Explorations in Complex Functions: 287

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