By Murali Rao
This can be a rigorous creation to the speculation of complicated services of 1 complicated variable. The authors have made an attempt to provide a few of the deeper and extra attention-grabbing effects, for instance, Picard's theorems, Riemann mapping theorem, Runge's theorem within the first few chapters. even if, the very easy thought is however given a radical therapy in order that readers should not think misplaced. After the 1st 5 chapters, the order might be tailored to fit the path. every one bankruptcy finishes with workouts.
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Additional info for Complex Analysis: An Invitation: A Concise Introduction to Complex Function Theory
Example text
We now turn to the question of regularity of logarithms and arguments. The following theorem tells in particular that any branch of log z automatically is holomorphic. But more is true: In Chapter IV we shall present a generalization in which we replace exp by any non-constant holomorphic function (Proposition IV. 18). 25 Chapter 3. The Exponential Function, the Logarithm and the Winding Number Theorem S. Let f be holomorphic on an open subset Q of the complex plane, and let F be a continuous logarithm off in Then F is also holomorphic in fl, and F' = 1,/f.
The example shows in particular that the index of a but also on how the moving of the curve curve depends not just on the image point 7(t) traces the image 7. A trivial example: A constant curve has index 0. We have in the above examples pedantically computed the index analytically, although the results are geometrically evident. From now on we will not bother to do so. If a closed curve has so simple a look that it is geometrically obvious what its index is, then we will not elaborate on the matter.
N(i) is a a constant. Taking the condition a continuous function [0,11 , hence we see that the constant is 0. We have thus shown that so the collection continuous, integer valued = = into account Then N(y) we have and agree on their common domain of definition, and z E X patches together to a continuous function 9: X x [0, 1] —' which is the desired logarithm of C, F. 2nd step. ,n-—1 in the following way: 29 log z E X. By chapter 3. e. C such that exp(Ao) = F on N x [O,t1] Noting that exp(Ao(x,O)) = F(x,O) = exp(44x)) so that Ao(x,O)) = 1 we define 9o(x,t) := Ao(x,t) — Ao(z,i) + çt(z) Then : N x [0,t1) —.