4.II.13B

Let $\Delta^{*}=\{z: 0<|z|<r\}$ be a punctured disc, and $f$ an analytic function on $\Delta^{*}$. What does it mean to say that $f$ has the origin as (i) a removable singularity, (ii) a pole, and (iii) an essential singularity? State criteria for (i), (ii), (iii) to occur, in terms of the Laurent series for $f$ at 0 .

Suppose now that the origin is an essential singularity for $f$. Given any $w \in \mathbb{C}$, show that there exists a sequence $\left(z_{n}\right)$ of points in $\Delta^{*}$ such that $z_{n} \rightarrow 0$ and $f\left(z_{n}\right) \rightarrow w$. [You may assume the fact that an isolated singularity is removable if the function is bounded in some open neighbourhood of the singularity.]

State the Open Mapping Theorem. Prove that if $f$ is analytic and injective on $\Delta^{*}$, then the origin cannot be an essential singularity. By applying this to the function $g(1 / z)$, or otherwise, deduce that if $g$ is an injective analytic function on $\mathbb{C}$, then $g$ is linear of the form $a z+b$, for some non-zero complex number $a$. [Here, you may assume that $g$ injective implies that its derivative $g^{\prime}$ is nowhere vanishing.]

Part IB

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