Explorations in Complex Functions

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