MMfE - Classification of Singularities

Classification of Singularities

Mathematical_Methods_for_EngineeringTheorem Given an holomorphic function $f(z)$, if such function has a pole in $z_0$, so it can be written as:

$$f(z)=\frac{g(z)}{(z-z_0{)}^k}$$

Where $k$ is the multiplicity of the pole $z_0$ .

Then suppose we can write it with the Laurent Series such as:

$$f(z)=\sum _{k\in \mathbb{Z}}{c}_k(z-z_0{)}^k$$

Then we can classify the singularity $z_0$ as:

• REMOVABLE: if $k\ge 0$ (The Lauren Series has no negative exponents)
• POLE: if $k\in \mathbb{R},\text{and}k<0$
• ESSENTIAL: if $k=-\infty$

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Another way:

Another way to classify a singularity is to solve the following limit:

$$\underset{z\to z_0}{\lim }|f(z)|$$

Then we can classify the singularity $z_0$ as:

• REMOVABLE: if the limit exist and it’s finite
• POLE: if the limit is equal to $\pm \infty$
• ESSENTIAL: if the limit does not exist

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