Closed-contour integral of a derivative

Question

Suppose a function $f(z)$ is single-valued everywhere and holomorphic inside a closed contour $C$, except for one pole. The derivative $ g(z) = \frac{ \partial f}{\partial z}$ also has the same pole. I have a naive question about the integral

$$ \int_C dz g(z) $$

This is the integral around a closed contour of the derivative of a single valued function, so it should be zero. But if done with the residues it should be nonzero because of the pole inside $C$.

So is it zero or not?

Answer 1

You cannot say that that integral is non-zero simply because there is a pole. For example the integral of $\frac 1 {z^{2}}$ over the unit circle is $0$. Answer: YES, the integral is $0$.