Poles, branch cuts, and zeros From what I understand, these three concepts all describe the points where the function is not continuous. How to tell them apart? Thanks!

*

*If $f(z)$ is holomorphic/analytic on $0 < |z-z_0| < r$ then $z_0$ is an isolated singularity. From the Cauchy integral formula in an annulus you have the Laurent series $f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n$ converging on $0 < |z-z_0| < r$, and two cases are possible : 


*

*$a_n = 0 $ for $n < -k$ so that $(z-z_0)^{k}f(z)$ is analytic on $|z-z_0| < r$ and $z=z_0$ is a pole of order $k$. If $k\le 0$ then $z_0$ was in fact a removable singularity

*otherwise $z= z_0$ is an essential singularity of $f(z)$


*Other types of  singularities are non-isolated and include :


*

*branch points : a point around which you can continue analytically $f(z)$, but $f(z_0+e^{2i \pi}(z-z_0)) \ne f(z)$)

*and frontiers : $f(z) = \sum_{n=1}^\infty z^{2^n}$ is analytic on $|z| < 1$ but $\lim_{r \to 1^-} f(r e^{2 \pi i m/2^k}) = \infty$ whenever $m,k \in \mathbb{N}$, so you can't continue analytically $f(z)$ beyond $|z| < 1$
