Linear and Complex Analysis for Applications John P. D’Angelo DEPT. OF MATHEMATICS,UNIV. OF ILLINOIS, 1409 W. GREEN ST., URBANA IL 61801 E-mail address: [email protected] c 2014 by John P. D’Angelo Contents Chapter 1. Linear algebra 9 1. Linear equations 9 2. Matrices and row operations 15 3. Vector spaces 21 4. Dimension 25 5. Eigenvalues and eigenvectors 29 6. Bases and the matrix of a linear map 32 7. Determinants 37 8. Diagonalization and generalized eigenspaces 41 9. Characteristic and minimal polynomials 44 10. Similarity 46 Chapter 2. Complex numbers 49 1. Basic definitions 49 2. Limits 51 3. The exponential function and trig 53 4. Subsets of C 58 5. Complex logarithms 59 6. Linear fractional transformations 63 7. Linear fractional transformations and matrices 66 8. The Riemann sphere 69 Chapter 3. Vector analysis 73 1. Euclidean geometry 73 2. Differentiable functions 75 3. Line integrals and work 80 4. Surface integrals and flux 84 5. The divergence theorem 88 6. The classical Stokes’ theorem 91 7. A quick look at Maxwell’s equations 94 8. Differential forms 96 9. Inverse and implicit functions 101 Chapter 4. Complex analysis 105 1. Open subsets of C 105 2. Complex differentiation 106 3. Complex integration 109 4. The Cauchy theory 110 3 4 CONTENTS 5. Polynomials and power series 114 6. Singularities 117 7. Residues 119 8. More residue computations 123 Chapter 5. Transform methods 131 1. Laplace transforms 131 2. Generating functions and the Z-transform 139 3. Fourier series 148 4. Fourier transforms on R 152 Chapter 6. Hilbert spaces 161 1. Inner products and norms 161 2. Orthonormal expansion 164 3. Linear transformations 166 4. Adjoints 167 5. Hermitian and unitary operators 170 6. L2 spaces 176 7. Sturm-Liouville equations 179 8. The Green’s function 182 9. Additional techniques 186 10. Spectral theory 189 11. Orthonormal expansions in eigenvectors 191 Chapter 7. Examples and applications 195 1. Least squares 195 2. The wave equation 199 3. Legendre polynomials 203 4. Probability 206 5. Quantum mechanics 211 6. The Dirichlet problem and conformal mapping 217 References 223 Index 225 CONTENTS 5 Preface This book is based on a course taught in Spring 2014 at the University of Illinois. The course, Elec- trical and Computer Engineering (ECE) 493, is cross-listed as Math 487. Twenty-one students, including several graduate students, completed the class. Areas of specialization were electrical, mechanical, and civil engineering, engineering physics, and mathematics. Background and interests varied considerably; communicating with all these students at the same time and in the same notation was challenging. The course began with three reviews. We discussed elementary linear algebra and differential equations, basic complex variables, and some multi-variable calculus. The students took two diagnostic exams on these topics. The first three chapters include these reviews but also introduce additional material. The author feels that spiral learning works well here; in the classroom he prefers (for example) to introduce differential forms when reviewing line and surface integrals rather than to start anew. Thus the course used these reviews to gradually seduce the student into a more abstract point of view. This seduction is particularly prominent in the material on Hilbert spaces. This book aims to develop understanding of sophisticated tools by using them rather than by filling in all the details, and hence it is not completely rigorous. The review material helped students relearn things better while creating a desire to see powerful ideas in action. After the reviews, the course began anew, with about 4 weeks of complex analysis, 1 week on the Laplace transform, 3 weeks on Hilbert spaces and orthonormal expansion. We discussed Sturm-Liouville theory to give examples of orthonormal expansion, and we briefly discussed generating functions (and the so-called Z-transform) to provide a discrete analogue of the Laplace transform. The unifying theme of the class remained linearity and related spectral methods. The course sketched a proof of the Sturm-Liouville theorem, but filling in the details was not possible because student backgrounds in analysis were so varied. The book itself includes much more material than was discussed in the course, and it is quite flexible as a potential text. Mathematical prerequisites include three semesters of calculus, a beginning course in differential equations, and some elementary linear algebra. Complex variable theory is developed in the book, and hence is not a prerequisite. First year courses in Physics are useful but not strictly necessary. Several possible courses can be based upon this book. Covering the first three chapters in the first half of the semester and then choosing topics from the last four chapters in the second half make a nice course at the junior or senior level. Such a course should appeal to faculty in engineering who want an integrated treatment of linear algebra and complex analysis, including