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Evaluate $\int(\overline z + 2z)dz$ along the circle $x^2+y^2 = 1.$
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$x^2+ y^2 = 1$ is a circle with center $(0,0)$ and radius $= 1$ $$\text { put }z=reiθ = 1 eiθ$$

$$∴dz = eiθ . idθ $$

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and, $= e^{-iθ} $

$$∴ = . ie^{iθ}dθ$$

$$= i . dθ$$

$$= i [1θ +]_0^{2π}$$

$$= i ]$$

Now, $$= \cos 4π + i \sin 4π = 1 + i(0) = 1$$

$$∴ = i{ 2π + - }$$

$$∴ = 2iπ$$

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