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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The booklet includes thirteen articles, a few of that are survey articles and others learn papers. Written through eminent mathematicians, those articles have been awarded on the overseas Workshop on complicated research and Its functions held at Walchand collage of Engineering, Sangli. all of the contributing authors are actively engaged in study fields concerning the subject of the publication.

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

**Example text**

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).

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

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