applications and also reviewing vector analysis. After covering the first four chapters, students in mathematics and physics can continue with the Hilbert space chapter and conclude with the sections on probability and quantum mechanics. The first five chapters together with the last section of Chapter 7 make a coherent applied complex variables course. Students with strong backgrounds can read the first three chapters on their own as review, and take a one semester course based on the last four chapters. Such a course would be ideal for many graduate students in applied mathematics, physics, and engineering. More than 300 exercises of considerably varying difficulty and feeling appear here. They are placed at the ends of sections, rather than at the end of chapters, to encourage the reader to solve them as she reads. Many of the exercises are routine and a few are difficult. Others introduce new material or new settings where the techniques can be used. Most of the exercises come from mathematics; users of the book can provide additional exercises coming from engineering and physics. The text weaves abstract mathematics, routine computational problems, and applications into a coherent whole, whose unifying theme is linear systems. The author hopes that readers from diverse backgrounds will appreciate the mathematics, while recognizing that not all readers will find the many topics equally compelling. The book itself has three primary goals. One goal is to develop enough linear analysis and complex variable theory to prepare students in engineering or applied mathematics for advanced work. The second goal is to unify many distinct and seemingly isolated topics. The third goal is to reveal mathematics as both 6 CONTENTS interesting and useful, especially via the juxtaposition of examples and theorems. We give some examples of how we achieve these goals: • A Fourier series is a particular type of orthonormal expansion; we include many examples of orthonormal expansion as well as the general theory. • Many aspects of the theory of linear equations are the same in finite and infinite dimensions; we glimpse both the similarities and the differences in Chapter 1 and develop the ideas in detail in Chapter 6. • One recurring theme concerns functions of operators. We regard diagonalization of a matrix and taking Laplace or Fourier transforms as similar (pun intended) techniques. Early in the course, and somewhat out of the blue, the author asked the class whatp would it mean to take “half of a derivative”. One bright engineering student said “multiply by s”. • The idea of choosing coordinates and notation in which computations are easy is a major theme in this book. Complex variables illustrate this theme throughout. For example, we use complex exponentials rather than trig functions nearly always. • We state the spectral theorem for compact operators on Hilbert space as part of our discussion of Sturm-Liouville equations, hoping that many readers will want to study more operator theory. • We express Maxwell’s equations in terms of both vector fields and differential forms, hoping that many readers want to study differential geometry. Many of the examples will be accessible yet new to nearly all readers. The author hopes the reader appreci- ates these examples as both science and art. This book contains many topics and examples not found in other books on engineering mathematics. Here are some examples. To illustrate the usefulness of changing basis, we determine which polynomials with rational coefficients map the integers to the integers. We mention the scalar field consisting of two elements, and include several exercises about lights-out puzzles to illuminate linear algebra involving this field. We compute line and surface integrals using both differential forms and the classical notation of vector analysis, and we include a section on Maxwell’s equations expressed using differential forms. To illustrate sine series, we prove the Wirtinger inequality relating the L2 norms of a function and its derivative. Throughout the book we develop connections between finite and infinite dimensional linear algebra, and many of these ideas will be new to most readers. For example, we compute ζ(2) by finding the trace of the Green’s operator for the second derivative. We introduce enough probability to say a bit about quantum mechanics; for example, we prove the Heisenberg uncertainty principle. We also include exercises on dice with strange properties. Our discussion of generating functions (Z-transforms) includes a derivation of PN p a general formula for the sums n=1 n (using Bernoulli numbers) as well as (the much easier) Binet’s formula for the Fibonacci numbers. The section on generating functions contains several unusual exercises, including a test for rationality and the notion of Abel summability. There we also discuss the precise meaning −1 of the notorious but dubious formula 1 + 2 + 3 + ::: = 12 , and we mention the concept of zeta function regularization.