Dedekind Eta Function

Author

Eric W. Weisstein
March 20, 2009

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

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

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

Sums

(DedekindEta[q])==(∑_(n=1)^∞)(((-1))^n) ((q^((1/2) n (3 n-1)))+(q^((1/2) n (3 n+1))))

Solve[q==Exp[2I π τ],τ]

(q^(1/24))Sum[(((-1))^n)(q^(n(3n-1)/2)),{n,-∞,∞}]

Sum[(((-1))^n)((q^(n(3n-1)/2))+(q^(n(3n+1)/2))),{n,∞}]

V7

(q^(1/24))Sum[(((-1))^n)(q^(n(3n-1)/2)),{n,-∞,∞}]

Sum[(((-1))^n)((q^(n(3n-1)/2))+(q^(n(3n+1)/2))),{n,∞}]

V8

Sum[(((-1))^n)((q^(n(3n-1)/2))+(q^(n(3n+1)/2))),{n,∞}]

Products

(q^(1/24))Product[1-(q^n),{n,∞}]

(q^(1/24))Product[1-(q^n),{n,∞}]

(e^((π i z)/12)) (∏_(k=1)^∞)(1-(e^(2 π i k z)))

v5.2

Exp[π I z/12]Product[1-Exp[2π I k z],{k,∞}]

v6

Exp[π I z/12]Product[1-Exp[2π I k z],{k,∞}]

V7

Exp[π I z/12]Product[1-Exp[2π I k z],{k,∞}]

FullSimplify[%,Im[z]>0&&Abs[z]<1]

FullSimplify[%-DedekindEta[z]]

Length[l=Select[Table[RandomComplexDisk[],{200}],Im[#]>0&]]

Subtract@@@({DedekindEta[z],(E^((I π z)/12)) QPochhammer[(E^(2 I π z))]}/.z->#&/@l)//Chop//Union

V8

Exp[π I z/12]Product[1-Exp[2π I k z],{k,∞}]

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

Plot[DedekindEta[x I],{x,0,10},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,DedekindEta[x I]}]

Complex Plane

ComplexPlot3D[DedekindEta[τ],{τ,-1+.01I,1+I},PlotPoints->200]//Timing

ComplexContourPlot[DedekindEta[τ],{τ,-1+.01I,1+I}]//Timing

Special Values

<<Utilities`Typesetting`

η(i)==((Γ((1/4)))/(2 (π^(3/4))))

Numeric

N[DedekindEta[I],30]//Chop

With[{q=Exp[2π I I]},(q^(1/24))NSum[(((-1))^n)(q^(n(3n-1)/2)),{n,-∞,∞},WorkingPrecision->30]]//Chop

v5.2

DedekindEta[I]

v6

DedekindEta[I]

N[%,30]

V7

DedekindEta[I]

-(((((DedekindEta[τ])/((Sqrt[2]) (DedekindEta[2 τ])))))^24)==(Subscript[((x^3)-x-1), 1]^24)/;τ==(1/2) (1+(Sqrt[23]) i)

((Root[x^3-x-1,1])^24)//N

With[{τ=(1+I(Sqrt[23]))/2},-(((((DedekindEta[τ])/((Sqrt[2])DedekindEta[2τ]))))^24)]//N

(((DedekindEta[τ]) (e^(π i/24)))/((Sqrt[2]) (DedekindEta[2 τ])))==(Subscript[((x^3)-x-1), 1])/;τ==(1/2) ((Sqrt[23]) i+1)

Root[x^3-x-1,1]//N[#,12]&

With[{τ=(1+I(Sqrt[23]))/2},((DedekindEta[τ](E^(I π/24)))/((Sqrt[2])DedekindEta[2τ]))]//N

Derivative

v5.2

DedekindEta'[t]

v6

DedekindEta'[t]

expr=Subtract@@{D[Log[DedekindEta[-(1/t)]],t],D[Log[DedekindEta[t]],t]+(1/2)D[Log[-I t],t]};

Table[expr/. t->Random[Complex,5{-1,1+I},50],{10}]//Chop//Union

Beter Derivative

expr=Subtract@@{D[Log[dedekindEta[-(1/t)]],t],D[Log[dedekindEta[t]],t]+(1/2)D[Log[-I t],t]}/.dedekindEta'[t_]:>(I/π)DedekindEta[t]WeierstrassZeta[1,WeierstrassInvariants[{1,t}]]/.dedekindEta:>DedekindEta

