[ModularLambda[τ+2]-ModularLambda[τ],Im[τ]>0]

FunctionExpand[%]

Table[ModularLambda[τ+2]-ModularLambda[τ]/.τ->Random[Complex,{-5,5+5I}],{10}]//Chop

(ModularLambda[(τ/(2 τ+1))])==(ModularLambda[τ])

FullSimplify[ModularLambda[τ/(2τ+1)]-ModularLambda[τ],Im[τ]>0]

FunctionExpand[%]

Table[ModularLambda[τ/(2τ+1)]-ModularLambda[τ]/.τ->Random[Complex,{-5,5+5I}],{10}]//Chop

Plot

<<Utilities`Plot`

ComplexPlot3D[ModularLambda[τ],{τ,-3+.01I,3+3I},PlotPoints->100]//Timing

ComplexContourPlot[ModularLambda[τ],{τ,-3+.01I,3+3I}]//Timing

Limits

Limit[ModularLambda[x],x->∞]

Limit[ModularLambda[x I],x->∞]

Special Values

λ(τ)

ModularLambda/@{0,-1,1,I,∞,I ∞}

ModularLambda/@{0,-1,1,I,∞,I ∞}//N

FullSimplify[Divide@@(EllipticTheta[#,0,Exp[I Pi I]]^4&/@{2,3})]

FunctionExpand[%]//FullSimplify

N[%,20]

ListPlot[EllipticLambda/@Range[20]//N,Joined->True]

(SuperStar[λ])(1)==(1/(Sqrt[2]))

EllipticLambda[1]

(SuperStar[λ])(2)==(Sqrt[2])-1

EllipticLambda[2]

(SuperStar[λ])(3)==(1/4)(Sqrt[2])((Sqrt[3])-1)

EllipticLambda2[3]-EllipticLambda[3]//FullSimplify//N//Chop

EllipticLambda[3]-(1/4)(Sqrt[2])((Sqrt[3])-1)//FullSimplify

(SuperStar[λ])(4)==3-2(Sqrt[2])

EllipticLambda2[4]-EllipticLambda[4]//FullSimplify//N//Chop

EllipticLambda[4]-(3-2(Sqrt[2]))//FullSimplify

(SuperStar[λ])(12)==((((Sqrt[3])-(Sqrt[2])))^2) ((((Sqrt[2])-1))^2)

EllipticLambda[12]-((((Sqrt[3])-(Sqrt[2])))^2)((((Sqrt[2])-1))^2)//FullSimplify

EllipticLambda[12]

(SuperStar[λ])(17)==(1/2) (Sqrt[2]) (Sqrt[1-2 (Sqrt[-100 (Sqrt[17])+5 (Sqrt[3298 (Sqrt[17])+13598])-412])])

EllipticLambda[17]

%//N

EllipticLambda2[17]

N[%]//Chop

Wrong

(1/4)(Sqrt[2])((Sqrt[42+10(Sqrt[17])])-13(Sqrt[-3+(Sqrt[17])])(Sqrt[5+(Sqrt[17])])-3(Sqrt[17])(Sqrt[-3+(Sqrt[17])])(Sqrt[5+(Sqrt[17])])-(Sqrt[-38-10(Sqrt[17])+13(Sqrt[-3+(Sqrt[17])])])(Sqrt[5+(Sqrt[17])])+3(Sqrt[17])(Sqrt[-3+(Sqrt[17])])(Sqrt[5+(Sqrt[17])]))//N

Also Wrong

(Sqrt[(1/2)+(1/2)B(Sqrt[5A-16])])/.{A->(Sqrt[2+2(Sqrt[17])]),B->(Sqrt[103+25(Sqrt[17])])}//N

f[e_]:=1000Count[e,_Root,{0,Infinity}]+ByteCount[e]

EllipticLambda[17]//ToRadicals//FullSimplify[#,ComplexityFunction->f]&

FullSimplify[(Sqrt[1649+400 (Sqrt[17])-20 (Sqrt[13598+3298 (Sqrt[17])])])((Sqrt[2-4 (Sqrt[-412-100 (Sqrt[17])+5 (Sqrt[13598+3298 (Sqrt[17])])])]))/√(Expand[(2-4 (Sqrt[-412-100 (Sqrt[17])+5 (Sqrt[13598+3298 (Sqrt[17])])]))(2+4 (Sqrt[-412-100 (Sqrt[17])+5 (Sqrt[13598+3298 (Sqrt[17])])]))]),ComplexityFunction->f]

N[%]

((Sqrt[2])/2)(Sqrt[1-2 (Sqrt[-412-100 (Sqrt[17])+5 (Sqrt[13598+3298 (Sqrt[17])])])])//N

FullSimplify[((Sqrt[2])/2)(Sqrt[1-2 (Sqrt[-412-100 (Sqrt[17])+5 (Sqrt[13598+3298 (Sqrt[17])])])])-EllipticLambda[17]]

Table[(((5+(Sqrt[17])))^n),{n,10}]//Expand

Table[(((5+(Sqrt[17])))^n),{n,10}]//Expand

EllipticLambda[17]//N

EllipticLambda2[17]//N//Chop

(SuperStar[λ])(29)==(Subscript[(4096 (q^24)-24576 (q^22)+356108288 (q^20)-1780316160 (q^18)+3216322304 (q^16)-2183662592 (q^14)+127790848 (q^12)-217611008 (q^10)+872674032 (q^8)-369772256 (q^6)-20122168 (q^4)-1390808 (q^2)+1), 3])

