# Classifying Singularities

## Zeroes

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

## Singularities

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, $$(b_n=0\forall n)$$ 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 $$b_n\neq 0$$ 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 $$\infty$$, then the singularity at $$z_0$$ is called an essential singularity (E.g. Picard’s theorem).

Created: 2023-06-25 Sun 02:25

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