$$\int _ C \log z\,dz$$ where $C$ is a full circle in positive direction with radius $R$. I substitute $z=Re^{it}$, $dz=Rie^{it}dt$
$$\int _ 0 ^{2\pi} \log (Re^{it})Rie^{it}\,dt$$ $$\int _ 0 ^{2\pi} \log (R)Rie^{it}\,dt +\int _ 0 ^{2\pi} itRie^{it}\,dt$$
First integral is zero.
$$-R\int _ 0 ^{2\pi} te^{it}\,dt$$
$$+R2\pi i$$
Am I doing something wrong? Why does the integral depend on R? Isn't the branch cut supposed to be the same value for all magnitudes of R?
Also wolframalpha find the antiderivative (for $R=2$) as $2 e^{i t} \left(-1+\log \left(2 e^{i t}\right)\right)$ which, it then evaluates to be $-4\pi i$. Why is it negative?