18.04 Complex analysis with applications Spring 2019 lecture notes Instructor: J¨ornDunkel This PDF is an adaption and extension of the original by Andre Nachbin and Jeremy Orloff. Credit for course design and content should go to them; responsibility for typos and errors lies with me. 1 Contents 1 Brief course description 7 1.1 Topics needed from prerequisite math classes . .7 1.2 Level of mathematical rigor . .7 1.3 Speed of the class . .7 2 Complex algebra and the complex plane 8 2.1 Motivation . .8 2.2 Fundamental theorem of algebra . .8 2.3 Terminology and basic arithmetic . .9 2.4 The complex plane . 10 2.4.1 The geometry of complex numbers . 10 2.4.2 The triangle inequality . 10 2.5 Polar coordinates . 11 2.6 Euler's Formula . 12 2.6.1 eiθ behaves like a true exponential . 12 2.6.2 Complex exponentials and polar form . 13 2.6.3 Complexification or complex replacement . 15 2.6.4 Nth roots . 16 2.6.5 The geometry of Nth roots . 17 2.7 Inverse Euler formula . 18 2.8 de Moivre's formula . 18 2.9 Representing complex multiplication as matrix multiplication . 18 3 Complex functions 19 3.1 The exponential function . 19 3.2 Complex functions as mappings . 20 3.3 The function arg(z)............................... 25 3.3.1 Many-to-one functions . 25 3.3.2 Branches of arg(z)............................ 26 3.3.3 The principal branch of arg(z)..................... 28 3.4 Concise summary of branches and branch cuts . 29 3.5 The function log(z)................................ 29 3.5.1 Figures showing w = log(z) as a mapping . 30 3.5.2 Complex powers . 32 4 Analytic functions 33 4.1 The derivative: preliminaries . 33 4.2 Open disks, open deleted disks, open regions . 34 4.3 Limits and continuous functions . 35 4.3.1 Properties of limits . 36 4.3.2 Continuous functions . 36 4.3.3 Properties of continuous functions . 37 4.4 The point at infinity . 38 4.4.1 Limits involving infinity . 38 2 4.4.2 Stereographic projection from the Riemann sphere . 39 4.5 Derivatives . 40 4.5.1 Derivative rules . 40 4.6 Cauchy-Riemann equations . 41 4.6.1 Partial derivatives as limits . 41 4.6.2 The Cauchy-Riemann equations . 42 4.6.3 Using the Cauchy-Riemann equations . 43 4.6.4 f 0(z) as a 2 2 matrix . 44 × 4.7 Geometric interpretation & linear elasticity theory . 45 4.8 Cauchy-Riemann all the way down . 46 4.9 Gallery of functions . 47 4.9.1 Gallery of functions, derivatives and properties . 47 4.9.2 A few proofs . 51 4.10 Branch cuts and function composition . 52 4.11 Appendix: Limits . 54 4.11.1 Limits of sequences . 54 4.11.2 lim f(z)................................. 56 z!z0 4.11.3 Connection between limits of sequences and limits of functions . 56 5 Line integrals and Cauchy's theorem 56 5.1 Complex line integrals . 57 5.2 Fundamental theorem for complex line integrals . 58 5.3 Path independence . 59 5.4 Examples . 60 5.5 Cauchy's theorem . 61 5.6 Extensions of Cauchy's theorem . 64 6 Cauchy's integral formula 67 6.1 Cauchy's integral for functions . 67 6.2 Cauchy's integral formula for derivatives . 69 6.2.1 Another approach to some basic examples . 70 6.2.2 More examples . 70 6.2.3 The triangle inequality for integrals . 73 6.3 Proof of Cauchy's integral formula . 75 6.3.1 A useful theorem . 75 6.3.2 Proof of Cauchy's integral formula . 76 6.4 Proof of Cauchy's integral formula for derivatives . 77 6.5 Consequences of Cauchy's integral formula . 78 6.5.1 Existence of derivatives . 78 6.5.2 Cauchy's inequality . 78 6.5.3 Liouville's theorem . 79 6.5.4 Maximum modulus principle . 80 3 7 Introduction to harmonic functions 82 7.1 Harmonic functions . 83 7.2 Del notation . 83 7.2.1 Analytic functions have harmonic pieces . 83 7.2.2 Harmonic conjugates . 85 7.3 A second proof that u and v are harmonic . 85 7.4 Maximum principle and mean value property . 86 7.5 Orthogonality of curves . 86 8 Two dimensional hydrodynamics and complex potentials 90 8.1 Velocity fields . 91 8.2 Stationary flows . 91 8.3 Physical assumptions, mathematical consequences . 92 8.3.1 Physical assumptions . 92 8.3.2 Examples . 93 8.3.3 Summary . 94 8.4 Complex potentials . 94 8.4.1 Analytic functions give us incompressible, irrotational flows . 94 8.4.2 Incompressible, irrotational flows always have complex potential func- tions . 95 8.5 Stream functions . 96 8.5.1 Examples . 97 8.5.2 Stagnation points . 99 8.6 More examples . 99 9 Taylor and Laurent series 102 9.1 Geometric series . 103 9.1.1 Connection to Cauchy's integral formula . 104 9.2 Convergence of power series . 104 9.2.1 Ratio test and root test . 105 9.3 Taylor series . 106 9.3.1 Order of a zero . 106 9.3.2 Taylor series examples . 107 9.3.3 Proof of Taylor's theorem . 111 9.4 Singularities . 112 9.5 Laurent series . 112 9.5.1 Examples of Laurent series . 115 9.6 Digression to differential equations . 117 9.7 Poles . 118 9.7.1 Examples of poles . 119 9.7.2 Residues . 120 4 10 Residue Theorem 121 10.1 Poles and zeros . 121 10.2 Words: Holomorphic and meromorphic . 122 10.3 Behavior of functions near zeros and poles . 122 10.3.1 Picard's theorem and essential singularities . 123 10.3.2 Quotients of functions . 123 10.4 Residues . 124 10.4.1 Residues at simple poles ..