Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2013

[Date Index] [Thread Index] [Author Index]

Search the Archive

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]]





  • Prev by Date: Converting a magnified image of a line to a single pixel smoothed
  • Next by Date: Re: Plot with axes exchanged
  • Previous by thread: Re: Converting a magnified image of a line to a single pixel smoothed
  • Next by thread: programmatically save as PDF (SaveRename problem)