Complex Analysis/development in Laurent series

Let ${\displaystyle G\subseteq \mathbb {C} }$ be a domain, ${\displaystyle z_{0}\in G}$, and ${\displaystyle f\colon G\setminus {z_{0}}\to \mathbb {C} }$ a holomorphic function. A Laurent expansion of ${\displaystyle f}$ around ${\displaystyle z_{0}}$ is a representation of ${\displaystyle f}$ as a Laurent Series:

${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n}}$

with ${\displaystyle a_{n}\in \mathbb {C} }$, which converges on a punctured disk (i.e., excluding the center ${\displaystyle z_{0}}$) around ${\displaystyle z_{0}}$.

A more general case than the above is the following: Let ${\displaystyle 0\leq r_{1}<r_{2}}$ be two radii (the expansion around a point corresponds to ${\displaystyle r_{1}=0}$), and let ${\displaystyle A_{r_{1},r_{2}}:=\{z\in \mathbb {C} :r_{1}<|z-z_{0}|<r_{2}\}}$ be an annulus around ${\displaystyle z_{0}}$. If ${\displaystyle f\colon A_{r_{1},r_{2}}\to \mathbb {C} }$ is a holomorphic function, then the Laurent Series

${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n}}$

with ${\displaystyle a_{n}\in \mathbb {C} }$ is a Laurent expansion of ${\displaystyle f}$ on ${\displaystyle A_{r_{1},r_{2}}}$, provided the series converges for all ${\displaystyle z\in A_{r_{1},r_{2}}}$.

Every holomorphic function on ${\displaystyle A_{r_{1},r_{2}}}$ possesses a Laurent expansion around ${\displaystyle z_{0}}$. Coefficients ${\displaystyle a_{n}}$ with ${\displaystyle n\in \mathbb {Z} }$ exist and can be calculated with

${\displaystyle a_{n}={\frac {1}{2\pi i}}\int _{|z-z_{0}|=r}{\frac {f(z)}{(z-z_{0})^{n+1}}}\,dz}$

for any radius ${\displaystyle r}$ with ${\displaystyle r_{1}<r<r_{2}}$.

The coefficients are uniquely determined by:

${\displaystyle a_{n}={\frac {1}{2\pi i}}\int _{|z-z_{0}|=r}{\frac {f(z)}{(z-z_{0})^{n+1}}}\,dz}$

The uniqueness follows from the identity theorem for Laurent Series. For existence, choose ${\displaystyle r}$ with ${\displaystyle r_{1}<r<r_{2}}$ and ${\displaystyle R_{1},R_{2}}$ such that ${\displaystyle r_{1}<R_{1}<r<R_{2}<r_{2}}$. Let ${\displaystyle z\in A_{R_{1},R_{2}}}$ be arbitrary. "Cut" the annulus ${\displaystyle A_{R_{1},R_{2}}}$ at two points using radii ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ such that the cycle ${\displaystyle \partial K_{R_{2}}-\partial K_{R_{1}}}$ can be expressed as the sum of two closed, null-homotopic curves ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$. Choose ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ such that ${\displaystyle z}$ is enclosed by ${\displaystyle C_{1}}$. By the Cauchy Integral Theorem, we have:

${\displaystyle f(z)={\frac {1}{2\pi i}}\int _{C_{1}}{\frac {f(w)}{w-z}}\,dw}$

and

${\displaystyle 0={\frac {1}{2\pi i}}\int _{C_{2}}{\frac {f(w)}{w-z}}\,dw}$

since ${\displaystyle C_{2}}$ does not enclose ${\displaystyle z}$. Hence, due to ${\displaystyle C_{1}+C_{2}=\partial K_{R_{2}}-\partial K_{R_{1}}}$, we obtain:

${\displaystyle f(z)={\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{2}}{\frac {f(w)}{w-z}}\,dw-{\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{1}}{\frac {f(w)}{w-z}}\,dw}$

We now expand ${\displaystyle {\frac {1}{w-z}}}$ for ${\displaystyle |w-z_{0}|=R_{2}}$ using:

${\displaystyle {\begin{array}{rl}{\frac {1}{w-z}}&={\frac {1}{(w-z_{0})-(z-z_{0})}}\\&={\frac {1}{w-z_{0}}}\cdot {\frac {1}{1-{\frac {z-z_{0}}{w-z_{0}}}}}\\&={\frac {1}{w-z_{0}}}\sum _{n=0}^{\infty }{\frac {(z-z_{0})^{n}}{(w-z_{0})^{n}}}\end{array}}}$

This series converges absolutely for ${\displaystyle |z-z_{0}|<|w-z_{0}|}$, yielding:

${\displaystyle {\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{2}}{\frac {f(w)}{w-z}}\,dw=\sum _{n=0}^{\infty }\left({\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{2}}{\frac {f(w)}{(w-z_{0})^{n+1}}}\,dw\right)(z-z_{0})^{n}.}$

Similarly, for the inner circle ${\displaystyle |w-z_{0}|=R_{1}}$, we expand and calculate analogously. The final result shows that for ${\displaystyle z\in A_{R_{1},R_{2}}}$, the Laurent series converges, proving the existence of the Laurent expansion.

• Example Computation with Laurent Series
• Laurent Series
• Holomorphic function

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This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/Laurententwicklung - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung

• Date: 1/1/2025