Suppose $f(z)$ is analytic on $A=\{z: 0<|z|<\delta\}$ with an essential singularity at 0 such that its Laurent expansion is
$ \sum_{n=-\infty}^\infty a_nz^n $ Ahlfors in his book "Complex Analysis" seems to say that $f(z)-\frac{a_{-1}}{z}$ need not be the derivative of an analytic function on $A$.
Does anyone know an example? If such example exists, I suppose that $ \int_C f(z)\,dz $ need not be equal to $2\pi i a_{-1}.$ Am I right?