Nome

Author

Eric W. Weisstein
February 8, 2009

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

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

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

Plots

<<Utilities`Plot`

Real Axis

Plot[EllipticNomeQ[k^2],{k,0,1},AxesLabel->{k,EllipticNomeQ[k]},PlotStyle->Red,PlotRange->{0,.5},ImageSize->250]

Complex Plane

ComplexPlot3D[EllipticNomeQ[k^2],{k,-2,2},
ComplexPlotLabel->EllipticNomeQ[k],AxesLabel->TraditionalForm/@{Subscript[k,x],Subscript[k,y],""}]//Timing

ComplexContourPlot[EllipticNomeQ[k^2],{k,-2,2},
ComplexPlotLabel->EllipticNomeQ[k],FrameLabel->TraditionalForm/@{Subscript[k,x],Subscript[k,y]}]//Timing

Series

About 0

Series[EllipticNomeQ[m],{m,0,10}]

Through[{Numerator,Denominator}[CoefficientList[%,m]]]

SeriesCoefficient[EllipticNomeQ[m],{m,0,n}]

FunctionExpand[EllipticNomeQ[m]]

SeriesCoefficient[(E^(-((π EllipticK[1-m])/(EllipticK[m])))),{m,0,n}]

About 1

Series[EllipticNomeQ[m],{m,1,2}]//FullSimplify

Special Values

EllipticNomeQ/@{0,1/2,1}

ODE

eqn=(-1+m)^2 m^2 q'[m]^4-q[m]^2 ((1-m+m^2) q'[m]^2+3 (-1+m)^2 m^2 q''[m]^2-2 (-1+m)^2 m^2 q'[m] q'''[m]);

DSolve[eqn==0,q[m],m]

DSolve[{eqn==0,q[0]==0,q[1/2]==Exp[-Pi],q[1]==1},q[m],m]

Simplify[eqn/.q->EllipticNomeQ]

Derivative

v5.2

EllipticNomeQ'[m]

FullSimplify[%]

v6

EllipticNomeQ'[m]

FullSimplify[Subtract@@{-(π (EllipticK[1-m] (EllipticE[m]-EllipticK[m])+EllipticE[1-m] EllipticK[m]) EllipticNomeQ[m])/(2 (-1+m) m ((EllipticK[m])^2)),-(((π^2) EllipticNomeQ[m])/(4 (-1+m) m ((EllipticK[m])^2)))}]

Table[Subtract@@{-(π (EllipticK[1-m] (EllipticE[m]-EllipticK[m])+EllipticE[1-m] EllipticK[m]) EllipticNomeQ[m])/(2 (-1+m) m ((EllipticK[m])^2)),-(((π^2) EllipticNomeQ[m])/(4 (-1+m) m ((EllipticK[m])^2)))}/.m->Random[Real,{-1,1},20],{100}]//Chop//Union

V7

EllipticNomeQ'[m]

FullSimplify[%]

Limit

Limit[EllipticNomeQ[m],m->1]

Integral

v5.2

Integrate[EllipticNomeQ[m],m]

v6

Integrate[EllipticNomeQ[m],m]

V7

Integrate[EllipticNomeQ[m],m]