Ramanujan g- and G-Functions

Author

Eric W. Weisstein
January 25, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Ramanujang-andG-Functions.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Ramanujang-andG-Functions.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

(Subscript[G, n])==((Product[(e^(π (-k) (Sqrt[n])))+1,{k,1,∞,2}])/((2^(1/4)) (e^(-(1/24) (π (Sqrt[n]))))))==(1/((2^(1/12)) ((-((ModularLambda[i (Sqrt[n])])-1) (ModularLambda[i (Sqrt[n])]))^(1/24))))

V7

((Product[1+Exp[-k π(Sqrt[n])],{k,1,∞,2}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))

(1/((2^(1/12)) ((((λ^2)-(λ^4)))^(1/24))))/.λ->EllipticLambda2[n]//FullSimplify

(1/((2^(1/12)) ((((λ^2)-(λ^4)))^(1/24))))/.λ->EllipticLambda2[5]//N//Chop

With[{n=5},((Product[1+Exp[-k π(Sqrt[n])],{k,1,10001,2}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))]//N

(Subscript[G, n])==(((∏_k^∞)((e^(π (-(2 k-1)) (Sqrt[n])))+1))/((2^(1/4)) (e^(-(1/24) (π (Sqrt[n]))))))==(1/((2^(1/12)) ((-((ModularLambda[i (Sqrt[n])])-1) (ModularLambda[i (Sqrt[n])]))^(1/24))))

((Product[1+Exp[-(2k-1) π(Sqrt[n])],{k,∞}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))

With[{n=5},((Product[1+Exp[-(2k-1) π(Sqrt[n])],{k,1000}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))]//N

(Subscript[g, n])==((Product[1-exp(π (-k) (Sqrt[n])),{k,1,∞,2}])/((2^(1/4)) exp(-((π (Sqrt[n]))/24))))==((((1-(ModularLambda[i (Sqrt[n])]))/(Sqrt[(ModularLambda[i (Sqrt[n])])]))^(1/12))/(2^(1/12)))

V7

((Product[1-Exp[-k π(Sqrt[n])],{k,1,∞,2}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))

(((((1/λ)-λ))^(1/12))/(2^(1/12)))/.λ->EllipticLambda2[n]//FullSimplify

(((((1/λ)-λ))^(1/12))/(2^(1/12)))/.λ->EllipticLambda2[5]//N//Chop

With[{n=5},((Product[1-Exp[-k π(Sqrt[n])],{k,1,10001,2}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))]//N

(Subscript[g, n])==(((∏_k^∞)(1-exp(π (-(2 k-1)) (Sqrt[n]))))/((2^(1/4)) exp(-((π (Sqrt[n]))/24))))==((((1-(ModularLambda[i (Sqrt[n])]))/(Sqrt[(ModularLambda[i (Sqrt[n])])]))^(1/12))/(2^(1/12)))

((Product[1-Exp[-(2k -1)π(Sqrt[n])],{k,∞}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))

With[{n=5},((Product[1-Exp[-(2k -1)π(Sqrt[n])],{k,1000}])/((2^(1/4))Exp[-π(Sqrt[n])/24]))]//N

Solve λ(G) for G

G

soln=FullSimplify[Solve[λ==(1/2)((Sqrt[1+(G^(-12))])-(Sqrt[1-(G^(-12))])),G],λ>0]

g

soln=FullSimplify[Solve[λ==(g^6)((Sqrt[(g^12)+(g^(-12))])-(g^6)),g],λ>0]