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