title: Cauchy Principal Value and Kramers-Kronig relations
body: <p>I am fiddling around with Kramers-Kronig relations, and for that I need to use the Principal Value. I have the following notebook, where I take the dispersion <code>disp</code> and from that find the absorption using the Kramers-Kronig relation.</p>
  
  <p>When I compare the resulting absorption to the analytical expression for the absorption, I see that the widths of are not the same after normalizing - which they should be. Is there a setting/parameter I am missing?</p>
  
  <pre><code>\[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 -&gt; PrincipalValue, Exclusions -&gt; Automatic, 
       MaxRecursion -&gt; 100] // Quiet;
  
  
  max = \[CapitalGamma];
  step = 100;
  
  
  absVals = {}; dispVals = {};
  For[i = -step, i &lt; step + 1, i++,
    \[Delta] = max*i/step;
    absVals = Append[absVals, {\[Delta], absKK[\[Delta]]}]];
  
  Show[
   ListLinePlot[absVals, PlotRange -&gt; Full, PlotStyle -&gt; {Red, Dashed}],
   Plot[-6.5 abs[\[CapitalDelta]], {\[CapitalDelta], -\[CapitalGamma], \
  \[CapitalGamma]}, PlotRange -&gt; Full]]
  </code></pre>