MMfE - Laurent Series

• Think of the Laurent Series as Taylor Series for complex numbers.
• Online Resources: Youtube REALLY GOOD
• Online Resources: Youtube

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Def.: Laurent Series

Mathematical_Methods_for_EngineeringDefinition

Objective:

Transform the function $f(z)$ in this form:

$$f(z)=\sum _{n=0}^{\infty }{a}_n\cdot {\left(z-z_0\right)}^n+\sum _{n=1}^{\infty }{a}_{-n}\cdot {\left(z-z_0\right)}^{-n}$$

Useful Functions to Remember

$$\begin{array}{rl}& \frac{1}{1-cz}=\sum _{n=0}^{\infty }{\left(cz\right)}^n\text{for}|z|<\frac{1}{|c|}\\ & \frac{1}{1-\frac{b}{z}}=\sum _{n=0}^{\infty }{\left(1-\frac{b}{z}\right)}^n\text{for}|z|>|b|\end{array}$$

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~Ex.:

$f(z)=\frac{1}{(z-1)(z-2)}$

We can have it in 3 different forms, each with its own MMfE - Region of Holomorphicity

1. Full Circle:

$$\begin{array}{rl}f(z)=\frac{1}{(z-1)(z-2)}& =\frac{A}{(z-1)}+\frac{B}{(z-2)}\\ (A=-1,B=1)\to & |\\ & =\frac{1}{(z-2)}-\frac{1}{(z-1)}\\ \frac{1}{z-2}=-{\sum }_{n=0}^{\infty }\frac{z^n}{2^{n+1}}\text{for}|z|<2\to & |\\ & \\ \frac{1}{z-1}=-{\sum }_{n=0}^{\infty }{z}^n\text{for}|z|<1\to & |\\ & =-{\sum }_{n=0}^{\infty }{z}^n\left(1+\frac{1}{2^{n+1}}\right)\text{for}|z|<1\end{array}$$
3. **Donut**: ![[Pasted image 20220605121501.png]] $$ \begin{array}{rl} f(z) = \Large \frac{1}{(z-1)(z-2)} &= \Large \frac{1}{(z-2)} - \frac{1}{(z-1)} \\[8px] \large \frac{1}{z-2} = -\sum_{n=0}^{\infty} \kern3px \frac{z^n}{2^{n+1}} \kern10px \normalsize \text{for}\ |z| < 2 \to &\ | \\[1px] \\[1px] \large \frac{1}{z-1} = -\frac{1}{z} \cdot \sum_{n=0}^{\infty} \kern3px \left(\frac{1}{z}\right)^n \kern10px \normalsize \text{for}\ |z| > 1 \to &\ | \\[1px] &= \Large -\sum_{n=1}^{\infty} (z)^{-n} - \sum_{n=0}^{\infty} \kern3px \frac{z^n}{2^{n+1}} \kern10px \normalsize \text{for}\ 1 < |z| < 2 \end{array} $$
5. **Open Circle**: ![[Pasted image 20220605122038.png]] $$ \begin{align} &\ldots \\[7px] &f(z) = \large -\sum_{n=1}^{\infty} (z)^{-n} - 2\cdot\sum_{n=1}^{\infty} \kern3px \left(\frac{z}{2}\right)^{-n} \kern10px \normalsize \text{for}\ |z| > 2 \end{align} $$

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Original Files:

Laurent Series_220511_144338.pdf