Rutherford Constant

Author

Eric W. Weisstein
September 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/RutherfordConstant.nb.

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

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

Value

Symbolic

Integrate[Cos[u]/(2 Pi)/Sqrt[x-Cos[u]],{u,If[-1<x<1,ArcCos[x],0],2 Pi-If[-1<x<1,ArcCos[x],0]},Assumptions->x≥-1]

Integrate[1/(2 Pi)/Sqrt[x-Cos[u]],{u,If[-1<x<1,ArcCos[x],0],2 Pi-If[-1<x<1,ArcCos[x],0]},Assumptions->x≥-1]

Numeric

R[x_?NumericQ]:=NIntegrate[Cos[u]/(2 Pi)/Sqrt[x-Cos[u]],{u,If[-1<x<1,ArcCos[x],0],2 Pi-If[-1<x<1,ArcCos[x],0]}]

S[x_?NumericQ]:=NIntegrate[1/(2 Pi)/Sqrt[x-Cos[u]],{u,If[-1<x<1,ArcCos[x],0],2 Pi-If[-1<x<1,ArcCos[x],0]}]

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,Infinity}]//Timing

Plot[R[x]^2/S[x],{x,-1,10},PlotRange->All]

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1,Infinity}]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1,Infinity},WorkingPrecision->10]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1,Infinity},WorkingPrecision->15]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1,Infinity},WorkingPrecision->20]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1,10^10},WorkingPrecision->20]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,10^10,∞},WorkingPrecision->20]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,Infinity},Method->"DoubleExponential"]//Timing

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,Infinity},Method->"GaussKronrod"]//Timing

Left half

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1},WorkingPrecision->20]

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,-1,1},WorkingPrecision->20,MaxRecursion->10]

Right half

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,1,∞},WorkingPrecision->20]

Sqrt[2] NIntegrate[R[x]^2/S[x],{x,1,∞},WorkingPrecision->20,MaxRecursion->10]