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Lecture Notes for Math 185 (Follow Stein & Shakarchi's Book) Rui Wang

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Lecture Notes for Math 185 (Follow Stein & Shakarchi's Book) Rui Wang Lecture Notes for Math 185 (Follow Stein & Shakarchi’s book) Rui Wang Contents Chapter 1. Preliminaries to Complex Analysis 5 1. Complex numbers and the complex plane 5 1.1. Sequences and convergence in C 6 1.2. Open sets in C 6 2. Functions on C 7 2.1. Continuous functions 7 2.2. Holomorphic functions 8 2.3. Complex-valued functions as mappings 8 3. Power series 10 4. Integration along curves 13 Chapter 2. Cauchy’s Theorem and Its Applications 19 1. Cauchy’s theorem and Goursat’s theorem 19 2. Application of Cauchy’s theorem 22 3. Cauchy’s integral formulas 25 3.1. Cauchy’s integral formula and Cauchy’s inequality 25 3.2. Holomorphic functions are analytic 29 4. Important Applications of Cauchy’s thoerem 31 4.1. Morera’s theorem 31 4.2. Sequence of holomorphic functions 32 4.3. Holomorphic functions defined in terms of integrals 33 4.4. Schwarz reflection principle 34 4.5. Runge’s approximation theorem 35 Chapter 3. Meromorphic functions and the logarithm 39 1. Zeros and Poles 39 2. The residue formula 42 3. Singularities and meromorphic functions 44 4. The argument principle and applications 48 5. The complex logarithm 51 5.1. Homotopy of paths 51 5.2. The complex logarithm 53 Chapter 4. Conformal mappings 57 1. Conformal equivalence and examples 57 1.1. The unit disk and upper half plane 58 1.2. More examples 59 2. Automorphism of D and H 59 3 4 CONTENTS 2.1. Automorphism of D 59 2.2. Automorphism of H 62 3. Riemann mapping theorem 64 3.1. Normal family and Montel’s theorem 65 3.2. The proof of the Riemann mapping theorem 67 CHAPTER 1 Preliminaries to Complex Analysis 1. Complex numbers and the complex plane Set of complex numbers is the same as R2 and is denoted by C = ^z = x + iyðx; y Ë R`: In z = x + it, the x is called the real part and the y is called the imaginary part of z, and write as x = Re.z/; y = Im.z/: One can extend addition + and multiplication ⋅ over R to C (as real part) via z + w = .Re.z/ + Re.w// + i.Im.z/ + Im.w//; and z ⋅ w = .Re.z/Re.w/ * Im.z/Im.w// + i.Re.z/Im.w/ + Im.z/Re.w//: In particular, i2 := i ⋅ i = *1. It is easy to check .C; +; ⋅/ is a field as a field extension of R. It is much easier to express the multiplication using polar coordinates. For each r g 0; Ë R, rei := .r cos / + i.r sin /Ë C: Conversely, every nonzero complex number z Ë C<, there is a unique r > 0 and []Ë R_2 so that z = rei. Such .r; / is called a polar coordinate of z. i Assume zk = rke k , k = 1; 2. Then i.1+2/ z1 ⋅ z2 = .r1r2/e : The geometric meaning of r is the norm of z, i.e., the distance between the origin and z, as ù ðzð := Re.z/2 + Im.z/2: The geometric meaning of is the argument of z, i.e., the angle between the real axis and the radical line ⃗oz, as y tan = : x Another operation over C is the conjugation, which is the reflection about the real axis, i.e., ̄z = Re.z/* iIm.z/: It is easy to check 2 1 ̄z ðzð = z ̄z = ̄zz; = ; z ðzð2 and z + ̄z z * ̄z Re.z/ = ; Im.z/ = : 2 2i The norm ð ⋅ ð introduces a natural distance d.z; w/ = ðz * wð 5 6 1. PRELIMINARIES TO COMPLEX ANALYSIS over C. As a metric space, .C; d/ behaves exactly the same as R2. We now review some basic properties for this metric space. 1.1. Sequences and convergence in C. A sequence ^zn` is called convergent in C, if there is some w Ë C so that lim zn * w = 0: n→∞ ð ð Such w, if exists, must be unique, and we denote such convergence as limn→∞ zn = w, or zn → w; as n → ∞: LEMMA 1.1. limn→∞ zn = w if and only if lim Re.zn/ = Re.w/; lim Im.zn/ = Im.w/: n→∞ n→∞ PROOF. Details are left to you. A sequence ^zn` is called a Cauchy sequence, if for any > 0, there exists some N > 0 so that any m; n > N, ðzn * zmð < . Clearly, every convergent sequence is a Cauchy sequence. The converse is also true for C. In another word, we have THEOREM 1.2. C is complete. PROOF. ^zn` is a Cauchy sequence if and only if ^Re.zn/`; ^Im.zn/` are both