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Chapter8

# Chapter8 - Chapter 8 Taylor and Laurent Series We think in...

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Chapter 8 Taylor and Laurent Series We think in generalities, but we live in details. A. N. Whitehead 8.1 Power Series and Holomorphic Functions We will see in this section that power series and holomorphic functions are intimately related. In fact, the two cornerstone theorems of this section are that any power series represents a holomorphic function, and conversely, any holomorphic function can be represented by a power series. We begin by showing a power series represents a holomorphic function, and consider some of the consequences of this: Theorem 8.1. Suppose f ( z ) = k 0 c k ( z - z 0 ) k has radius of convergence R . Then f is holo- morphic in { z C : | z - z 0 | < R } . Proof. Given any closed curve γ ⊂ { z C : | z - z 0 | < R } , we have by Corollary 7.27 Z γ X k 0 c k ( z - z 0 ) k dz = 0 . On the other hand, Corollary 7.26 says that f is continuous. Now apply Morera’s theorem (Corol- lary 5.17). A special case of the last result concerns power series with infinite radius of convergence: those represent entire functions. Now that we know that power series are holomorphic (i.e., differentiable) on their regions of convergence we can ask how to find their derivatives. The next result says that we can simply differentiate the series “term by term.” Theorem 8.2. Suppose f ( z ) = k 0 c k ( z - z 0 ) k has radius of convergence R . Then f 0 ( z ) = X k 1 k c k ( z - z 0 ) k - 1 , and the radius of convergence of this power series is also R . 82

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CHAPTER 8. TAYLOR AND LAURENT SERIES 83 Proof. Let f ( z ) = k 0 c k ( z - z 0 ) k . Since we know that f is holomorphic in its region of conver- gence we can use Theorem 5.1. Let γ be any simple closed curve in { z C : | z - z 0 | < R } . Note that the power series of f converges uniformly on γ , so that we are free to interchange integral and infinite sum. And then we use Theorem 5.1 again , but applied to the function ( z - z 0 ) k . Here are the details: f 0 ( z ) = 1 2 πi Z γ f ( w ) ( w - z ) 2 dw = 1 2 πi Z γ k 0 c k ( w - z 0 ) k ( w - z ) 2 dw = X k 0 c k · 1 2 πi Z γ ( w - z 0 ) k ( w - z ) 2 dw = X k 0 c k · d dw ( w - z 0 ) k w = z = X k 0 k c k ( z - z 0 ) k - 1 . The last statement of the theorem is easy to show: the radius of convergence R of f 0 ( z ) is at least R (since we have shown that the series converges whenever | z - z 0 | < R ), and it cannot be larger than R by comparison to the series for f ( z ), since the coefficients for ( z - z 0 ) f 0 ( z ) are bigger than the corresponding ones for f ( z ). Naturally, the last theorem can be repeatedly applied to f 0 , then to f 00 , and so on. The various derivatives of a power series can also be seen as ingredients of the series itself. This is the statement of the following Taylor 1 series expansion . Corollary 8.3. Suppose f ( z ) = k 0 c k ( z - z 0 ) k has a positive radius of convergence. Then c k = f ( k ) ( z 0 ) k ! . Proof. For starters, f ( z 0 ) = c 0 . Theorem 8.2 gives f 0 ( z 0 ) = c 1 . Applying the same theorem to f 0 gives f 00 ( z ) = X k 2 k ( k - 1) c k ( z - z 0 ) k - 2 and f 00 ( z 0 ) = 2 c 2 . We can play the same game for f 000 ( z 0 ), f 0000 ( z 0 ), etc.
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