By Murali Rao

ISBN-10: 9810203756

ISBN-13: 9789810203757

ISBN-10: 9810203764

ISBN-13: 9789810203764

It is a rigorous creation to the idea of advanced capabilities of 1 complicated variable. The authors have made an attempt to give the various deeper and extra attention-grabbing effects, for instance, Picard's theorems, Riemann mapping theorem, Runge's theorem within the first few chapters. despite the fact that, the very uncomplicated conception is however given a radical therapy in order that readers should not consider misplaced. After the 1st 5 chapters, the order should be tailored to fit the direction. every one bankruptcy finishes with routines.

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Additional info for Complex Analysis: An Invitation: A Concise Introduction to Complex Function Theory

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Cos S + j Sin S := exp (ix) maps the real line onto the From this we see in particular that unit circle {zECIIzI=1} most important property of the exponential function is that it couples the additive and multiplicative structures of C as follows: The Theorem I (The Addition Theorem). b E C Proof. When we differentiate g(z) := exp(z)exp(a + b — z) we get g'(z) = so g is constant. The contents of the theorem are g(0) = g(a). 0 0 Note that exp never vanishes (by The Addition Theorem). The power series for exp has real coefficients.

Let be a topological space. (a) Two closed curves 70,72 [a, b] —. ii are said to be homotopic in Q, jf there exists a continuous map (a homotopy) r: [a, b] x [0, 1] —. homotopic, 4f it is homotopic to a constant curve. The geometrical contents of (a) are that the curve 70 continuously is deformed into 71 as the parameters increases from 0 to 1. The curves t 1'(t, s) are closed curves for each fIxed 3 E (0,1] and they form intermediate steps in the deformation of 70 into 71• Theorem 17 (Properties of the number).

Either + 1 or —1. 0 Proposition 20. Let 1: —' C\ {O) be a holomorphic function defined on an open subset Q of the complex plane. Then any continuous square root off is holomorphic on ft Remark. It is actually superfluous to assume that f never vanishes. 18. Proof: Let r be any continuous square root of f in Let B be any open disc in Q. Now f has a holomorphic logarithm F in B (Theorems 10 and 5), and so exp(F/2) is a holomorphic square root of f in B. But rIB is by Proposition 19 either exp (F/2) or exp(—F/2).

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Complex Analysis: An Invitation: A Concise Introduction to Complex Function Theory by Murali Rao

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