Topics in Complex Analysis M. Bertola‡1 ‡ Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montr´eal, Qu´ebec, Canada H4B 1R6 [email protected] Contents 1 Introduction 4 1 Complex numbers, complex plane ................................ 4 2 Sequences, series and convergence ................................ 6 3 Functions of one complex variable ............................... 7 2 Calculus 8 1 Holomorphic functions ...................................... 8 2 Power series ............................................ 11 3 Integration ............................................ 12 3.1 Cauchy’s Theorem .................................... 15 4 Cauchy’s integral formula .................................... 18 4.1 Taylor Series ....................................... 19 4.2 Morera’s theorem ..................................... 21 5 Maximum modulus and mean value .............................. 22 5.1 Mean value principle ................................... 24 6 Laurent series and isolated singularities ............................ 26 7 Residues .............................................. 28 7.1 Zeroes of holomorphic functions ............................. 28 7.2 Meromorphic functions ................................. 29 7.3 Application of the residue theorem to multiplicities of zeroes ............ 29 3 Applications of complex integration 31 1 Summation of series ....................................... 31 1.1 Fourier series ....................................... 33 2 Integrals .............................................. 33 2.1 Case1 ........................................... 33 2.2 Case2 ........................................... 34 3 Jordan’s lemma and applications ................................ 34 1 4 Analytic continuation 36 1 Analytic continuation ...................................... 36 1.1 Monodromy theorem ................................... 39 2 Schwartz reflection principle ................................... 41 5 One-dimensional complex manifolds 42 1 Definition ............................................. 42 2 One-forms and integration .................................... 44 2.1 Zeroes and poles: residues ................................ 46 3 Coverings and Riemann surfaces ................................ 47 4 Fundamental group and universal covering ........................... 49 4.1 Intersection number ................................... 51 5 Universal covering ........................................ 52 6 Riemann’s sphere ......................................... 53 6.1 One forms ......................................... 55 6.2 Automorphisms ...................................... 55 7 The unit disk .......................................... 57 7.1 Automorphisms ...................................... 58 8 Complex tori and elliptic functions ............................... 59 8.1 One-forms ......................................... 62 8.2 Elliptic functions ..................................... 62 8.3 Automorphisms and equivalence of complex tori ................... 75 9 Modular group and modular forms ............................... 76 9.1 Modular functions and forms .............................. 80 10 Classification of complex one-dimensional manifolds ..................... 86 11 Algebraic functions and algebraic curves ............................ 87 11.1 Manifold structure on the locus P (w,z)=0 ...................... 89 11.2 Surgery .......................................... 90 11.3 Hyperelliptic curves ................................... 91 11.4 Abelian Integrals ..................................... 94 11.5 Symplectic basis in the homology ............................ 95 6 Harmonic functions 97 1 Harmonic conjugate ....................................... 97 2 Mean and maximum value theorems .............................. 98 2 7 Riemann mapping theorem 101 1 Statement ............................................. 101 2 Topologyof ( ) ........................................ 104 H D 3 Proof ................................................ 108 4 Extensions of the theorem .................................... 109 4.1 Some conformal mappings ................................ 110 5 Exercises ............................................. 110 8 Picard’s theorems 111 1 Picard’s Little Theorem ..................................... 111 2 ProofofThm. 8.1.0 ....................................... 114 3 Picard’s great theorem ...................................... 