Classifying Singularities

If $f(z)$ is holomorphic at $z_0$ and the first $n$-terms of its Taylor series are 0. I.e. $f(z)=a_n(z-z_0{)}^n+a_{n+1}(z-z_0{)}^{n+1}+\cdots =(z-z_0{)}^n(a_n+a_{n+1}(z-z_0)+\cdots )=(z-z_0{)}^{n}g(z)$. Then $f(z)$ has 0 of order $n$ at $z_0$ and $g(z_0)=a_n$ is non-zero and is holomorphic.

Zeros of holomorphic functions are isolated or the function is constant.

There is an isolated singularity at $z_0$ if $f(z)$ is holomorphic in a neighborhood of $z_0$ but not at $z_0$.

If Laurent series has 0 principle part, $(b_n=0\mathrm{\forall }n)$ but the function is not holomorphic at $z_0$, then $z_0$ is called a removable singularity.

Example: Rational function, piecewise function defined with discontinuity at $z_0$, $g(z)=\frac{\exp (z)-1}{z}$ (we can patch this by defining a new function with $g(0)=1$).

If the highest power of $\frac{1}{z-z_0}$ with $b_n\ne 0$ is $n$, we say $f(z)=\sum _m{a}_m(z-z_0{)}^m+\sum _{m=1}^n\frac{b_m}{(z-z_0{)}^m}$ has a pole of order $n$ at $z_0$. If $n=1$ we call this a simple pole.

If the principal part of the Laurent expansion goes to $\mathrm{\infty }$, then the singularity at $z_0$ is called an essential singularity (E.g. Picard’s theorem).