Principal Values and NIntegrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg130508] Principal Values and NIntegrate*From*: Niles <niels.martinsen at gmail.com>*Date*: Wed, 17 Apr 2013 02:31:01 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

I am fiddling around with Kramers-Kronig relations, and for that I need to use the Principal Value. I have the following notebook (shown at the end), where I take the dispersion "disp" and from that find the absorption using the Kramers-Kronig relation. When I compare the resulting absorption to the analytical expression "abs", I see that the widths of are not the same after normalizing - which they should be. Is there a setting/parameter I am missing? \[CapitalGamma] = 50 10^3; disp[\[CapitalDelta]_] := 1/\[Pi] \[CapitalDelta]/(\[CapitalDelta]^2 + (\[CapitalGamma]/(4 \ \[Pi]))^2/4); abs[\[CapitalDelta]_] := 1/\[Pi] (\[CapitalGamma]/(4 \[Pi]))/(\[CapitalDelta]^2 + (\ \[CapitalGamma]/(4 \[Pi]))^2); absKK[\[CapitalDelta]_] := -NIntegrate[disp[x]/( x - \[CapitalDelta]), {x, -Infinity, \[CapitalDelta], Infinity}, Method -> PrincipalValue, Exclusions -> Automatic, MaxRecursion -> 100] // Quiet; max = \[CapitalGamma]; step = 100; absVals = {}; dispVals = {}; For[i = -step, i < step + 1, i++, \[Delta] = max*i/step; absVals = Append[absVals, {\[Delta], absKK[\[Delta]]}]]; Show[ ListLinePlot[absVals, PlotRange -> Full, PlotStyle -> {Red, Dashed}], Plot[-6.5 abs[\[CapitalDelta]], {\[CapitalDelta], -\[CapitalGamma], \ \[CapitalGamma]}, PlotRange -> Full]]