Inverse Nome

Author

Eric W. Weisstein
September 20, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

q on Real Line

<<Utilities`Plot`

V5

Plot[InverseEllipticNomeQ[x],{x,0,1},PlotRange->All,
PlotStyle->Red,AxesLabel->TraditionalForm/@{x,InverseEllipticNomeQ[x]}]

V6

$Version

Plot[InverseEllipticNomeQ[x],{x,0,1},PlotRange->All,AxesLabel->{x,InverseEllipticNomeQ[x]}]

Plot[InverseEllipticNomeQ[x],{x,-.1,1},PlotRange->{{-.1,1},{-.5,1}},AxesLabel->{q,m[q]}]

Plot[InverseEllipticNomeQ[x],{x,-.2,1},PlotRange->{{-.1,1},{-2,1}},AxesLabel->{q,m[q]}]

V7

Plot[InverseEllipticNomeQ[x],{x,0,1},PlotRange->All,AxesLabel->{x,InverseEllipticNomeQ[x]}]

Plot[InverseEllipticNomeQ[x],{x,-.1,1},PlotRange->{{-.1,1},{-.5,1}},AxesLabel->{q,m[q]}]

Plot[InverseEllipticNomeQ[x],{x,-.2,1},PlotRange->{{-.1,1},{-2,1}},AxesLabel->{q,m[q]}]

Unit Disk

V5

Show[GraphicsArray[Function[f,Graphics[{
RadialContourPlot[f[InverseEllipticNomeQ[#]]&,{r,0,.995},PlotPoints->{300,200},ColorFunction->(Hue[Random[]]&)][[1]],
Text[TraditionalForm[f[InverseEllipticNomeQ[z]]],{0,1.2}]},AspectRatio->Automatic]]/@{Re,Im}]]

V6

DensityPlot[Piecewise[{{Re[InverseEllipticNomeQ[x+I y]],x^2+y^2<1}}],{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&),ColorFunctionScaling->True,PlotRange->All,PlotPoints->20,MaxRecursion->3]

...crash...

DensityPlot[Piecewise[{{Re[InverseEllipticNomeQ[x+I y]],x^2+y^2<1}}],{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&),ColorFunctionScaling->False,PlotRange->All,PlotPoints->20,MaxRecursion->3]

DensityPlot[Piecewise[{{Re[InverseEllipticNomeQ[x+I y]],x^2+y^2<1}}],{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&),ColorFunction->Hue,ColorFunctionScaling->False,PlotRange->All,PlotPoints->20,MaxRecursion->3]

DensityPlot[Piecewise[{{Re[InverseEllipticNomeQ[x+I y]],x^2+y^2<1}}],{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&),ColorFunction->Hue,ColorFunctionScaling->False,PlotRange->All,PlotPoints->25,MaxRecursion->4]

...crash...

DensityPlot[Piecewise[{{Im[InverseEllipticNomeQ[x+I y]],x^2+y^2<1}}],{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&),ColorFunction->Hue,ColorFunctionScaling->False,PlotRange->All,PlotPoints->20,MaxRecursion->3]

Special Values

InverseEllipticNomeQ/@{0,Exp[-Pi],1}

Numerical Evaluation

InverseEllipticNomeQ[-0.99`3]

InverseEllipticNomeQ[-0.99]

InverseEllipticNomeQ[-0.8]

InverseEllipticNomeQ[-0.3]

Table[{q,InverseEllipticNomeQ[q]},{q,-.1,-.9,-.05}]//Chop//TableForm

Do[Print[{n,N[InverseEllipticNomeQ[1-(10^(-n))]]}//Timing],{n,8}]

Do[Print[{n,TimeConstrained[N[InverseEllipticNomeQ[-1+(10^(-n))]],60]//Timing}],{n,8}]

Do[Print[{n,TimeConstrained[N[InverseEllipticNomeQ[-1+(10^(-n))],10],60]//Timing}],{n,8}]

Limits

(lim_(q->(SuperMinus[1]))) (q^(-1))(q)==1

Limit[InverseEllipticNomeQ[q],q->0]

