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MA 530 Complex Analysis: Review MA 530 Complex Analysis: Review Yingwei Wang ∗ Department of Mathematics, Purdue University, West Lafayette, IN, USA Contents 1 How to prove analytic (holomorphic, complex differentiable)? 3 1.1 Definition (Difference Quotient) . .. 3 1.2 Cauchy-RiemannEquations . 3 1.3 Integration along closed curves equals zeros . ....... 3 1.4 Powerseries...................................... 4 1.4.1 From holomorphic to power series . 4 1.4.2 From power series to holomorphic . 5 1.4.3 Differentiate.................................. 5 2 Integration along curves 6 2.1 Preliminaries ..................................... 6 2.2 FundamentalTheoremofCalculus . .. 7 2.3 CauchyTheorem ................................... 8 2.4 CauchyIntegralFormula . .. .. .. 10 2.5 Liouville’s theorem . 13 2.6 FundamentalTheoremofAlgebra . ... 13 3 Useful properties of holomorphic functions 13 3.1 Isolatedzeros ..................................... 13 3.2 Identitytheorem ................................... 14 3.3 Averagingproperty ................................. 15 3.4 Maximumprinciple.................................. 15 3.5 Rouche’sTheorem .................................. 16 3.6 ArgumentPrinciple.................................. 17 ∗This is based on the lecture notes of Prof Buzzard (Fall 2010) and Prof Bell (Spring 2011). 1 Yingwei Wang Complex Analysis 4 Harmonic functions 18 4.1 Definition ....................................... 18 4.2 Harmonicconjugate................................. 19 4.3 Poissonintegralformula .............................. 19 4.4 Mean Value Property and Maximum Principle . 20 4.5 Zerosofharmonicfunctions . .. .. .. 21 5 Isolated singularity 21 5.1 Definition ....................................... 21 5.1.1 Removable singularity . 21 5.1.2 Pole ...................................... 21 5.1.3 Essential singularity . 22 5.1.4 Fromtheviewofpowerseries . .. .. 22 5.2 Riemann removable singularities theorem . ... 23 5.2.1 Method I: integral formulas + estimates . 23 5.2.2 MethodII:constructfunction . 24 5.3 PoleTheorem..................................... 24 5.4 Casorati-WeierstrassTheorem . .... 25 5.5 Meromorphicfunctions ............................... 25 5.5.1 Polynomials.................................. 25 5.5.2 Rationalfunctions .............................. 28 6 Uniform convergence 29 6.1 Hurwitz’Theorem .................................. 29 6.2 MontelTheorem ................................... 30 6.2.1 NormalFamily ................................ 30 6.2.2 Uniformboundedness............................. 30 6.2.3 Equicontinuity ................................ 30 6.2.4 MontelTheorem ............................... 30 6.3 Openmappingtheorem................................ 31 7 Univalence 31 7.1 Localunivalence.................................... 31 7.2 Globalunivalence................................... 32 7.3 Limits univalence . 33 8 Conformal mappings 33 8.1 SchwartzLemma ................................... 33 8.2 Automorphismofthedisc .............................. 34 8.3 From upper half plan to the unit disc . 34 9 Roots of functions 35 2 Yingwei Wang Complex Analysis 1 How to prove analytic (holomorphic, complex differ- entiable)? Note: let Ω be an open set in C and f be a complex-valued function on Ω. 1.1 Definition (Difference Quotient) Definition 1.1. Say f is complex differentiable (holomorphic) at z0 ∈ Ω, if f(z + h) − f(z ) DQ = 0 0 h ′ converges to a limit when h → 0. Call the limit f (z0). If f is complex differentiable at all points in Ω, then call f holomorphic on Ω. Remark 1.1. It should be emphasized that in the above limit, h is a complex number that may approach 0 from any direction. Remark 1.2. A holomorphic function will actually be infinitely many times complex differen- tiable, that is, the existence of the first derivative will guarantee the existence of derivatives of any order. For proof, see Section 1.4.3 and Theorem 2.4. For more application of this method, see Lemma 2.3, Theorem 2.2, Theorem 2.4. 1.2 Cauchy-Riemann Equations Theorem 1.1 (C-R ⇒ analytic). Suppose u, v ∈ C1(Ω) and satisfy the C-R equations u = v , x y (1.1) u = −v , y x then f(x, y)= u(xy)+ iv(x, y) is analytic on Ω. Remark 1.3. Actually, by Looman-Monchoff Theorem, we just need that u, v are continuous and all their first partial derivatives exit (may be not continuous) and satisfy the C-R equations, then f = u + iv is analytic. 