EllipticLambda1[29]//FunctionExpand//FullSimplify

EllipticLambda[29]//FunctionExpand

(Sqrt[ModularLambda[I (Sqrt[29])]])//N//Chop

Options[RootApproximant]

RootApproximant[EllipticLambda[29]]

N[%]

N[EllipticLambda[29]-Root[(1-1390808 (q^2)-20122168 (q^4)-369772256 (q^6)+872674032 (q^8)-217611008 (q^10)+127790848 (q^12)-2183662592 (q^14)+3216322304 (q^16)-1780316160 (q^18)+356108288 (q^20)-24576 (q^22)+4096 (q^24)),3],10^4]

EllipticLambda[29]//N[#,100]&

RootApproximant[EllipticLambda[29]//N[#,200]&]

Factor[Resultant[1/x^24-4(q^2-q^4),x^24-9x^20+5x^16+2x^12-5x^8-9x^4-1,x]]

Root[(1-1390808 (q^2)-20122168 (q^4)-369772256 (q^6)+872674032 (q^8)-217611008 (q^10)+127790848 (q^12)-2183662592 (q^14)+3216322304 (q^16)-1780316160 (q^18)+356108288 (q^20)-24576 (q^22)+4096 (q^24)),3]//N

Analytic Recognition

<<Utilities`Simplify`
Off[N::"meprec"];

ks={};
Do[Print[{n,Last[AppendTo[ks,TimeConstrained[RootReduce[ToExact[LambdaPrime[n],Weight->5]],10]]]//Timing}],
{n,50}
]

Do[Print[{n,Last[AppendTo[ks,TimeConstrained[RootReduce[ToExact[LambdaPrime[n],Weight->5]],10]]]//Timing}],
{n,51,100}
]

degs=If[AtomQ[#],#,If[FreeQ[#,ModularLambda],2,0]]&/@(ks/.Root[r_,n__]:>Exponent[r[x],x])

Flatten[Position[%,0]]

Try to Get 29

RootApproximant[N[LambdaPrime[29],100]]//Timing

RootApproximant[N[LambdaPrime[29],200]]//Timing

ToExact[LambdaPrime[29]^2,Weight->10^4]//Timing

ToExact[LambdaPrime[29]^2,Weight->10^4,MaxOrder->32]//Timing

Rational Values

RootApproximant[N[LambdaPrime[2/29],50]]

ToRadicals[%]//TraditionalForm

RootReduce[(13(Sqrt[58])-99)((((Sqrt[2])+1))^6)]

ToExact[LambdaPrime[1/5],Weight->10^4]//FullSimplify//Timing

ToExact[LambdaPrime[2/5]]//FullSimplify//Timing

RootReduce[((Sqrt[10])-3)((((Sqrt[2])+1))^2)]

ToExact[LambdaPrime[3/5],Weight->10^4]//FullSimplify//Timing

ToExact[LambdaPrime[4/5],Weight->10^4]//FullSimplify//Timing

ToExact[LambdaPrime[1/4]]//FullSimplify//Timing

RootApproximant[N[LambdaPrime[3/4],50]]

RootApproximant[N[LambdaPrime[3/4],100]]

ToRadicals[%]//TraditionalForm

ToExact[LambdaPrime[1/3]]//FullSimplify//Timing

ToExact[LambdaPrime[2/3]]//FullSimplify//Timing

RootReduce[%]

RootReduce[(2-(Sqrt[3]))((Sqrt[2])+(Sqrt[3]))]

ToExact[LambdaPrime[1/2]]//FullSimplify//Timing

λ'(τ)

LambdaPrime[r_]:=Sqrt[ModularLambda[Sqrt[r]I]]

kr=Table[EllipticTheta[2,0,#]^2/EllipticTheta[3,0,#]^2&[Exp[-Pi Sqrt[n]]],{n,5}]

kr//N

Table[LambdaPrime[r],{r,5}]//N//Chop

N[(EllipticK[1-#]/EllipticK[#]&/@(kr^2))^2,20]//N

Derivative

V5.2

ModularLambda'[q]//TraditionalForm

FullSimplify[%]//TraditionalForm

V6

ModularLambda'[q]//TraditionalForm

FullSimplify[%]//TraditionalForm

Series

Solve[q==Exp[I π t],t]

Series

Via EllipticTheta

λ[t_]:=EllipticTheta[2,0,Exp[I*Pi*t]]^4/EllipticTheta[3,0,Exp[I*Pi*t]]^4

Series[λ[τ],{τ,0,1}]

With[{τ=-((I Log[q])/π)},Series[λ[τ],{q,0,10}]]

CoefficientList[%,]

With[{τ=-((I Log[q])/π)},Series[(16/(λ[τ])),{q,0,10}]]

CoefficientList[%-1/q,q]

v5.2

Series[ModularLambda[-((I Log[q])/π)],{q,0,3}]

Series[ModularLambda[-((I Log[q])/π)],{q,∞,1}]

Series[ModularLambda[q],{q,I ∞,1}]

v6

Series[ModularLambda[-((I Log[q])/π)],{q,0,5}]

Series[ModularLambda[-((I Log[q])/π)],{q,∞,5}]

Series[ModularLambda[q],{q,I ∞,5}]