Let be a punctured disc, and an analytic function on . What does it mean to say that 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 at 0 .
Suppose now that the origin is an essential singularity for . Given any , show that there exists a sequence of points in such that and . [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 is analytic and injective on , then the origin cannot be an essential singularity. By applying this to the function , or otherwise, deduce that if is an injective analytic function on , then is linear of the form , for some non-zero complex number . [Here, you may assume that injective implies that its derivative is nowhere vanishing.]