115 3 Chapter 1 Introduction 1 Complex numbers, complex plane We define the following structure (=set + operations) C := z := (a,b) a,b R (= R2) (1.1.1) { | ∈ } (a,b) + (c, d) := (a + c,b + d) (1.1.2) (a,b) (c, d) := (ac bd, ad + bc) (1.1.3) · − Lemma 1.1.3 The strucuture (C, +, ) satisfies the axioms of field with the multiplicative inverse given · as follows: 1 a b 1 z = (a,b) =0=:(0, 0) , z− = , − , z z− =1=:(1, 0). (1.1.4) 6 a2 + b2 a2 + b2 · Remark 1.1 For which n’s we can endow the set Rn with a structure of division algebra? Only for n = 1, 2, 4, 8. This was proved by J. Adams in 1962 using methods of algebraic topology. We will use the following shortccut for complex numbers z = a + ib, i2 = 1, i := (0, 1). (1.1.5) − We also define the following real-valued functions (z) := a , (z)= b (1.1.6) ℜ ℑ and the following operation : C C (1.1.7) → (1.1.8) z z := a ib 7→ − 4 We then have the following properties: z + z z z (z)= , (z)= − (1.1.9) ℜ 2 ℑ 2i z w = z w (1.1.10) ± ± z w = z w (1.1.11) · · [From now on we omit the to denote the multiplication of complex numbers] · The modulus and argument are defined by z := √zz = a2 + b2 0 and = 0 iff z =0. (1.1.12) | | ≥ (z) (z) arg(z)= φ(mod 2π) s.t. cos φ = ℜ , sin φ = ℑ . (1.1.13) z z | | | | Lemma 1.1.13 [Exercise] Prove that z w z + w z + w (1.1.14) | | | − | ||≤| | ≤ | | | | Exercise 1.1 Find all complex numbers satisfying the following relations z = z (1.1.15) 1 z z− = , z =0. (1.1.16) z 2 6 | | Examples 1.1 . 1. The map z z is the reflection about the Real axis. 7→ 2. The set z z = R is the circle of radius R centered at z . {| − 0| } 0 3. The set arg(z) θ <ǫ is a wedge of width 2ǫ with bisecant forming an agle θ with the positive {| − 0| } 0 real axis. Lemma 1.1.16 [Exercise] zw = z w (1.1.17) | | | | | | arg(zw)= arg(z)+ arg(w) mod 2π (1.1.18) Corollary 1.1.18 (Exercise) The map z λz with λ C× (C× := C 0 ) is a rotation of angle 7→ ∈ \{ } arg(λ) followed by a dilation by λ . | | Example 1.1 Using the tautological identification of C R2, find the matrix that represents the linear ≃ map T (z) := λz : C C as a linear transformation T˜ : R2 R2. Characterize all linear maps R2 R2 → → → that can be represented by multiplication by a complex number. 5 Lemma 1.1.18 [Euler’s formula] Let z = x + iy, then ex+iy =ex cos(y)+ i sin(y) (1.1.19) Exercise 1.2 Using Euler’s formula prove that Tn(x) := cos(nθ) , x := cos(θ) (1.1.20) is a polynomial in x of degree n (Tchebicheff’s polynomial). [Hint: cos(nθ)= (exp(inθ))]. ℜ 2 Sequences, series and convergence The topology of C is inherited from the tautological identification with R2 (which is a metric space) i.e. Definition 2.1 A sequence zn : N C converges limn zn = w iff limn zn w =0. → →∞ →∞ | − | Since R2 is complete, so is C, namely Corollary 1.2.0 Every Cauchy sequence in C has a limit in C, or, in more detail: A sequence zn n N converges iff { } ∈ ǫ> 0 N s.t. n,m>N z z <ǫ (1.2.1) ∀ ∃ ∀ | n − m| As for real numbers, convergence for series is defined according to the convergence of the partial sums, i.e. the series ∞ zn (1.2.2) n=0 X n converges iff the sequence of its partial sums sn := j=0 zj converges. Lemma 1.2.2 [Exercise] If the series P ∞ z (1.2.3) | n| n=0 X converges then the series n∞=0 zn converges as well and P ∞ ∞ zn zn (1.2.4) ≤ | | n=0 n=0 X X Definition 2.2 If the series ∞ zn converges we say that the series ∞ zn is absolutely con- n=0 | | n=0 vergent. P P 6 Example 2.1 Suppose we have ∞ zn and ∞ wn two absolutely convergent series, with sum respec- tively Z and W . Prove (with an ǫ δ proof) that P− P azn + bwn = aZ + bW (1.2.5) X ∞ ∞ znwm = ZW (1.2.6) n=0 m=0 X X 3 Functions of one complex variable Definition 3.1 A subset of C is called a domain if it is open and connected. D Recall that a set Y in a topological space (X, τ) is said to be connected if the following condition applies Y = Y Y , Y and Y open subsets , Y = ,Y, Y Y = . (1.3.1) 1 ∪ 2 1 2 j 6 ∅ 1 ∩ 2 ∅ We now begin the study of the properties of functions f : C (1.3.2) D → Limits and continuity are defined as for any maps between topological (actually metric) spaces. Differ- entiability can be defined using the tautological identification C R2, however we will need a refinement ≃ of this notion which is ”compatible” with the structure of the complex plane. We will use the notations z = x + iy f(z)= u(x, y)+ iv(x, y). (1.3.3) Exercise 3.1 Formulate the continuity requirement and prove that a function f = u+iv is continuous (at a point/ on a domain) iff both its real and imaginary parts (u and v) are. Prove that the product, linear combination, ratio (iff the denominator does not vanish) of two continuous functions f,g is continuous. Examples 3.1 . 1. f(z)= z 2. f(z)= z N M n m 3. f(x)= n=0 m=0 anmz z Remark 3.1P[Exercise]P If D⊂ C is closed and bounded ( and hence compact) then a continuous function f is also uniformly continous, namely ∀ǫ> 0 ∃δ > 0 s.t. ∀z, w ∈ D, |z − w| < δ ⇒ |f(z) − f(w)| < ǫ . (1.3.4) 7 Chapter 2 Calculus 1 Holomorphic functions Definition 1.1 Given a function f(z) : C we say that it has complex derivative at z if the D → 0 following limit exists finite f(z) f(z0) f ′(z0) := lim − . (2.1.1) z z0 z z → − 0 Note that the ratio is the ratio of complex numbers and the limit is taken in C. Example 1.1 The function f(z)= zz does not have complex derivative in the above sense: however it is differentiable when seen as a function R2 R2. → Exercise 1.1 Prove that the functions f(z) = zn, n Z have complex derivative at all points in C ∈ (excluding z =0 for n< 0) and that n 1 f ′(z)= nz − . (2.1.2) Definition 1.2 A function f : C defined on the domain is called holomorphic if it has complex D → D derivative at all points of the domain.