1.3 Integration along closed curves equals zeros Theorem 1.2 (Morera). Suppose f is continuous on an open set Ω and for any triangle T contained in Ω, f(z)dz =0, ZT then f is holomorphic. 3 Yingwei Wang Complex Analysis 1.4 Power series 1.4.1 From holomorphic to power series Lemma 1.1. Let N SN (z)=1+ z + ··· + z , (1.2) zN+1 E (z)= , (1.3) N 1 − z then 1 = S (z)+ E (z). 1 − z N N ρN+1 Furthermore, if |z| <ρ< 1, then |EN (z)| ≤ 1−ρ . Theorem 1.3 (holomorphic ⇒ analytic). Suppose f is holomorphic on Ω and Dr(z0) ⊂ Ω. Then f has a power series expansion in Dr(z0), ∞ n f(z)= an(z − z0) , ∀z ∈ Dr(z0), n=0 X (n) f (z0) 1 f(w) with a = = n+1 dw. n n! 2πi Cr(z0) (w−z0) Proof. Without loss ofR generality, we can take z0 = 0 and ρ<r. By Cauchy formula (Thm 2.3) and Lemma 1.1, 1 f(w) f(z) = dw 2πi w − z ZCr 1 f(w) 1 = dw 2πi w 1 − z ZCr w 1 f(w) 1 f(w) = S (z/w) dw + E (z/w) dw 2πi w N 2πi w N ZCr ZCr N 1 f(w) = dw zn + ε (z), 2πi wn+1 N n=0 Cr X Z where 1 sup |f| (ρ/r)N+1 |ε (z)| ≤ Cr (2πr) → 0. N 2π r 1 − ρ/r as N →∞. Besides, 1 f(w) f (n)(0) a = dw = . n 2πi wn+1 n! ZCr 4 Yingwei Wang Complex Analysis 1.4.2 From power series to holomorphic Definition 1.2. Say {fn} converges uniformly on compact subsets of Ω to f, if for any compact subsect K ⊂ Ω, and ∀ε> 0, there is an N such that |fn(z) − f(z)| <ε, ∀z ∈ K, n > N. Remark 1.4. Power series converge uniformly on compact subsects inside the circle. Theorem 1.4 (uniformly limit ⇒ analytic). Suppose {fn} analytic on Ω and converges uni- formly on compact subsets of Ω to f. Then f is analytic. Consequently, power series converge to analytic functions. Proof. Let D be any disc whose closure is contained in Ω and T be any triangle in that disc. Then, since each fn is holomorphic, Goursat’s theorem (Lemma 2.4) implies fn(z) dz =0, ∀n. ZT By assumption, fn → f uniformly on D¯, so f is continuous and fn(z)dz → f(z) dz. ZT ZT As a result, we find T f(z)dz = 0 for ∀T ⊂ D. By Morera theorem (Thm 1.2), we conclude that f is holomorphic in D. Since this conclusion is true for every D whose closure is contained R in Ω, we find that f is holomorphic in all of Ω. 1.4.3 Differentiate Theorem 1.5 (Can differentiate power series term by term). Suppose ∞ n f(z)= an(z − z0) n=0 X with radius of convergence R. Then power series for f ′ has the same radius of convergence and ∞ ′ n−1 f (z)= nan(z − z0) . n=0 X Theorem 1.6 (Derivative convergence). Suppose {fn} analytic on Ω and converges uniformly (k) on compact subsets of Ω to f. Then {fn } converges uniformly on compact subsets of Ω to f (k). 5 Yingwei Wang Complex Analysis 2 Integration along curves 2.1 Preliminaries Definition 2.1. Given a curve γ ∈ C with para z(t):[a, b] → C. Suppose f : γ → C is continuous. Then b N−1 ′ f(z)dz = f(z(t))z (t)dt = lim f(zj)(zj+1 − zj), |∆z|→0 γ a j=0 Z Z X where ∆zj = zj+1 − zj, |∆z| = max |∆zj|. j Lemma 2.1 (Basic estimate). f(z)dz ≤ length (γ) sup |f(z)| Zγ z∈γ b ′ where length (γ)= a |z (t)| d t. Lemma 2.2 (ReverseR orientation). If γ− is γ with reverse orientation, then f(z)dz = − f(z)dz. − Zγ Zγ Proof. Let γ be parameterized by z(t):[a, b] → C and γ− parameterized by z−(t):[a, b] → C. The relationship between z(t) and z−(t) is z−(t)= z(a + b − t). b−a Let i =0, ··· , n, ∆x = n , xi = a + i∆x, x0 = a, xn = b, and yi = a + b − xi, ∆y = ∆x, y0 = b, yn = a. By the definition of integration along curves (Def 2.1), b f(z)dz = f(z−(t))(z−)′(t)dt γ− a Z Z n − − ′ = lim f(z (xi))(z ) (xi)∆x n→∞ i=1 Xn ′ = lim f(z(yi))(−z (yi)) ∆y n→∞ i=1 bX = − f(z(t))z′(t)dt Za = − f(z). Zγ 6 Yingwei Wang Complex Analysis 2.2 Fundamental Theorem of Calculus Definition 2.2. A primitive for f on Ω is a function F that is holomorphic on Ω and such that F ′(z)= f(z), ∀z ∈ Ω. Theorem 2.1 (Fundamental Theorem of Calculus #2). Let Ω be an open set in C and γ be a curve (or path) in Ω that begins at w1 and ends at w2. Version I: If f is analytic on Ω, then ′ f (z)dz = f(w2) − f(w1). Zγ Version II: If f has a primitive in Ω, then f(z)dz = F (w2) − F (w1). Zγ Proof. Chain Rule + Fundamental Theorem of Calculus # 1. Corollary 2.1. If γ is a closed curve and f is holomorphic, then γ f(z)dz =0. Corollary 2.2. Any two primitives of f (if they exist) differ by aR constant. Proof. Suppose both F and G are the primitives of function f. According to Thm 2.1 (version II), we know that if γ is a curve in Ω from w0 to w, then f(z)dz = F (w) − F (w0)= G(w) − G(w0). Zγ Fix w0, then ∀w ∈ Ω, F (w) − G(w)= F (w0) − G(w0) = constant.
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