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)
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• Delivered-to: l-mathgroup@wolfram.com
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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]]

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