In complex analysis we say that for fn's like lnz we apply a branch cut along positive x-axis to make sure it's single valued. i.e restrict theta s.t 0<=theta<2Pi but we never allow theta to equal 2Pi as this would make lnz take on 2nd value.(adsbygoogle = window.adsbygoogle || []).push({});

Let us integrate around a contour which goes from origin to x= infinity, then goes anticlockwise around a circle of infinite radius back to the positive x-axis at plus infinity, then we go back to the origin along the x-axis, and then go clockwise around a circle of zero radius, hence avoiding crossing the branch cut. we say that theta = 0 as we go from the origin out to x=infinity, and then we have to say (in order to get the answer right) that theta = 2pi when we go from x-infinity back to the origin. But surely this is a contradiction, since we assumed that theta could not equal 2 Pi?

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# Branch cut contradiction

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