Lecture Note for Math 220A Complex Analysis of One Variable Song-Ying Li University of California, Irvine Contents 1 Complex numbers and geometry 2 1.1 Complex number field . 2 1.2 Geometry of the complex numbers . 3 1.2.1 Euler's Formula . 3 1.3 Holomorphic linear factional maps . 6 1.3.1 Self-maps of unit circle and the unit disc. 6 1.3.2 Maps from line to circle and upper half plane to disc. 7 2 Smooth functions on domains in C 8 2.1 Notation and definitions . 8 2.2 Polynomial of degree n ...................... 9 2.3 Rules of differentiations . 11 3 Holomorphic, harmonic functions 14 3.1 Holomorphic functions and C-R equations . 14 3.2 Harmonic functions . 15 3.3 Translation formula for Laplacian . 17 4 Line integral and cohomology group 18 4.1 Line integrals . 18 4.2 Cohomology group . 19 4.3 Harmonic conjugate . 21 1 5 Complex line integrals 23 5.1 Definition and examples . 23 5.2 Green's theorem for complex line integral . 25 6 Complex differentiation 26 6.1 Definition of complex differentiation . 26 6.2 Properties of complex derivatives . 26 6.3 Complex anti-derivative . 27 7 Cauchy's theorem and Morera's theorem 31 7.1 Cauchy's theorems . 31 7.2 Morera's theorem . 33 8 Cauchy integral formula 34 8.1 Integral formula for C1 and holomorphic functions . 34 8.2 Examples of evaluating line integrals . 35 8.3 Cauchy integral for kth derivative f (k)(z) . 36 9 Application of the Cauchy integral formula 36 9.1 Mean value properties . 36 9.2 Entire holomorphic functions . 38 9.3 Liouville's Theorem . 38 9.4 Application of Liouville's theorem . 38 9.5 Entire functions with sub-linear growth . 39 9.6 Polynomial growth entire holomorphic functions . 39 10 Fundamental theorem of algebra 40 10.1 Lower bound for holomorphic polynomials . 40 10.2 Fundamental theorem of algebra . 41 11 Homework 3 42 12 Series of complex numbers 44 12.1 Definition of the series of complex numbers . 44 12.1.1 Tests for convergent series . 44 12.1.2 Examples of Dirichlet's test . 46 12.2 Power series . 46 12.2.1 Radius of convergence for a power series . 46 12.2.2 Examples of power series . 47 2 12.2.3 Examples of functions with power series expansions . 49 12.2.4 Differentiation of power series . 49 12.3 Analytic Functions . 50 12.3.1 Definition of an analytic function . 50 12.3.2 Connection between holomorphic and analytic functions 51 13 Homework 4 53 13.1 Uniqueness for analytic functions . 55 14 Uniqueness theorem and applications 58 15 Homework 5 61 16 Midterm Review 63 17 Uniqueness Theorem and Applications 67 18 Isolated Singularities 70 18.1 Definition of an isolated singularity . 70 18.2 Riemann removable singularity lemma . 72 18.3 Poles and essential singularity . 73 18.3.1 Characterization for poles . 73 18.3.2 Order of the pole . 73 18.3.3 Behavior of f near an essential singularity . 74 18.4 Homework 6 . 76 19 Laurent Series 78 19.1 Definition and examples of Laurent Series . 78 19.2 Examples of the convergent annulus . 78 19.3 Laurent series expansion . 79 19.4 Laurent series and isolated singularities . 81 20 Meromorphic functions 83 20.1 Construction of certain meromorphic functions . 84 20.2 Singularity at 1 ......................... 87 20.3 Rational functions . 88 20.4 Stereographic projection . 89 20.5 Homework 7 . 90 3 21 Residues and the Residue Theorem 91 21.1 Definition of a Residue . 91 21.2 Formulae for Computing Residue of f . 92 21.3 Residue at z = 1 ......................... 94 21.4 Residue Theorems . 96 21.5 Winding number and Residue Theorem . 98 21.6 Homework 8 . 102 22 Final Review 104 List of Figures 1 Disc Chain . 56 Acknowledgement: I would like to thank J. N. Treuer for reading through the first draft and making some revisions. 