By Raghavan Narasimhan

This publication provides advanced research in a single variable within the context of recent arithmetic, with transparent connections to a number of complicated variables, de Rham thought, genuine research, and different branches of arithmetic. therefore, protecting areas are used explicitly in facing Cauchy's theorem, actual variable equipment are illustrated within the Loman-Menchoff theorem and within the corona theorem, and the algebraic constitution of the hoop of holomorphic capabilities is studied.

Using the original place of complicated research, a box drawing on many disciplines, the ebook additionally illustrates robust mathematical principles and instruments, and calls for minimum heritage fabric. Cohomological tools are brought, either in reference to the lifestyles of primitives and within the learn of meromorphic functionas on a compact Riemann floor. The evidence of Picard's theorem given the following illustrates the powerful regulations on holomorphic mappings imposed by means of curvature conditions.

New to this moment version, a set of over a hundred pages worthy of routines, difficulties, and examples provides scholars a chance to consolidate their command of advanced research and its kinfolk to different branches of arithmetic, together with complicated calculus, topology, and genuine applications.

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**Example text**

Thus a simple zero is a zero of order 1. Definition 8. Leta E C, r > 0 and let f be holomorphic on {z E qo < Iz - al < r}. We say that a is an essential singularity of f (or that f has an essential singularity at a) if, in the Laurent expansion 00 L n=-oo Cn (z - a)n of f at a, there are infinitely many n < 0 with Cn =1= o. This is equivalent to saying that f is not meromorphic on D(a, r). Theorem 4 (The Casorati-Weierstrass Theorem). Let a E C, r > 0, D* = {z E q Iz - al < r}. Let f E H(D*) and suppose that a is an essential singularity of f· Then f(D*) is dense in C.

Since Cn does not depend on p, we have 00 f(w) = LcnW n, 0< Iwl < r. (D) (Weierstrass' theorem). Clearly FID* = f. Another proof, not using the Laurent expansion, runs as follows. 40 Chapter I. Elementary Theory of Holomorphic Functions If f E H(D*) and zf(z) -+ 0 as z -+ 0, z i= 0, define a function g on D by g(z) = Z2 f (z), z i= 0, g(O) = O. Then g is (:-differentiable at 0 with g' (0) = 0; in fact I -(g(S) - g(O») s = l;f(S) -+ 0 as Since clearly g is (:-differentiable on D*, we have g since g(O) = g'(O) = O.

Let Q be a connected open set in C and let f, g {z E Qlf(z) = g(z)} E 'H(Q). If the set =f. 0 Chapter 1. Elementary Theory of Holomorphic Functions 24 has a point of accumulation in n, then f == g. This is simply Theorem 2 applied to f - g. We now pass on to the maximum principle and the open mapping theorem which are of fundamental importance. We begin with a very simple result which we shall use again later (in Chapter 4). Lemma 2. Let I be an open set in ]R and