Complex Analysis Math 147—Winter 2008 Bernard Russo March 14, 2008 Contents 1 Monday January 7—Course information; complex numbers; Assignment 1 1 1.1 Course information . 1 1.2 Complex numbers . 2 2 Thursday January 10–Polar form. Assignment 2 2 2.1 Polar coordinates . 2 3 Friday January 11 2 3.1 Complex functions . 2 4 Monday, January 14—Limits and continuous functions 3 5 Wednesday January 16—The Cauchy-Riemann equations; Assignment 3 3 5.1 Cauchy-Riemann equations . 3 5.2 The exponential function . 3 6 Friday January 18—Cauchy-Riemann equations-revisited 4 7 Monday January 21—Holiday 4 8 Wednesday January 23—Trigonometric functions-(Prof. R. Reilly) 4 9 Friday January 25—The complex logarithm and complex powers;-(Prof. A. Figotin) 4 10 Monday January 28—-Line integrals, Assignment 4 4 11 Wednesday January 30—-Line integrals, Independence of the path 5 12 Friday February 1—Review for Midterm 6 12.1 Review Problems . 6 12.2 Highlights of Assignments 1,2,3 . 6 13 Monday February 4—First Midterm 6 i 14 Wednesday February 6—-More on Independence of the path 6 15 Friday February 8—Homotopic curves 6 16 Monday February 11—Cauchy’s theorem; Assignment 5 7 17 Wednesday February 13 7 18 Friday February 15 7 19 Monday February 18—holiday 7 20 Wednesday February 20 7 21 Friday February 22 8 22 Monday February 25—Maximum Modulus Theorem 8 22.1 Announcements . 8 22.2 Maximum modulus theorem . 9 23 Wednesday February 27 9 24 Friday February 29 9 24.1 Schwarz’s lemma . 9 24.2 Taylor’s Theorem . 10 25 Monday March 3 10 25.1 Announcements . 10 25.2 Power series . 10 25.3 The Identity Theorem . 10 26 Wednesday March 5—Identity theorem; proof and examples 11 26.1 List of Named Theorems . 11 26.2 Completion of the proof of the identity theorem . 11 26.3 Examples . 12 27 Friday March 7 12 27.1 An application of the identity theorem . 12 27.2 A corollary to the Identity theorem . 12 27.3 Riemann’s Removable Singularity Theorem . 12 28 Monday March 10 13 28.1 More on the fundamental theorem of algebra. Exercise 12 of chapter 6 . 13 28.2 Completion of the proof of Riemann’s removable singularity theorem . 13 29 Tuesday March 11-Completion of the proof of Riemann’s removable sin- gularity theorem and of Schwarz’s lemma 14 29.1 completion of the proof of Triangulated Morera theorem . 14 ii 29.2 completion of the proof of Riemanns’ removable singularity theorem . 14 29.3 completion of the proof of Schwarz’s lemma . 14 30 Wednesday March 12—Classification of Singularities 15 30.1 The order of a pole . 15 30.2 Classification of Singularities . 15 30.3 The range of an analytic function on C − {z0} . 15 31 Friday March 14—Laurent’s theorem 16 31.1 Proof of the Casorati-Weierstrass theorem (Part (c) of Proposition 30.2) . 16 31.2 More properties of singularities . 16 31.3 About final exam week . 17 iii 1 Monday January 7—Course information; complex num- bers; Assignment 1 1.1 Course information • Course: Mathematics 147 MWF 11:00–11:50 ET201 • Prerequisite: Math 140AB or consent of the instructor (if you have had 140A and are taking 140B concurrently, that is acceptable) • Instructor: Bernard Russo MSTB 263 Office Hours MW 10:00-10:40 or by appoint- ment (a good time for short questions is right after class just outside the classroom) • There is a link to this course on Russo’s web page: www.math.uci.edu/∼ brusso • Discussion section: TuTh 11:00–11:50 HICF 100M • Teaching Assistant: Kenn Huber • Homework: There will be approximately 10-12 assignments with at least one week notice before the due date. First midterm February 1 (Friday of week 4) 20 percent Second midterm February 29 (Friday of week 8) 20 percent • Grading: Final Exam March 21 (Friday 8:00-10:00 am) 40 percent Homework approximately 12 assignments 20 percent • Holidays: January 21, February 18 • Text: George Cain “Complex Analysis”, Freely available on the web (see Russo’s web page or go directly to http://www.math.gatech.edu/ cain/winter99/complex.html) • Material to be Covered: All of the text with the possible exception of chapters 8 and 11. However, there will be some material that is not in the text. • Catalog description: Rigorous treatment of basic complex analysis: complex numbers, analytic functions, Cauchy integral theory and its consequences (Morera’s Theorem, The Argument Principle, The Fundamental Theorem of Algebra, The Maximum Mod- ulus Principle, Liouville’s Theorem), power series, residue calculus, harmonic