1. Cauchy's Integral theorem

Quick question

im trying to calculate the integral around the closed curve C of

z(bar) dz where z(bar) is the complex conjugate of z

where C is the unit circle.

Using the subsitution z = e^it i calculated the integral to be 0.
=> z(bar) = e^-it

However after finding the question in a maths book the answer is given to be 2pi(i).

Can any of ye tell me which solution is correct

2. $\mathop\oint\limits_{z=e^{it}}\overline{z}dz=\int_ 0^{2\pi}e^{-it}ie^{it}=2\pi i$

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integrate zbar dz around z=2 using complex cauchy integratio and cauchy theorem

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