Limit[InverseEllipticNomeQ[q],q->1]

Limit[InverseEllipticNomeQ[q],q->1,Direction->1]

(lim_(q->0)) (q^(-1))(q)==0

Limit[InverseEllipticNomeQ[q],q->0]

Limit[InverseEllipticNomeQ[q],q->0,Direction->#]&/@{-1,1,Automatic}

(lim_(q->-1)) (q^(-1))(q)==-∞

Limit[InverseEllipticNomeQ[q],q->-1]

Limit[InverseEllipticNomeQ[q],q->-1,Direction->#]&/@{-1,1,Automatic}

Table[{q,InverseEllipticNomeQ[q]},{q,-.1,-.9,-.05}]//Chop//TableForm

Derivative

v5.2

(d=FullSimplify[D[InverseEllipticNomeQ[q],q]])//TraditionalForm

v6

(d=D[InverseEllipticNomeQ[q],q])//TraditionalForm

V7

(d=D[InverseEllipticNomeQ[q],q])//TraditionalForm

Numerical

N[d/.q->9/10,100]

N[d/.q->99/100,100]

Plot[d,{q,0,1},PlotRange->All,PlotStyle->Red]

Limit

v5.2

Limit[D[InverseEllipticNomeQ[q],q],q->0]

Limit[D[InverseEllipticNomeQ[q],q],q->1]

v6

Limit[D[InverseEllipticNomeQ[q],q],q->0]

Limit[D[InverseEllipticNomeQ[q],q],q->1]

V7

Limit[D[InverseEllipticNomeQ[q],q],q->0]

Limit[D[InverseEllipticNomeQ[q],q],q->1]

Identities

(q^(-1))(q)==((((Subscript[ϑ, 2])(0,q))^4)/(((Subscript[ϑ, 3])(0,q))^4))

FullSimplify[InverseEllipticNomeQ[q]-(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4))]

FunctionExpand[%]

l=Select[Table[RandomComplex[],{100}],Abs[#]<1&];

Subtract@@@(({(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4)),InverseEllipticNomeQ[q]}/.q->#)&/@l)//Chop//Union

((16 (((DedekindEta[(τ/2)])^8) ((DedekindEta[2 τ])^16)))/((DedekindEta[τ])^24))==(q^(-1))(q)

Series[16((((DedekindEta[τ/2])^8)((DedekindEta[2τ])^16))/((DedekindEta[τ])^24))/.τ->-((I Log[q])/π),{q,0,6}]

Series[%,{q,0,6}]

Series[Exp[x]Exp[2x],{x,0,10}]

FullSimplify[16((((DedekindEta[τ/2])^8)((DedekindEta[2τ])^16))/((DedekindEta[τ])^24))/.τ->-((I Log[q])/π)]

FullSimplify[expr=((16 ((DedekindEta[-((I Log[q])/(2 π))])^8) ((DedekindEta[-((2 I Log[q])/π)])^16))/((DedekindEta[-((I Log[q])/π)])^24))]==expr

l=Select[Table[RandomComplex[],{100}],Abs[#]<1&];

Subtract@@@(({InverseEllipticNomeQ[q],((16 ((DedekindEta[-((I Log[q])/(2 π))])^8) ((DedekindEta[-((2 I Log[q])/π)])^16))/((DedekindEta[-((I Log[q])/π)])^24))}/.q->#)&/@l)//Chop//Union

Series

About 0

V6

Series[InverseEllipticNomeQ[q],{q,0,3}]

V7

Series[InverseEllipticNomeQ[q],{q,0,5}]

CoefficientList[%,q]

SeriesCoefficient[InverseEllipticNomeQ[q],{q,0,n}]

Series[(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4)),{q,0,4}]

SeriesCoefficient[(((EllipticTheta[2,0,q])^4)/((EllipticTheta[3,0,q])^4)),{q,0,n}]

About 1

V4

Series[InverseEllipticNomeQ[q],{q,1,3}]

V7

Series[InverseEllipticNomeQ[q],{q,1,3}]

Series