Week 8 Worksheet

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Laurent Series

1. For each of the following functions determine its Laurent series on the given annulus.
   1. ${\displaystyle \frac{1}{z-3}}$ on $\mathsf{Ann}(0,3,\mathrm{\infty })$
   2. ${\displaystyle \frac{1}{z(1-z)}}$ on $\mathsf{Ann}(0,0,1)$
   3. $\cos (1/z)\phantom{{\displaystyle \frac{1}{z}}}$ on $\mathsf{Ann}(0,0,\mathrm{\infty })$
2. Define $f:\mathbb{C}\setminus \{-1,3\}$ by $f(z)={\displaystyle \frac{1}{z+1}}+{\displaystyle \frac{1}{z-3}}$. Calculate its Laurent series on $\mathsf{Ann}(0,0,1)$, $\mathsf{Ann}(0,1,3)$ and $\mathsf{Ann}(0,3,\mathrm{\infty })$.
3. Define $f:\mathbb{C}\setminus \{0,1\}\to \mathbb{C}$ by $f(z)={\displaystyle \frac{1}{z^2(z-1)}}$.
   1. Calculate its Laurent series on $\mathsf{Ann}(0,0,1)$.
   2. Calculate its Laurent series on $\mathsf{Ann}(1,0,1)$.
4. Define $f:\mathbb{C}\setminus \{1\}\to \mathbb{C}$ by $f(z)={\displaystyle \frac{1}{(z-1{)}^2}}$.
   1. Calculate its Laurent series on $\mathsf{Ann}(1,0,\mathrm{\infty })$.
   2. Calculate its Laurent series on $\mathsf{Ann}(0,0,1)$.
   3. Calculate its Laurent series on $\mathsf{Ann}(0,1,\mathrm{\infty })$.

Isolated Singularities

5. For each of the following functions determine all the poles and their orders.
   1. $f(z)={\displaystyle \frac{1}{z^2+1}}$
   2. $f(z)={\displaystyle \frac{1}{z^4+16}}$
   3. $f(z)={\displaystyle \frac{1}{z^4+2z^2+1}}$
   4. $f(z)={\displaystyle \frac{1}{z^2+z-1}}$
6. What type of singularity does each of the following functions have at the origin?
   1. $f(z)=\sin (1/z)$
   2. $f(z)={\displaystyle \frac{1}{z^3}}(\sin (z){)}^2$
   3. $f(z)={\displaystyle \frac{\cos (z)-1}{z^2}}$
7. Fix a domain $D$ and $b\in D$. Suppose $f$ is holomorphic and bounded on $D\setminus \{b\}$. Prove that $f$ has a removable singularity at $b$.