Cauchy sequences. Then the completeness of C follows from completeness of R. 1.2. Open sets in C. Denote by Dr.z0/ := ^z Ë Cððz * z0ð < r`; r > 0 the open disk of radius r centered at z0. (All open disks form a topology base of C. ) A closed disk is defined as Dr.z0/ := ^z Ë Cððz * z0ð f r`; r > 0: We use D to denote the unit (open) disk center at the origin. ◦ For a subset Ω Ï C, a point z ËΩ is called an interior point, if there is some Dr.z/ Ï Ω. We use Ω to denote the set of interior points of Ω and it the interior of Ω. The set Ω is open if and only if Ω = Ω◦. A set Ω is closed if Ωc is open. A set is closed if and only if it contains all limit points. For any set Ω, its closure Ω is defined as the union of itself with its limit points. It is closed. The boundary )Ω is defined as )Ω := Ω ⧵ Ω◦: The closed disk Dr.z0/ is the closure of the open disk Dr.z0/, and )Dr.z0/ = )Dr.z0/ = Cr.z0/ := ^z Ë Cððz * z0ð = r0` the circle of radius r centered at z0. A subset Ω in C is called bounded, if there exists some M > 0 so that ðzð f M for any z ËΩ. For a bounded set Ω, define its diameter as diam(Ω) := sup ðz * wð; z;w∈Ω which is a finite number. Then the following statements are equivalent for a subset Ω of C: ∙Ω is compact; 2. FUNCTIONS ON C 7 ∙Ω is both closed and bounded; ∙Ω is sequentially compact, i.e., every sequence in Ω has a convergent subsequence. The following proposition is useful in proving the Goursat’s theorem later. PROPOSITION 1.3. Assume Ω1 Ð Ω2 Ð 5 is a sequence of nested subsets of C, with each one nonempty, compact and lim diam(Ωn/ = 0: n→∞ Then their intersection contains a unique point z0 Ë C. PROOF. We first prove that ãnΩn is not empty, i.e., it contains some point z0 Ë C. For this, take zn ËΩn for each n = 1; 2; 5 and we obtain a Cauchy sequence ^zn` in C since limn→∞ diam(Ωn/ = 0. By the completeness of C, the sequence ^zn` converges to some point z0 Ë C. The limit z0 lives in each Ωn since each Ωn is (sequentially) compact. This proves z0 Ë ãnΩn. z z¨ Next we show 0 is the only point in ãnΩn. Assume there is another point 0 Ë ãnΩn. Then their distance ¨ ðz0 * z0ð f diam(Ωn/; n = 1; 2; 5 : n z z¨ z¨ z Take → ∞, there must be ð 0 * 0ð = 0 and then 0 = 0. An open subset Ω of C is called connected, if there is no way to write Ω as a union of two disjoint nonempty open sets in C. This is equivalent to say Ω is path-connected, i.e., for any two points z0; z1 Ë Ω, there is a continuous path in Ω which connects z0 and z1. We call Ω a region in C if it is both open and connected. 2. Functions on C 2.1. Continuous functions. Assume Ω is a subset of C and z0 ËΩ. A function f :Ω → C is called continuous at z0, if for any > 0, there exists some > 0 so that ðf.z/* f.z0/ð < ; for any ðz * z0ð < ; z ËΩ: The following equivalence is left as a homework problem: f is continuous at z0 if and only if f.zn/ → f.z0/ for any sequence zn → z0 as n → ∞: Clearly, f is continuous at z0 if and only if both Re.f/ and Im.f/ are continuous at z0. 0 We write f Ë C (Ω) if f is continuous on every z0 ËΩ. The definition of continuity and be generalized to functions defined over any metric spaces, for example, over C × C. Then it is easy to check addition, subtraction, multiplication, division, norm are all continuous functions on the corresponding natural domains. The following property for continuous function defined over a compact domain is very useful. THEOREM 2.1. If Ω is a compact subset in C and f Ë C0(Ω), then f is bounded and can obtain its maximum and minimum on Ω. 8 1. PRELIMINARIES TO COMPLEX ANALYSIS 2.2. Holomorphic functions. Now there comes the key concept in complex analysis. DEFINITION 2.2. Assume Ω is an open subset in C. A function f :Ω → C is called holomorphic at z0 ËΩ, if the function f.z0 + h/* f.z0/ h is convergent as h → 0 (with z0 + h ËΩ). The limit is called the derivative of f at z0 and is denoted by ¨ f .z0/. ¨ If f .z0/ exists for every z0 ËΩ, then we say the function f is a holomorphic function over Ω.A holomorphic function f defines a derivative function f ¨ :Ω → C.
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