Elliptic Logarithm

Author

Eric W. Weisstein
June 27, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheoreticFunctions/EllipticLogarithm.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Plot`

Plot[EllipticLog[{z,Sqrt[z^3+z^2+z]},{1,1}],{z,0,5},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,(Subscript[eln, 1,1])[z]}]

ComplexPlot3D[EllipticLog[{z,Sqrt[z^3+z^2+z]},{1,1}],{z,-3,3},ComplexPlotLabel->(Subscript[eln, 1,1])[z],
PlotPoints->100
]//Timing

ComplexContourPlot[EllipticLog[{z,Sqrt[z^3+z^2+z]},{1,1}],{z,-3,3},ComplexPlotLabel->(Subscript[eln, 1,1])[z]]//Timing

Definition

i=FullSimplify[Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x}]]/2

v5.0

FullSimplify[Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x},Assumptions->(a|b)∈Reals]/2]

i=FullSimplify[Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x},GenerateConditions->False]/2]

Subtract@@@Table[N[{i,EllipticLog[{x,Sqrt[x^3+a x^2+b x]},{a,b}]}/.{x->Random[Complex],a->Random[Complex],b->Random[Complex]}],{20}]//Chop

v4.2

With[{a=1.,b=1.},Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x}]
]

v5.0

With[{a=1,b=1},Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x},GenerateConditions->False]
]

Chop[Table[Subtract@@{NIntegrate[1/Sqrt[t^3+t^2+t],{t,Infinity,x}],(1/Sqrt[x*(1+x+x^2)])*(2*(-1)^(2/3)*Sqrt[1-(-1)^(2/3)/x]*x^(3/2)*Sqrt[((-1)^(1/3)+x)/x]*EllipticF[(-I)*ArcSinh[(-1)^(5/6)/Sqrt[x]],(-1)^(2/3)])}/.x->Random[Complex],{10}]]

With[{a=1,b=1},Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x}]
]

(1/(Sqrt[x+(x^2)+(x^3)]))(2 (((-1))^(2/3)) (Sqrt[1-((((-1))^(2/3))/x)]) (x^(3/2)) (Sqrt[(((((-1))^(1/3))+x)/x)]) EllipticF[-I ArcSinh[((((-1))^(5/6))/(Sqrt[x]))],(((-1))^(2/3))])/.x->0

With[{a=1,b=1},Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,0}]
]

With[{a=1.,b=1.},Integrate[1/Sqrt[t^3+a t^2+b t],{t,∞,x}]
]

Crash