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MathGroup Archive 2006

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Integrate failure

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71428] Integrate failure
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Sat, 18 Nov 2006 04:40:59 -0500 (EST)

Hello to all.

Consider the following function.

f[z_]:=Log[z^2-1]

Here is plots of the real and imaginary part

Show[MapIndexed[Plot[#1[f[z]], {z, -3, 3}, PlotStyle -> Hue[#2[[1]]/3],
DisplayFunction -> Identity] & , {Re, Im}],
  DisplayFunction -> $DisplayFunction, Frame -> {True, True, False,
False}, Axes -> False]

Here is contour plots of the real and imaginary part

Show[GraphicsArray[ContourPlot[#[f[x+I
y]],{x,-2,2},{y,-2,2},PlotPoints®100,Contours®50,ContourShading®False,DisplayFunction®Identity]&/@{Re,Im},Frame®True],ImageSize®600]

Here is plots of the real and imaginary part in the complex domain.

Show[GraphicsArray[(Plot3D[#1[f[x + I*y]], {x, -2, 2}, {y, -2, 2},
PlotPoints -> 40, DisplayFunction -> Identity] & ) /@
    {Re, Im}, Frame -> True], ImageSize -> 600]

The integrand clearly has a branch cut between -1 and 1.

Integrate fails to give the right answer as a quick check with
NIntegrate confirms.

Integrate[Log[z^2 - 1], {z, 1/10 - I, 1/10 + I}]
(-(1/5))*I*(ArcTan[20/199] - 5*(-4 + Pi + ArcTan[400/39999] +
Log[40001/10000]))
{N[%], NIntegrate[Sqrt[z^2 - 1], {z, 1/10 - I, 1/10 + I}]}
{0. + 0.5178786980875206*I, 0. + 0.08305882917250686*I}

Apparently the problem arises from the branch cut.

How can we get the correct answer within Mathematica (version 5.2)?

Thanks a lot for any response
Dimitris


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