36 CHAPTER 2. COMPLEX-VALUED FUNCTIONS. In this case, we denote lim z z0 f(z) = α.

Transcription

1 36 CHAPTER 2. COMPLEX-VALUED FUNCTIONS In this case, we denote lim z z0 f(z) = α. A complex-valued function f defined in A is called continuous at z 0 A if lim z z 0 f(z) = f(z 0 ). Theorem Let A C be compact and f : A C be a continuous complex-valued function. Then (i) f is bounded and f(z) attains the maximum. 2 (ii) f is ormly continuous, i.e., ǫ > 0, δ > 0 such that f(z) f(w) < ǫ, z, w A with z w < δ. Proof: (i) a A, (a, δ a ) such that which implies f(z) f(a) < 1, z (a, δ a ) (since f is continuous, we take ǫ = 1) f(z) 1 + f(a), z (a, δ a ). = We have an open covering { (a, δ a )} a A of A. A is compact, a finite subcovering of A, say, = z A, (a 1, δ a1 )... (a n, δ an ) A. f(z) max 1 j n ( 1 + f(aj ) ). In fact, z A = z (a j, δ aj ) for some j so that f(z) 1 + f(a j ). Hence f is bounded. Taking a sequence of points z n A such that f(z n ) max A f. Assume z n a. Since A is compact (i.e., closed and bounded), it infers that a A and f(a) = max A f. (ii) To show: ǫ > 0, δ > 0 such that f(z) f(w) < ǫ, z w < δ, z, w A. 2 Since we cannot compare big or less for complex numbers, we cannot talk about the maximum for the complex-valued f(z) but it makes sense for the real-valued function f(z). For the boundedness, the proof here is similar to the one in the example beneath the Heine-Borel Theorem

2 2.1. COMPLEX-VALUED FUNCTIONS 37 Since f is continuous, a A, δ a > 0 such that f(z) f(a) < ǫ 2, z (a, 2δ a). (2.1) = We have an open covering { (a, δ a )} a A of A. = By the compactness, a finite subcovering: (a 1, δ a1 )... (a n, δ an ) A. = For the given ǫ, take δ := min{δ a1,..., δ an }. = z, w A with z w < δ, we must habe z (a j, δ aj ) for some j so that w (a j, 2δ aj ). = We have f(z) f(w) f(z) f(a j ) + f(w) f(a j ) < ǫ 2 + ǫ 2 because both f(z) f(a j ) and f(w) f(a j ) satisfy (2.1).

3 38 CHAPTER 2. COMPLEX-VALUED FUNCTIONS 2.2 Sequence of Complex-valued Functions Uniform Convergence Let A C and {f n (z)} n=1 be a sequence of complex-valued functions defined on a set A C. We say that {f n } is convergent if z A, lim n f n (z) exists. In this case, we denote f(z) = lim n f n (z), or f = lim f n, or f n f. We say that {f n } ormly converges to a function f(z) on A if ǫ > 0, n 0 N such that In this case, we denote f n f on A. f n (z) f(z) < ǫ, n n 0, z A. Theorem (i) (Cauchy criterion) f n f on A if and only if ǫ > 0, n 0 N such that f m (z) f n (z) < ǫ, m, n n 0, z A. (ii) If f n are continuous and f n f on A, then f is continuous on A. In 1821 A. L. Cauchy published a wrong theorem: The pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counterexamples in the context of Fourier series. Dirichlet then analyzed Cauchy s proof and found the mistake: the notion of pointwise convergence had to be replaced by orm convergence. The concept of orm convergence was probably first used by Christoph Gudermann, who was the teacher of Karl Weierstrass, in 1838 where he used the phrase convergence in a orm way. Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century. [Example] Let f n (x) = x n be continuous defined on [0, 1]. Then its limit function f = lim f n becomes { 0, if x [0, 1); f(x) := 1, if x = 1, which is not continuous. Ascoli-Arzela Theorem Recall Theorem 3.1, any bounded sequence has a convergent subsequence. In the theory of functions, in order to obtain continuous limit function, one naturally asks: For a sequence of bounded continuous functions, is there any ormly convergent subsequence (so that its limit function is also continuous)? This question was answered by the following theorem.

4 2.2. SEQUENCE OF COMPLEX-VALUED FUNCTIONS 39 Theorem (Ascoli-Arzela) Let A C be a compact subset, and f n functions defined on A that are ormly bounded and equicontinuous. Then {f n } has a ormly convergent subsequence. Here we explain the conditions used in the above theorem: (1) A sequence of complex-valued functions {f n } on A is called ormly bounded if M > 0 such that f n (z) M, z A, n. (2) A family 3 F of complex-valued functions on A is called equicontinuous if ǫ > 0, δ > 0 such that f(z) f(w) < ǫ, z, w A with z w < δ, f F. We skip its proof of the Ascoli-Arzela theorem which is standard in real analysis. Notice the set A in Ascoli-Arzela theorem is required to be compact (e.g. [a, b] [c, d]). However, in general case, we would consider A as an open subset in C. As a result, we should have the following notion and result (the proof is a simple consequence of the Ascoli-Arzela theorem). Let D C be an open subset, and {f n } defined on D. We say that {f n } converges ormly on compact subsets on D if for any compact subset A D, the restriction {f n } converges ormly. In application, we take a sequence of open subsets U 1 U 2... D such that k=1 U k = D. For U 1, {f n } has a ormly convergent subsequence {f (1) f (1) 1, f(1) 2,...,. For U 2, {f (1) n } has a ormly convergent subsequence {f (2) f (2) 1, f (2) 2, For U k, {f n (k 1) } has a ormly convergent subsequence {f (k) f (k) 1, f (k) 2,... We take a subsequence {f (k) k } k and then it ormly converges on every U j. Then {f (k) k } converges to a continuous function f which is continuous on every U k and hence continuous on D. 3 countable or uncountable

5 40 CHAPTER 2. COMPLEX-VALUED FUNCTIONS Theorem (i) If f n are continuous on an open subset D C, and {f n } n=1 converges ormly on compact subsets of D, then f := lim n f n is continuous on D. (ii) {f n } n=1 converges ormly on compact subsets of D if and only if a D, r > 0 such that {f n } n=1 converges ormly on (a; r) D. Theorem If {f n } n=1 on D C is ormly bounded and equicontinuous on compact subsets of D (i.e., they hold on any fixed relatively compact subset of D), then {f n } n=1 has a subsequence that converges ormly on compact subsets of D.