Table[expr/. t->Random[Complex,5{-1,1+I},50],{20}]//Chop//Union

In Terms of Elliptic

D[((EllipticTheta[2,(π/6),(q^(1/6))])/(Sqrt[3])),q]//FullSimplify

FullSimplify[D[With[{q=Exp[2π I τ]},((EllipticTheta[2,(π/6),(q^(1/6))])/(Sqrt[3]))],τ]]

expr1=Subtract@@{D[Log[dedekindEta[-(1/t)]],t],D[Log[dedekindEta[t]],t]+(1/2)D[Log[-I t],t]}/.dedekindEta'[t_]:>((I ((((E^(2 I π t))))^(1/6)) π Derivative[0,0,1][EllipticTheta][2,(π/6),((((E^(2 I π t))))^(1/6))])/(3 (Sqrt[3])))/.dedekindEta:>DedekindEta

Table[expr1/.t->Random[Complex,5{-1,1+I},50],{20}]//Chop//Union

Identities

(DedekindEta[τ])==(e^(i π τ/12)) (QPochhammer1[(e^(2 i π τ))])

{DedekindEta[τ],Exp[I π τ/12]QPochhammer[Exp[2π I τ]]}/.τ->.1I

η((OverBar[q]))==(1/(Sqrt[3]))(Subscript[ϑ, 2])((π/6),((OverBar[q])^(1/6)))

Solve[q==Exp[2I π τ],τ]

FullSimplify[id=Subtract@@{DedekindEta[-((I Log[q])/(2π))],((EllipticTheta[2,(π/6),(q^(1/6))])/(Sqrt[3]))}]

Developer`ZeroQ[id]

Cases[Table[id/.q->Random[Complex],{1000}]//Chop//Union,_?NumericQ]

η(t+1)==(e^(i π/12)) η(t)

FullSimplify[Subtract@@{DedekindEta[t+1],Exp[I Pi/12]DedekindEta[t]},Im[t]>0]

Table[Subtract@@{DedekindEta[t+1],Exp[I Pi/12]DedekindEta[t]}/.t->Random[Complex,{-2,2+2I},20],{100}]//Chop//Union

η(n+t)==(e^(i π n/12)) η(t)

Table[Subtract@@{DedekindEta[t+n ],Exp[I Pi n/12]DedekindEta[t]}/.{n->Random[Integer,{-100,100}],t->Random[Complex,{-2,2+2I},20]},{100}]//Chop//Union

FullSimplify[Subtract@@{DedekindEta[-1/t],Sqrt[-I t]DedekindEta[t]},Im[t]>0]

η(-(1/t))==(Sqrt[-i t]) η(t)

FullSimplify[Subtract@@{DedekindEta[-1/t],Sqrt[-I t]DedekindEta[t]},Im[t]>0]

Table[Subtract@@{DedekindEta[-1/t],Sqrt[-I t]DedekindEta[t]}/.t->Random[Complex,{-2,2+2I},20],{100}]//Chop//Union

(Subscript[ϑ, 3])(0,(e^(π i t)))==(((η(((t+1)/2)))^2)/(η(t+1)))

expr=FullSimplify[Subtract@@{EllipticTheta[3,0,Exp[π I t]],(((DedekindEta[(1/2)(t+1)])^2)/(DedekindEta[t+1]))}]

FunctionExpand[expr]

Table[expr/.t->Random[Complex,5{-1,1+I},50],{100}]//Chop//Union

Series

v5.2

Series[DedekindEta[t],{t,0,2}]

v6

Series[DedekindEta[t],{t,0,5}]

Solve[q==Exp[2I π τ],τ]

Series[DedekindEta[t],{t,0,5}]/.t->-((I Log[q])/(2π))

Series[DedekindEta[-((I Log[q])/(2π))],{q,0,5}]

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

Series[Sum[(((-1))^n)(q^(n(3n-1)/2)),{n,-100,100}],{q,0,50}]

Series[1+Sum[(((-1))^n)((q^(n(3n-1)/2))+(q^(n(3n+1)/2))),{n,100}],{q,0,50}]

CoefficientList[%,q]