Explorations in Complex Functions: 287

Description

This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book. Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method. Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout. Preface Contents 1 Basics 1.1 The Cauchy–Riemann equations and Cauchy's integral theorem 1.2 The Cauchy integral formula and applications 1.3 Change of contour, isolated singularities, residues 1.4 The logarithm and powers 1.5 Infinite products 1.6 Reflection principles 1.7 Analytic continuation 1.8 The Stieltjes integral 1.9 Hilbert spaces 1.10 Lp spaces Remarks and further reading 2 Linear Fractional Transformations 2.1 The Riemann sphere 2.2 The cross-ratio and mapping properties of linear fractional transformations 2.3 Upper half plane and unit disk Exercises Remarks and further reading 3 Hyperbolic geometry 3.1 Distance-preserving transformations and ``lines'' 3.2 Construction of a distance function 3.3 The triangle inequality 3.4 Distance and area elements Exercises Remarks and further reading 4 Harmonic functions 4.1 Harmonic functions and holomorphic functions 4.2 The mean value property, the maximum principle, and Poisson's formula 4.3 The Schwarz reflection principle 4.4 Application: approximation theorems Exercises Remarks and further reading 5 Conformal maps and the Riemann mapping theorem 5.1 Conformal maps 5.2 The Riemann mapping theorem 5.3 Proof of Lemma 5.2.2; the Ascoli–Arzelà theorem 5.4 Boundary behavior of conformal maps 5.5 Mapping polygons: the Schwarz–Christoffel formula 5.6 Triangles and rectangles 5.7 Univalent functions Exercises Remarks and further reading 6 The Schwarzian derivative 6.1 The Schwarzian derivative as measure of curvature 6.2 Some properties of the Schwarzian 6.3 The Schwarzian and curves 6.4 The Riemann mapping function and the Schwarzian 6.5 Triangles and hypergeometric functions 6.6 Regular polygons and hypergeometric functions Exercises Remarks and further reading 7 Riemann surfaces and algebraic curves 7.1 Analytic continuation 7.2 The Riemann surface of a function 7.3 Compact Riemann surfaces 7.4 Algebraic curves: some algebra 7.5 Algebraic curves: some analysis 7.6 Examples: elliptic and hyperelliptic curves 7.7 General compact Riemann surfaces 7.8 Algebraic curves of higher genus Exercises References and further reading 8 Entire functions 8.1 The Weierstrass product theorem 8.2 Jensen's formula 8.3 Functions of finite order 8.4 Hadamard's factorization theorem 8.5 Application to Riemann's xi function 8.6 Application: an inhomogeneous vibrating string Exercises Remarks and further reading 9 Value distribution theory 9.1 The Nevanlinna characteristic and the first fundamental theorem 9.2 The first fundamental theorem and a modified characteristic 9.3 The second fundamental theorem 9.4 Applications 9.5 Further properties of meromorphic functions Exercises Remarks and further reading 10 The gamma and beta functions 10.1 Euler's product solution 10.2 Euler's integral solution and the beta function 10.3 Legendre's duplication formula 10.4 The reflection formula and the product formula for sine 10.5 Asymptotics of the gamma function Exercises Remarks and further reading 11 The Riemann zeta function 11.1 Properties of ζ 11.2 The functional equation of the zeta function 11.3 Zeros 11.4 ζ(2m) 11.5 The function ξ(s) Exercises References and further reading 12 L-functions and primes 12.1 Factorization and Dirichlet characters 12.2 Characters of finite commutative groups 12.3 Analysis of L-functions 12.4 Proof of Dirichlet's Theorem 12.5 Functional equations 12.6 Other L-functions: algebraic and automorphic Exercises Remarks and further reading 13 The Riemann hypothesis 13.1 Primes and zeros of the zeta function 13.2 von Mangoldt's formula for ψ 13.3 The prime number theorem 13.4 Density of the zeros 13.5 The Riemann hypothesis and Gauss's approximation 13.6 Riemann's 1859 paper 13.7 Inverting the Mellin transform of ψ Exercises References and further reading 14 Elliptic functions and theta functions 14.1 Elliptic functions: generalities 14.2 Theta functions 14.3 Construction of elliptic functions 14.4 Integrating elliptic functions Exercises Remarks and further reading 15 Jacobi elliptic functions 15.1 The pendulum equation 15.2 Properties of the map F 15.3 The Jacobi functions 15.4 Elliptic curves: Jacobi parametrization Exercises Remarks and further reading 16 Weierstrass elliptic functions 16.1 The Weierstrass function 16.2 Integration of elliptic functions 16.3 Elliptic curves: Weierstrass parametrization 16.4 Addition on the curve Exercises Remarks and further reading 17 Automorphic functions and Picard's theorem 17.1 The elliptic modular function 17.2 The modular group and the fundamental domain 17.3 A closer look at λ; Picard's theorem 17.4 Automorphic functions; the J function Exercises Addendum: Moonshine References and further reading 18 Integral transforms 18.1 Approximate identities and Schwartz functions 18.2 The Cauchy Transform and the Hilbert transform 18.3 The Fourier transform 18.4 The Fourier transform for L1(mathbbR) 18.5 The Fourier transform for L2(mathbbR) Exercises Remarks and further reading 19 Theorems of Phragmén–Lindelöf and Paley–Wiener 19.1 Phragmén–Lindelöf theorems 19.2 Hardy's uncertainty principle 19.3 The Paley–Wiener Theorem 19.4 An application Exercises Remarks and further reading 20 Theorems of Wiener and Lévy; the Wiener–Hopf method 20.1 The ring mathcalR 20.2 Convolution equations 20.3 The case of real zeros of 1-k"0362k Exercises Remarks and further reading 21 Tauberian theorems 21.1 Hardy's theorem 21.2 Abel, Tauber, Littlewood, and Hardy–Littlewood 21.3 Karamata's tauberian theorem 21.4 Wiener's tauberian theorem 21.5 A theorem of Malliavin and applications Exercises Remarks and Further Reading 22 Asymptotics and the method of steepest descent 22.1 The method of steepest descent 22.2 The Airy integral 22.3 The partition function and the Hardy–Ramanujan formula 22.4 Proof of the functional equation (22.3.6) Exercises Remarks and further reading 23 Complex interpolation and the Riesz–Thorin theorem 23.1 Interpolation: the complex method 23.2 Lp spaces 23.3 Application: the Riesz–Thorin theorem 23.4 Application to Fourier series Exercises Remarks and further reading References Index