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Complex Analysis Mario Bonk Complex Analysis Mario Bonk Course notes for Math 246A and 246B University of California, Los Angeles Contents Preface 6 1 Algebraic properties of complex numbers 8 2 Topological properties of C 18 3 Differentiation 26 4 Path integrals 38 5 Power series 43 6 Local Cauchy theorems 55 7 Power series representations 62 8 Zeros of holomorphic functions 68 9 The Open Mapping Theorem 74 10 Elementary functions 79 11 The Riemann sphere 86 12 M¨obiustransformations 92 13 Schwarz's Lemma 105 14 Winding numbers 113 15 Global Cauchy theorems 122 16 Isolated singularities 129 17 The Residue Theorem 142 18 Normal families 152 19 The Riemann Mapping Theorem 161 3 CONTENTS 4 20 The Cauchy transform 170 21 Runge's Approximation Theorem 179 References 187 Preface These notes cover the material of a course on complex analysis that I taught repeatedly at UCLA. In many respects I closely follow Rudin's book on \Real and Complex Analysis". Since Walter Rudin is the unsurpassed master of mathematical exposition for whom I have great admiration, I saw no point in trying to improve on his presentation of subjects that are relevant for the course. The notes give a fairly accurate account of the material covered in class. They are rather terse as oral discussions that gave further explanations or put results into perspective are mostly omitted. In addition, all pictures and diagrams are currently missing as it is much easier to produce them on the blackboard than to put them into print. 6 1 Algebraic properties of complex numbers 1.1. Intuitive idea. Complex numbers are expressions of the form a + bi with real numbers a and b. Here i is the imaginary unit. One computes (i.e., adds and multiplies) with complex numbers as usual, but sets i2 = i·i := −1. For example, (3 + 5i)(2 + i) = 6 + 10i + 3i + 5i2 = 6 + 13i − 5 = 1 + 13i: 1.2. Definition of the complex numbers. For a rigorous definition we let the set of complex numbers be C := f(a; b): a; b 2 Rg with the correspondence (a; b) ∼= a + bi in mind. Addition and multiplication are defined accordingly: (a; b) + (c; d) := (a + c; b + d); (a; b) · (c; d) := (ac − bd; ad + bc): One often omits the multiplication sign and writes zw := z · w for z; w 2 C. One also uses the convention that multiplication binds stronger than addition. So u + vw = u + (v · w) for u; v; w 2 C, etc. That one computes with complex numbers \as usual" is mathematically expressed by the following fact. Theorem 1.3. (C; +; ·) is a field, that is: 1. (C; +) is an abelian group, which means that 1.1 the addition + is associative, 1.2 there exists a neutral element 0 := (0; 0) 2 C with respect to addition, 1.3 every element (a; b) 2 C has an (additive) inverse (−a; −b) 2 C, 1.4 the addition + is commutative. 2. (C; +; ·) is a commutative ring, which means that 8 1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 9 2.1 (C; +) is an abelian group, 2.2 the multiplication · is associative and commutative, 2.3 the distributive law holds. 3. (C∗; ·) is a group, where C∗ := C n f0g. The proof is straightforward, but tedious, and so we skip it. Remark 1.4. If we write (for the moment),a ~ := (a; 0) 2 C for a 2 R, then a~ + ~b = (a; 0) + (b; 0) = (a + b; 0) = a]+ b; and a~ · ~b = (a; 0) · (b; 0) = (ab; 0) = ab:e This means that with expressionsa ~, ~b, etc., one can compute in exactly the same way as with the underlying real numbers a, b, etc. More precisely, the map ': R ! C; a 2 R 7! '(a) :=a; ~ is a field isomorphism of R onto its image in C. Accordingly, one \identifies” the image of R under ' with R, writes a instead ofa ~, and considers R as subset of C. Now let i := (0; 1). Then for a; b 2 R we have (a; b) = (a; 0) + (0; b) = (a; 0) + (b; 0) · (0; 1) =a ~ + ~bi = a + bi: So every complex number z 2 C can uniquely be written as z = a+bi, where a; b 2 R. Note also that we have i2 := i · i = (0; 1) · (0; 1) = (−1; 0) = −f1 = −1: Using these conventions we reconcile the precise definition of complex num- bers and their operations with the intuitive notion that we took as out start- ing point. 1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 10 Definition 1.5. Let z = a + bi 2 C, where a; b 2 R. We define (i) Re(z) := a (the real part of z), (ii) Im(z) := b (the imaginary part of z), (iii)z ¯ := a − bi (the complex conjugate of z), p (iv) jzj := a2 + b2 (the absolute value of z). 