Kac Formula

Author

Eric W. Weisstein
May 26, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/KacFormula.nb.

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

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

Definition

e[n_]:=(1/π)(Sqrt[(1/((((t^2)-1))^2))-(((((n+1))^2) (t^(2 n)))/((((t^(2 n+2))-1))^2))])

Plots

Table[NIntegrate[e[n],{t,-∞,1,∞}],{n,10}]

<<Utilities`Plot`
<<Graphics`Colors`

MakeBoxes[elabel,TraditionalForm]:=SubscriptBox["E","n"]
MakeBoxes[ilabel,TraditionalForm]:=RowBox[{SubscriptBox["I","n"],"(","t",")"}]

Show[
GraphicsArray[Block[{$DisplayFunction=Identity},
{
Plot[Evaluate[Table[(1/π)(Sqrt[(1/((((t^2)-1))^2))-(((((n+1))^2) (t^(2 n)))/((((t^(2 n+2))-1))^2))]),{n,5}]],{t,-5,5},PlotStyle->{Red,Yellow,Green,Blue,Violet},AxesLabel->TraditionalForm/@{t,ilabel}],
Show[
Plot[(2/π)Log[n]+.6257+(2/(π n)),{n,.1,20},PlotStyle->Blue,PlotRange->{0,2.5},AxesLabel->TraditionalForm/@{n,elabel}],
IntegerListPlot[Table[NIntegrate[e[n],{t,-∞,1,∞}],{n,20}]]
]
}
]],GraphicsSpacing->-.05]

Integral

n==1

Integrate[e[1],{t,-∞,∞}]

n=2

FullSimplify[e[2],t∈Reals]

NIntegrate[((Sqrt[1+4 (t^2)+(t^4)])/(π+π (t^2)+π (t^4))),{t,-∞,∞},WorkingPrecision->20]

v4.2

Integrate[((Sqrt[1+4 (t^2)+(t^4)])/(π+π (t^2)+π (t^4))),{t,-∞,∞}]//Timing

v5

Integrate[((Sqrt[1+4 (t^2)+(t^4)])/(π+π (t^2)+π (t^4))),{t,-∞,∞}]//Timing

v6

Integrate[((Sqrt[1+4 (t^2)+(t^4)])/(π+π (t^2)+π (t^4))),{t,-∞,∞}]//Timing

...crash..

Asymptotic Constant

<<Graphics`Colors`

Plot[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0.01,20},PlotStyle->Red]

Plot[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,100,200},PlotStyle->Red]

Integrate

NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity}]//Timing

v4.2

Integrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity}]//Timing

v5

Integrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity}]//Timing

v6

Integrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity}]//Timing

NIntegrate

2/Pi(Log[2]+NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity},WorkingPrecision->50])//Timing

2/Pi(Log[2]+NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity},WorkingPrecision->50,PrecisionGoal->20])//Timing

2/Pi(Log[2]+NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity},WorkingPrecision->100,PrecisionGoal->20])//Timing

2/Pi(Log[2]+NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,Infinity},WorkingPrecision->200,PrecisionGoal->110])//Timing

Plot[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,20}]

v4.2

(i1=NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,2},PrecisionGoal->120,MaxRecursion->20,WorkingPrecision->300])//Timing

(i1=NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,2},PrecisionGoal->130,MaxRecursion->20,WorkingPrecision->400])//Timing

(i2=NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,2,∞},Method->DoubleExponential,PrecisionGoal->120,WorkingPrecision->300,MaxRecursion->20])//Timing

(i2=NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,2,∞},Method->DoubleExponential,PrecisionGoal->130,WorkingPrecision->400,MaxRecursion->20])//Timing

(2/π)(Log[2]+i1+i2)

v5

NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,0,2},PrecisionGoal->120,MaxRecursion->20,WorkingPrecision->300]//Timing

NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,2,∞},Method->DoubleExponential,PrecisionGoal->100,WorkingPrecision->200]//Timing

...crash...

FullSimplify[ExpandAll[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1)],x>2]

NIntegrate[-(1/(1+x))+(Sqrt[(1/(x^2))-((Csch[x])^2)]),{x,2,∞},Method->DoubleExponential,PrecisionGoal->100,WorkingPrecision->200]//Timing

v6

NIntegrate[Sqrt[1/x^2-4 Exp[-2x]/(1-Exp[-2x])^2]-1/(x+1),{x,2,∞},Method->DoubleExponential,PrecisionGoal->100,WorkingPrecision->200]//Timing

...hangs; not interruptible...

NIntegrate[-(1/(1+x))+(Sqrt[(1/(x^2))-((Csch[x])^2)]),{x,2,∞},Method->DoubleExponential,PrecisionGoal->100,WorkingPrecision->200]//Timing

Series

Series[e[n],{n,∞,2}]