Further Analysis | Part IB, 2001

Let Δ={z:0<z<r}\Delta^{*}=\{z: 0<|z|<r\} be a punctured disc, and ff an analytic function on Δ\Delta^{*}. What does it mean to say that ff 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 ff at 0 .

Suppose now that the origin is an essential singularity for ff. Given any wCw \in \mathbb{C}, show that there exists a sequence (zn)\left(z_{n}\right) of points in Δ\Delta^{*} such that zn0z_{n} \rightarrow 0 and f(zn)wf\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 ff is analytic and injective on Δ\Delta^{*}, then the origin cannot be an essential singularity. By applying this to the function g(1/z)g(1 / z), or otherwise, deduce that if gg is an injective analytic function on C\mathbb{C}, then gg is linear of the form az+ba z+b, for some non-zero complex number aa. [Here, you may assume that gg injective implies that its derivative gg^{\prime} is nowhere vanishing.]

Part IB

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