Nicholson's Formula

Author

Eric W. Weisstein
November 29, 2008

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

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

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

Identity

V6

8/Pi^2 Integrate[BesselK[0,2z Sinh[t]]Cosh[2ν t],{t,0,∞}]

V7

8/Pi^2 Integrate[BesselK[0,2z Sinh[t]]Cosh[2ν t],{t,0,∞}]

Numerical Check

With[{prec=25},
Table[
With[{
ν=Random[Complex,{-2-2I,2+2I},prec],
z=Random[Complex,{-2I,2+2I},prec]
},Subtract@@{8/Pi^2 NIntegrate[BesselK[0,2z Sinh[t]]Cosh[2ν t],{t,0,20},MaxRecursion->10,WorkingPrecision->prec],
BesselJ[ν,z]^2+BesselY[ν,z]^2
}//Chop
],{10}]
]