Functions of a complex variable
Authors
Description
Since its creation at the end of the 18th century, the theory of functions of complex variables has proved to be one of the most fruitful in the global context of mathematics. It has made it possible, for example, to better understand functions defined by power series, to establish important relationships between elementary functions, to make sense of the statement “Every polynomial equation has at least one solution”, among other equally important achievements.
In this elementary book, the author aims to introduce some of the basic aspects of the theory, which are necessary for understanding more advanced aspects. It is aimed at undergraduates and masters students of all disciplines who use mathematics as an essential tool. The subjects are ordered in order of increasing difficulty, approaching them in the most elementary way possible, assuming that the reader only has a good knowledge of calculus, Topology of R n and notions of convergence in function spaces (uniform convergence).
Target audience
Higher education
Name: Functions of a complex variable
Author(s): e Alcides Lins Neto
Pages: 384
Publication: IMPA, 2025
ISBN: 978-85-244-0498-6
Edition: 4
1 The Body of Complex Numbers
1.1 Complex Numbers
1.1.1 The set of complexes as a body
1.1.2 Cartesian representation and polar representation
1.1.3 Distance and fundamental inequalities
1.1.4 Limits of sequences
1.1.5 Infinite limits
1.1.6 Fundamental notions of the topology of C
1.1.7 Limits of functions
1.2 Series of complex numbers
1.2.1 Cauchy criterion
1.2.2 Reordering series
1.2.3 Summable families and double series
1.3 Spaces of Continuous Functions
1.3.1 Uniform convergence
1.3.2 Uniform convergence in compacts
2 Analytic Functions
2.1 Holomorphic functions
2.1.1 Real derivative
2.1.2 Complex derivative, holomorphic functions
2.1.3 Compliant applications
2.1.4 The inverse function theorem
2.2 Power Series
2.2.1 Functions defined by power series
2.2.2 Operations with power series
2.3 Exponential and Logarithm
2.3.1 The exponential function
2.3.2 The complex logarithm
2.3.3 Roots and generalized powers
2.3.4 Complex trigonometric functions
2.4 Analytical functions of a complex variable
2.4.1 Definition and examples
2.4.2 Zeros of an analytic function
2.4.3 The ring of analytic functions
3 Integration in the complex plane
3.1 Differential forms
3.1.1 Definition and examples
3.1.2 Integration of differential forms along paths
3.1.3 Integration of exact and closed 1-forms
3.1.4 Integration of closed forms along continuous paths
3.2 Homotopy and Integration
3.2.1 Homotopy
3.2.2 Integration of closed forms along homotopic paths
3.2.3 Index of a closed path
3.3 Jordan’s and Green’s theorems
3.3.1 Regions bounded by Jordan curves
3.3.2 Green’s theorem
4 Cauchy Theory
4.1 The Cauchy-Goursat Theorem
4.2 Cauchy’s integral formula and applications
4.2.1 Cauchy’s integral formula
4.2.2 Analyticity of holomorphic functions
4.2.3 The maximum modulus theorem
4.2.4 Schwarz’s reflection principle
4.3 Laurent series
4.3.1 Analytic functions on a ring
4.3.2 Isolated singularities of analytic functions
4.4 Waste Theory
4.4.1 Definition and examples
4.4.2 The residue theorem
4.4.3 Poles and zeros of meromorphic functions
4.4.4 Calculating definite integrals
4.5 The Riemann sphere
4.5.1 Constructions of the Riemann sphere
4.5.2 Holomorphic functions of the Riemann sphere
4.5.3 Differential forms and the residue theorem in C
4.5.4 Rational functions
5 Sequences, Series and Products of Holomorphic and Meromorphic Functions
5.1 The spaces of holomorphic and meromorphic functions
5.1.1 The topology of uniform convergence in compact parts
5.1.2 Sequences of meromorphic functions
5.1.3 Series of meromorphic functions
5.2 Normal families of holomorphic and meromorphic functions
5.2.1 The Arzelà-Ascoli Theorem
5.2.2 Normal families of holomorphic functions
5.2.3 Normal families of meromorphic functions
5.3 Doubly periodic functions
5.3.1 Periods of a meromorphic function
5.3.2 Doubly periodic functions
5.3.3 The Weierstrass functionP
5.4 Infinite products and the Weierstrass theorem
5.4.1 Infinite number products
5.4.2 Infinite products of holomorphic functions
5.4.3 The Weierstrass Factorization Theorem
5.5 Gamma and Zeta functions
5.5.1 The Gamma function
5.5.2 The Riemann Zeta function
5.6 Approximation of analytic functions by rational functions
5.6.1 Runge’s Theorem
5.6.2 The Mittag-Leffler Theorem
6 Riemann’s Uniformization Theorem
6.1 Conformal equivalences
6.1.1 Notations and elementary properties
6.1.2 Examples
6.2 Automorphisms of C and the unit disk
6.2.1 Some properties of homographies
6.2.2 The cross ratio
6.2.3 Holomorphic automorphisms of the unit disk
6.2.4 Anti-holomorphic automorphisms of C
6.3 Riemann’s Theorem
6.3.1 Proof of Riemann’s Theorem
6.3.2 Classification of simply connected open subsets of C .
6.3.3 A characterization of simply connected open subsets of C
6.3.4 Proof of Lemma 6.1
Bibliography
Author Index
Index