Chapter5 - Chapter 5 Consequences of Cauchys Theorem If...

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Chapter 5 Consequences of Cauchy’s Theorem If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do. Richard Askey 5.1 Extensions of Cauchy’s Formula We now derive formulas for f 0 and f 00 which resemble Cauchy’s formula (Theorem 4.10). Theorem 5.1. Suppose f is holomorphic on the region G , w G , and γ is a positively oriented, simple, closed, smooth, G -contractible curve such that w is inside γ . Then f 0 ( w ) = 1 2 πi Z γ f ( z ) ( z - w ) 2 dz and f 00 ( w ) = 1 πi Z γ f ( z ) ( z - w ) 3 dz . This innocent-looking theorem has a very powerful consequence: just from knowing that f is holomorphic we know of the existence of f 00 , that is, f 0 is also holomorphic in G . Repeating this argument for f 0 , then for f 00 , f 000 , etc., gives the following statement, which has no analog whatsoever in the reals. Corollary 5.2. If f is differentiable in the region G then f is infinitely differentiable in G . Proof of Theorem 1.1. The idea of the proof is very similar to the proof of Cauchy’s integral formula (Theorem 4.10). We will study the following difference quotient, which we can rewrite as follows by Theorem 4.10. f ( w + Δ w ) - f ( w ) Δ w = 1 Δ w 1 2 πi Z γ f ( z ) z - ( w + Δ w ) dz - 1 2 πi Z γ f ( z ) z - w dz = 1 2 πi Z γ f ( z ) ( z - w - Δ w )( z - w ) dz . 53
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CHAPTER 5. CONSEQUENCES OF CAUCHY’S THEOREM 54 Hence we will have to show that the following expression gets arbitrarily small as Δ w 0: f ( w + Δ w ) - f ( w ) Δ w - 1 2 πi Z γ f ( z ) ( z - w ) 2 dz = 1 2 πi Z γ f ( z ) ( z - w - Δ w )( z - w ) - f ( z ) ( z - w ) 2 dz = Δ w 1 2 πi Z γ f ( z ) ( z - w - Δ w )( z - w ) 2 dz . This can be made arbitrarily small if we can show that the integral stays bounded as Δ w 0. In fact, by Proposition 4.4(d), it suffices to show that the integrand stays bounded as Δ w 0 (because γ and hence length( γ ) are fixed). Let M = max z γ | f ( z ) | and N = max z γ | z - w | . Since γ is a closed set, there is some positive δ so that the open disk of radius δ around w does not intersect γ ; that is, | z - w | ≥ δ for all z on γ . By the reverse triangle inequality we have for all z γ f ( z ) ( z - w - Δ w )( z - w ) 2 | f ( z ) | ( | z - w | - | Δ w | ) | z - w | 2 M ( δ - | Δ w | ) N 2 , which certainly stays bounded as Δ w 0. The proof of the formula for f 00 is very similar and will be left for the exercises (see Exercise 2). Remarks. 1. Theorem 1.1 suggests that there are similar formulas for the higher derivatives of f . This is in fact true, and theoretically one could obtain them one by one with the methods of the proof of Theorem 1.1. However, once we start studying power series for holomorphic functions, we will obtain such a result much more easily; so we save the derivation of formulas for higher derivatives of f for later (see Corollary 8.6). 2. Theorem 1.1 can also be used to compute certain integrals. We give some examples of this application next.
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