Suppose that is analytic on a closed curve (i.e., is analytic in a region that contains ). Show that

is purely imaginary.

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- April 12th 2009, 08:45 PMStigercomplex integration
Suppose that is analytic on a closed curve (i.e., is analytic in a region that contains ). Show that

is purely imaginary. - April 13th 2009, 11:22 AMOpalg
If the curve is smooth enough to have a differentiable parametrisation then this is just integration by parts. In fact, suppose that is given by a differentiable path , with . Then

Thus the integral is equal to the negative of its complex conjugate, so it is purely imaginary.