By A. Brigalia, C. Cilberto

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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