Complex Analysis/Laurent Expansion

Every holomorphic function on ${\displaystyle A_{r_{1},r_{2}}}$ has a Laurent expansion around ${\displaystyle z_{0}}$, and the coefficients ${\displaystyle a_{n}}$ in the expansion are given by:

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

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

The coefficients are uniquely determined by:

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

Uniqueness follows from the Identity Theorem for Laurent Series. To prove existence, choose a radius ${\displaystyle r}$ such that ${\displaystyle r_{1}<r<r_{2}}$ and choose ${\displaystyle R_{1},R_{2}}$ so that ${\displaystyle r_{1}<R_{1}<r<R_{2}<r_{2}}$. Let ${\displaystyle z\in A_{R_{1},R_{2}}}$ be arbitrary. "Cut" the annular region ${\displaystyle A_{R_{1},R_{2}}}$ at two points using radii ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ such that the cycle ${\displaystyle \partial K_{R_{2}}-\partial K_{R_{1}}}$ is represented as the sum of two closed curves ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ in ${\displaystyle A}$ that are null-homotopic. Choose ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ so that ${\displaystyle z}$ is encircled 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 encircle ${\displaystyle z}$. Thus, because ${\displaystyle C_{1}+C_{2}=\partial K_{R_{2}}-\partial K_{R_{1}}}$, we have:

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

For ${\displaystyle |w-z_{0}|=R_{2}}$, we have:

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

The series converges absolutely because ${\displaystyle |z-z_{0}|<|w-z_{0}|}$, and we obtain:

${\displaystyle {\begin{array}{rl}{\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{2}}{\frac {f(w)}{w-z}}\,dw&={\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{2}}{\frac {1}{w-z_{0}}}\cdot {\frac {f(w)(z-z_{0})^{n}}{(w-z_{0})^{n}}}\,dw\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }\int _{|w-z_{0}|=R_{2}}{\frac {f(w)}{(w-z_{0})^{n+1}}}\,dw\cdot (z-z_{0})^{n}\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }\int _{|w-z_{0}|=r}{\frac {f(w)}{(w-z_{0})^{n+1}}}\,dw\cdot (z-z_{0})^{n}\end{array}}}$

Now, consider the integral over the inner circle, which is analogous to the above for ${\displaystyle |w-z_{0}|=R_{1}}$:

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

Thus, due to ${\displaystyle R_{1}=|w-z_{0}|<|z-z_{0}|}$, the series converges, and we obtain:

${\displaystyle {\begin{array}{rl}-{\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{1}}{\frac {f(w)}{w-z}}\,dw&={\frac {1}{2\pi i}}\int _{|w-z_{0}|=R_{1}}{\frac {-1}{z-z_{0}}}\cdot {\frac {f(w)(w-z_{0})^{n}}{(z-z_{0})^{n}}}\,dw\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }\int _{|w-z_{0}|=R_{1}}{\frac {f(w)}{(w-z_{0})^{-n}}}\,dw\cdot (z-z_{0})^{-n-1}\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }\int _{|w-z_{0}|=r}{\frac {f(w)}{(w-z_{0})^{-n}}}\,dw\cdot (z-z_{0})^{-n-1}\end{array}}}$

Thus, it follows that for ${\displaystyle z\in A_{R_{1},R_{2}}}$:

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

which proves the existence of the claimed Laurent expansion.

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: 11/26/2024