1. **Problem statement:** Find the poles or removable singularities of the functions and determine their orders. 2. **Recall:** A pole of order $m$ at $z=z_0$ means the function behaves like $\frac{1}{(z-z_0)^m}$ near $z_0$. A removable singularity means the function can be redefined at $z_0$ to be analytic. --- **(a) $f(z) = (z^2 + 1)^{-3} (z - 1)^{-4}$** - Poles occur where the denominator is zero. - $z^2 + 1 = 0 \Rightarrow z = i, -i$ are poles of order 3 each. - $z - 1 = 0 \Rightarrow z=1$ is a pole of order 4. **Answer:** Poles at $z=i$ and $z=-i$ of order 3, and at $z=1$ of order 4. --- **(b) $f(z) = z \cot(z)$** - Recall $\cot(z) = \frac{\cos z}{\sin z}$. - $\cot(z)$ has simple poles at $z = k\pi$, $k \in \mathbb{Z}$. - At $z=0$, $\cot(z)$ has a simple pole, but multiplied by $z$, the singularity at 0 is removable since $\lim_{z \to 0} z \cot(z) = 1$. - Other poles at $z = k\pi$, $k \neq 0$ remain simple poles. **Answer:** Removable singularity at $z=0$, simple poles at $z = k\pi$, $k \neq 0$. --- **(c) $f(z) = z^{-5} \sin(z)$** - $z^{-5}$ has a pole of order 5 at $z=0$. - $\sin(z)$ has a zero of order 1 at $z=0$. - Near $z=0$, $f(z) \sim \frac{z}{z^5} = z^{-4}$, so a pole of order 4 at $z=0$. **Answer:** Pole of order 4 at $z=0$. --- **(d) $f(z) = \frac{1}{1 - \exp(z)}$** - Singularities where denominator zero: $1 - e^z = 0 \Rightarrow e^z = 1 \Rightarrow z = 2k\pi i$, $k \in \mathbb{Z}$. - Near $z=0$, $1 - e^z \approx -z$, so simple pole at $z=0$. - Similarly, at each $z=2k\pi i$, simple poles. **Answer:** Simple poles at $z = 2k\pi i$, $k \in \mathbb{Z}$. --- **(e) $f(z) = \frac{z}{1 - \exp(z)}$** - Same denominator zeros as (d), poles at $z=2k\pi i$. - At $z=0$, numerator zero cancels denominator zero: - $z \sim 0$, numerator $\sim z$, denominator $\sim z$, so limit finite. - So removable singularity at $z=0$. - At $z=2k\pi i$, $k \neq 0$, simple poles remain. **Answer:** Removable singularity at $z=0$, simple poles at $z=2k\pi i$, $k \neq 0$. --- **Summary:** - (a) Poles: $z=\pm i$ order 3, $z=1$ order 4 - (b) Removable at 0, simple poles at $k\pi$, $k \neq 0$ - (c) Pole order 4 at 0 - (d) Simple poles at $2k\pi i$ - (e) Removable at 0, simple poles at $2k\pi i$, $k \neq 0$