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Analysis-Sheet1

# Analysis-Sheet1 - Analysis Sheet 1 Most of the questions on...

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Analysis Sheet 1 Most of the questions on this sheet are on material introduced in the Moderations syllabus: complex numbers, equations of circles and lines in the complex plane, M¨ obius transformations, uniform convergence of series of functions and convergence of power series. It is recommended that you first review your Moderations lecture notes on this material and then read Priestley, Introduction to Complex Analysis , Second Edition, OUP. Chapters 1,2,3 pp 30-32,14 pp 168- 173 (A primer on uniform convergence). In particular, note the definitions of open and closed subsets of the real line and the real plane. Problems 1. Let z = x + i y with x, y R , | z | := x 2 + y 2 and z := x - i y . Let a C be such that | a | = 1. (a) Show that z = z - 1 ⇔ | z | = 1. (b) Show that | z - a | 2 = | z | 2 - az - a z + | a | 2 (c) Show that if | z | = 1 then | z - a | | 1 - az | = 1. (d) Calculate the inverse of the M¨ obius transformation z - a 1 - az . (e) Show that if | z - a | | 1 - az | = 1 then | z | = 1. (f) Let D denote the unit disc, D = { z C : | z | < 1 } . Let a, b D . Find M¨ obius tranformations mapping D

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