func- tions, conformal mapping. Students are expected to do proofs. • Math 147 is replacing the old Math 114B, and is intended for mathematics majors. The sequence 114A-147 is acceptable for the specialization in applied mathematics. You cannot take 114A after taking 147. • Some alternate texts that you may want to look at, in no particular order. There are a great number of such texts at the undergraduate and at the graduate level. Undergraduate Level 1. S. Fisher: Complex Variables 2. R. Churchill and J. Brown; Complex Variables and Applications 3. J. Marsden and M. Hoffman, Basic Complex Analysis 1 4. E. Saff and A. Snider: Fundamentals of Complex Analysis Graduate Level 1. L. Ahlfors; Complex Analysis 2. J. Conway; Functions of one Complex Variable 3. J. Bak and D. Newman; Complex Analysis 1.2 Complex numbers algebra: rational numbers, real numbers, complex numbers sum, product, difference, quotient of complex numbers geometry: modulus, conjugate, triangle inequality notation: real and imaginary parts of z relations: zz = |z|2, |Re z| ≤ |z|, |Im z| ≤ |z| Proposition 1.1 (Triangle inequality) For any two complex numbers z and w, we have |z + w| ≤ |z| + |w|. Assignment 1 (1A is due January 11; 1B is due January 16) 1A: Problems 1-6 of chapter 1 of Cain. 1B: Problems 7-12 of chapter 1 of Cain. 2 Thursday January 10–Polar form. Assignment 2 Proof of Proposition 1.1 Corollary 2.1 For any two complex numbers z and w, we have |z − w| ≥ | |z| − |w| |. Assignment 2 (2A is due January 18; 2B is due January 23) 2A: Problems 1-3 and 5-9 of chapter 2 of Cain. (You may skip section 2.1 except for exercises 1–3.) 2B: Problems 10-16 of chapter 2 of Cain. 2.1 Polar coordinates argument, principal argument, geometric interpretation of sum and product Proposition 2.2 For any two complex numbers z and w, we have |zw| = |z| · |w| and arg zw = arg z + arg w (modulo 2πZ). 3 Friday January 11 3.1 Complex functions Derivatives of functions: • f : R → R, f 0(x), graph is a curve • f : R2 → R, ∂f/∂x, ∂f/∂y, graph is a surface 2 • f : R → R2, f 0(t) = (x0(t), y0(t)), parametric equations of a curve in R2 • f : C → C, f 0(z) (new idea) Definition of derivative of a complex valued function of a complex variable: f(z + h) − f(z) f 0(z) = lim h→0 h where h represents a complex number. EXAMPLES: • f(z) = z is differentiable for every z and f 0(z) = 1 for all z ∈ C. • g(z) = z is not differentiable for any z. 4 Monday, January 14—Limits and continuous functions Continuous function at a point, on a set. Sums, products, differences, and quotients of continuous functions are continuous (whenever defined), We proved the first of these two propositions (and ignored the second one!): • If f = u + iv is a complex valued function of a complex variable (with real part u and imaginary part v, then f is continuous if and only if both u and v are continuous. • If limz→z0 f(z) = L1 and limz→z0 g(z) = L2, then limz→z0 f(z)g(z) exists and equals L1L2 We showed that the function f(z) = z is not differentiable at any point z0. 5 Wednesday January 16—The Cauchy-Riemann equations; Assignment 3 5.1 Cauchy-Riemann equations We proved the first of these two propositions and started the proof of the second one: • If f = u+iv is a complex valued function of a complex variable, and f is differentiable at z0 = x0 + iy0, then u and v satisfy the Cauchy Riemann equations at (x0, y0). • If f = u + iv is a complex valued function of a complex variable, and u and v satisfy the Cauchy Riemann equations at (x0, y0), and if ux and uy are continuous at (x0, y0), then f is differentiable at z0 = x0 + iy0, 5.2 The exponential function Definition: exp z = ex cos y + iex sin y for z = x + iy. We proved the following two propositions: d • dz exp z = exp z 3 • exp(z + w) = exp z exp w Assignment 3 (3A is due January 23; 3B is due January 28; 3C is due January 30) 3A: Problems 1-6 of chapter 3 of Cain. 3B: Problems 8-11 of chapter 3 of Cain. 3C: Problems 12-17 of chapter 3 of Cain. 6 Friday January 18—Cauchy-Riemann equations-revisited Finished the proof of the sufficiency of the Cauchy-Riemann equations. 7 Monday January 21—Holiday 8 Wednesday January 23—Trigonometric functions-(Prof. R. Reilly) Definition and properties of the functions sin z, cos z. 9 Friday January 25—The complex logarithm and complex powers;-(Prof. A. Figotin) Definition and properties of the multi-valued functions log z, zc (c a complex number).