4 I. Holomorphic and harmonic functions 1 Complex numbers and geometry 1.1 Complex number field In the field of the real numbers IR, we know that x2 + 1 has no roots. It would be ideal if every polynomial of degree n hadpn roots. To remedy this situation, we introduce an imaginary number i = −1 (i2 = 1). Then the polynomial z2 + 1 has two roots: ±i. Viewing a point (x; y) in xy-plane, we introduce a corresponding complex number (1:1) z =: x + iy; C = fz =: x + iy : x; y 2 IRg: Here x is the real part of z which is denoted by Re z and y is the imaginary part of z which is denoted by Im z. • Two operations on C. For z1 = x1 + iy1; z2 = x2 + iy2 2 C; we define two operations: • Addition: (1:2) z1 + z2 =: (x1 + x2) + i(y1 + y2) • Multiplication: (1:3) z1 · z2 =: (x1x2 − y1y2) + i(x1y2 + x2y1): It is easy to verify the following proposition. Proposition 1.1 (C; +; ·) forms a number field. It is called the complex number field. Question: Is (C; +; ·) an order field? For each z = x + iy 2 C, the complex conjugate z of z and the norm jzj of z are defined as follows: (1:4) z = x − iy and jzj2 = z · z = x2 + y2: Then 5 i. The additive inverse of z is −z = −x + i(−y) ii. When z 6= 0, the multiplicative inverse of z is 1 z x y =: = − i z jzj2 x2 + y2 x2 + y2 iii. jzj2 = z · z = x2 + y2; 1 1 iv. x =: Re z = 2 (z + z); y =: Im z = 2i (z − z): 1 EXAMPLE 1 What is the complex number 2−3i ? Solution. 1 2 + 3i 2 3 = = + i : 2 − 3i 22 + 32 13 13 More generally, one can compute 1=z as follows: 1 z x y (1:5) = = − i : z jzj2 jzj2 jzj2 1.2 Geometry of the complex numbers 1.2.1 Euler's Formula Let jzj be the distance from z = x + iy to 0 = 0 + i0, let θ be the angle between the line segment [0; z] and the x-axis. Then x = jzj cos θ; y = jzj sin θ; z = jzj(cos θ + i sin θ) = jzjeiθ where (1:6) eiθ = cos θ + i sin θ is Euler's formula. 6 e iθ sinθ . θ cosθ One can prove Euler's formula by noting that 1 (iθ)n eiθ = X n=0 n! 1 (iθ)2n 1 (iθ)2n+1 = X + X n=0 (2n)! n=0 (2n + 1)! 1 (−1)nθ2n 1 (−1)nθ2n+1 = X + i X n=0 (2n)! n=0 (2n + 1)! = cos θ + i sin θ: • Fact: Given two complex numbers z = jzjeiθ and w = jwjei', one has (1:7) zw = jzjjwjeiθei' = jzjjwjei(θ+') and jz wj = jzjjwj: Proposition 1.2 (Triangle Inequality) If z; w 2 C, then jz + wj ≤ jzj + jwj: Proof. jz + wj2 = (z + w)(z + w) = zz + zw + wz + ww = jzj2 + jwj2 + (zw + zw) = jzj2 + jwj2 + 2Re (zw¯) ≤ jzj2 + jwj2 + 2jzw¯j = jzj2 + jwj2 + 2jzw¯j = jzj2 + jwj2 + 2jzjjwj = (jzj + jwj)2 7 This implies that jz + wj ≤ jzj + jwj: Moreover, by mathematical induction, one can prove that if z1; ··· ; zn 2 C, then (1:8) jz1 + ··· + znj ≤ jz1j + ··· + jznj: Proposition 1.3 (Cauchy-Schwartz Inequality) If z1; ··· ; zn and w1; ··· ; wn 2 C, then 2 n n n X X 2 X 2 (1:9) z w¯ ≤ ( jz j )( jw j ): j j j j j=1 j=1 j=1 Proof. For any λ 2 C, one has n X 2 0 ≤ jzj − λwjj j=1 n X ¯ = (zj − λwj)(z ¯j − λw¯j) j=1 n X ¯ ¯ = (zjz¯j − λzjw¯j − λwjz¯j + λwjλw¯j) j=1 n 0 n n 1 n X 2 ¯ X X 2 X 2 = jzjj − @λ zjw¯j + λ wjz¯jA + jλj jwjj : j=1 j=1 j=1 j=1 Pn Pn 2 If we choose λ = j=1 zjw¯j=( j=1 jwjj ) then 2 2 n Pn Pn n j=1 zjw¯j j=1 zjw¯j 0 ≤ X jz j2 − 2 + X jw j2 j Pn 2 2 j j=1 jwjj Pn 2 j=1 j=1 jwjj j=1 2 n Pn z w¯ X 2 j=1 j j = jzjj − Pn 2 : j=1 j=1 jwjj This implies that 2 n n n X X 2 X 2 z w¯ ≤ ( jz j )( jw j ): j j j j j=1 j=1 j=1 8 1.3 Holomorphic linear factional maps 1.3.1 Self-maps of unit circle and the unit disc. Let (1:10) D(0; 1) = fz 2 C: jzj < 1g be the unit disk centered at the origin and (1:11) T = @D(0; 1) = fz 2 C: jzj = 1g = feiθ : θ 2 [0; 2π)g be the unit circle in the complex plane. For any a 2 D(0; 1), we define a − z (1:10) φ (z) = ; z 2 D(0; 1): a 1 − az¯ THEOREM 1.4 For any a 2 D(0; 1) we have i. φa(0) = a and φa(a) = 0; −1 ii. φa = φa; iii. φa : T ! T , so it is one-to-one and onto; iv. φa : D(0; 1) ! D(0; 1), so it is one-to-one and onto. Proof. For Part 1, it is easy to see that φa(0) = a and φa(a) = 0. For Part 2, since 2 a − φa(z) a(1 − az) − (a − z)) z(1 − jaj ) φa(φa(z)) = = = 2 = z: 1 − aφa(z) 1 − az − a(a − z) 1 − jaj −1 This proves φa = φa.