I need some clarification on the proof of the Laurent series development as presented by Conway in his book Functions of One Complex Variable I, second edition, pages 107/108.
The function $f_1(z)$ is defined as $$f_1(z) = -\frac1{2\pi i} \int_{\gamma_1} \frac{f(w)}{w-z}dw$$ where $f(z)$ is analytic in the annulus $\mbox{ann}(a,R_1,R_2)$ and $\gamma_1(t) = a+r_1 e^{2\pi i t}$, $0\leq t\leq 2\pi$, with $R_1 < r_1 < R_2$.
The author then states that $f_1(z)$ is analytic in $$G=\{z:|z-a|> R_1\}.$$ But $\{\gamma_1\} \subset G$, and the integrand is not defined when $z \in \{\gamma_1\}$.
I would rather say that $$G=\{z:|z-a|> r_1\}.$$
Is this reasoning correct?
Similarly (page 108) the function $$g(z) = \begin{cases} f_1\left(a+\frac1z\right) & (z\neq 0)\\ 0 & (z= 0)\end{cases}$$ is said to be analytic in $B(0;1/R_1)$, but I would say that it is analytic in $B(0;1/r_1)$.