Yesterday, I read the proof of Laurent’s Theorem in my complex variables book, and I was stuck at the following equation.^{1}
Suppose that $f$ is analytic on the annular region $R_1 < \abs{z  z_0} < R_2$. Let $C$ be any postivelyoriented closed curve surrounding point $z_0$. By Cauchy–Goursat Theorem, we have
Image size: 300
Source code: $\rm \LaTeX$, SVG
I was stuck at this point.
The function $f$ is analytic on the entire circle $\gamma$, including the circumference of $\gamma$. Then the boundary of the region where $f$ is analytic is the circumferences of $C_1$ and $C_2$. I wondered why the third term in $\eqref{int}$ was needed. I took me half an hour to figure out that the integrand is not simply $f(s)$, but with a denominator $s  z$. Since $z$ is the centre of the circle $\gamma$, I need to also include it as the boundary of the region of analyticity of the integrand.

Brown, J. W., Churchill, R. V., & Lapidus, M. (1996). Complex variables and applications (Vol. 7). (pp. 17). New York: McGrawHill. ↩