I'm going through Ahlfors, and in the section about the Residue theorem, and he says that about the poles, a function $f$ can be expanded to $$f(z)=B_h(z-a)^{-h}+\cdots+B_1(z-a)^{-1}+\varphi(z)$$ he then says when $B_1(z-a)^{-1}$ is omitted, the remainder is a derivative, which I can see. However, I'm not entirely sure why $B_1$ has been ommitted. I've come up with two explanations
- $\int_{\gamma}(z-a)^{-1}$ is proportional to the winding number, since the winding number is defined over the integral of a piecewise differentiable function. This is true regardless of if $\gamma$ is homologous to zero, and so this term is non-zero.
- Since the integral defines a region containing $a$, $\log(z-a)$ is multi-valued in the region containing $a$, and hence $(z-a)^{-1}$ is not the derivative of a single-valued function. Therefore, it has a non-zero residue.
Is either explanation correct? I think once he started talking about homologous paths and omitting points I got a bit confused as what the winding number says about analyticity.