Contour integral - Use an example to contredict an answer

Question

The answer that my teacher gives us is $2πi \log z_0$. I know that this answer is false according to the question $\int_C \frac{\log z}{z-z_0} dz$ - Cauchy theorem with $z_0$ outside the interior of $\gamma$ and the answer given by Ron Gordon. I think he has used the simple Cauchy formula with the primitive of $\log′ z$. Is there a contradiction in this answer I can mention him? To know that he gives us only the simple solution (not his resolution method).

Answer 1

Your teacher made a mistake. An easy way to see this is to fix the contour and then note that as $z_0\to \infty,$ the integrand $\to 0$ uniformly, hence the integral $\to 0.$ But your teacher's answer has the integral $\to \infty$ as $z_0\to \infty.$