1.6. Geometric interpretations. These concepts and also addition of complex numbers have obvious geometric interpretations if one identifies z = a + bi with the point (a; b) in the plane R2. 1.7. Subtraction and division. As in every field one can define a notion of subtraction and division of complex numbers. Namely, if z 2 C, one denotes the additive inverse of z by −z and defines w − z := w + (−z) for z; w 2 C. If z = a + bi, w = c + di, then −z = (−a) + (−b)i, and so w − z = (c − a) + (d − b)i: If z 6= 0, then we denote by z−1 the (multiplicative) inverse of z. For z = a + bi 6= 0 we have a b z−1 = − i: a2 + b2 a2 + b2 One defines z z=w = := w · z−1: w One can compute with fractions of complex numbers as usual (as in any field). Using the fact that zz¯ = jzj2, one can simplify fractions of complex numbers by multiplying in numerator and denominator by the complex conjugate of the denominator. For example, 2 + i (2 + i)(3 − 5i) = 3 + 5i (3 + 5i)(3 − 5i) (6 + 5) + (−10 + 3)i = 32 + 52 11 − 7i 11 7 = = − i: 34 34 34 1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 11 Theorem 1.8. Let z; w 2 C. Then 1 (i) Re(z) = 2 (z +z ¯), 1 (ii) Im(z) = 2i (z − z¯), (iii) Re(z + w) = Re(z) + Re(w), (iv) Im(z + w) = Im(z) + Im(w), (v) z 2 R iff Im(z) = 0 iff z =z ¯, (vi) z¯ = z, (vii) z + w =z ¯ +w ¯, z − w =z ¯ − w¯, (viii) z · w =z ¯ · w¯, z z¯ (ix) = , w w¯ (x) z · z¯ = jzj2, (xi) jzj = 0 iff z = 0, (xii) jz · wj = jzj · jwj, z jzj (xiii) = , w jwj (xiv) jz + wj ≤ jzj + jwj (triangle inequality). Proof. The proofs of these facts are straightforward, often tedious, and we omit the details; as an example, we will prove (xii). If z = a + bi and w = c + di, where a; b; c; d 2 R, then z · w = (ac − bd) + (ad + bc)i; 1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 12 and so jz · wj2 = (ac − bd)2 + (ad + bc)2 = a2c2 + b2d2 + a2d2 + b2c2 = (a2 + b2)(c2 + d2) = jzj2 · jwj2: Hence jz · wj = jzj · jwj: (For this conclusion it is important that the terms on both sides are non- negative). Definition 1.9 (The exponential function for complex arguments). For z = x + iy, where x; y 2 R, we define ez = exp(z) := ex(cos y + i sin y): Note that for z 2 R, when y = Im(z) = 0, this agrees with the usual exponential function. So the exponential function for complex arguments is an extension of the exponential function for real arguments. Choosing this particular extension seems arbitrary and unmotivated at this point. It will become more natural after we have introduced power series, because we will see that the complex exponential function can be represented by the usual power series for the real exponential function. A deeper reason for choosing this particular extension is that the complex exponential function is holomorphic and by the uniqueness theorem every function on R has at most one holomorphic extension to C. All this will be discussed later in the course. Theorem 1.10. The exponential function exp: C ! C has the following properties: (i) eit = cos t + i sin t for t 2 R (Euler-Moivre formula), (ii) ez+w = ez · ew for z; w 2 C (functional equation of exp), (iii) ez+2πi = ez for z 2 C (exp is 2πi-periodic), (iv) ez = 1 iff z = 2πik with k 2 Z, 1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 13 (v) ew = ez iff w = z + 2πik with k 2 Z. Proof. (i) Obvious from the definition. (ii) If z = x + iy and w = u + iv with x; y; u; v 2 R, then z + w = (x + u) + i(y + v): Note that cos(y + v) = cos y cos v − sin y sin v; sin(y + v) = sin y cos v + cos y sin v: Hence ez · ew = ex(cos y + i sin y)eu(cos v + i sin v); = ex+u(cos y cos v − sin y sin v) + i(sin y cos v + cos y sin v) = ex+u(cos(y + v) + i sin(y + v)) = e(x+u)+i(y+v) = ez+w: (iv) Let z = x + iy, x; y 2 R. Then ez = 1 , ex(cos y + i sin y) = 1 , ex cos y = 1 and ex sin y = 0; , ex cos y = 1 and sin y = 0; x , e cos y = 1 and y = nπ for n 2 Z; x n , e (−1) = 1 and y = nπ for n 2 Z; x , e = 1 and y = nπ for some n 2 Z even; , x = 0 and y = nπ for some n 2 Z even; , z = x + iy = 2πik for k 2 Z: (iii) Using (ii) and (iv) we have ez+2πi = ez · e2πi = ez · 1 = ez for z 2 C.
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