Functions of a complex variable
Authors
Description
Since its creation in the late 18th century, the theory of functions of complex variables has proven to be one of the most fruitful in the global context of Mathematics. Through it, it has been possible, for example, to better understand functions defined by power series, establish important relationships between elementary functions, give meaning to 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, necessary for understanding more advanced aspects. It is intended for undergraduate and graduate students in all disciplines that use mathematics as an essential tool. The topics are ordered in increasing order of 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
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 complex numbers as a field
1.1.2 Cartesian and polar representations
1.1.3 Distance and fundamental inequalities
1.1.4 Sequence Limits
1.1.5 Infinite Limits
1.1.6 Fundamental concepts of C topology
1.1.7 Limits of functions
1.2 Series of complex numbers
1.2.1 Cauchy Criterion
1.2.2 Series reordering
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 compact cars
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 Generalized roots and powers
2.3.4 Complex trigonometric functions
2.4 Analytic 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 into the complex plan
3.1 Differential Forms
3.1.1 Definition and examples
3.1.2 Integration of differential forms in paths
3.1.3 Integration of 1 – exact and closed forms
3.1.4 Integration of closed forms along continuous paths
3.2 Homeopia and Integration
3.2.1 Homeopia
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's Theory
4.1 The Cauchy–Goursat Theorem
4.2 Cauchy's complete formula and applications
4.2.1 Cauchy integral formula
4.2.2 Analyticity of holomorphic functions
4.2.3 The Maximum Modulus Theorem
4.2.4 Schwarz's principle of reflection
4.3 Laurent Series
4.3.1 Analytic functions in a ring
4.3.2 Isolated Singularities of Analytic Functions
4.4 Residue 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 Calculation of 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 Holomorphic and meromorphic function spaces
5.1.1 The topology of uniform convergence in compact parts
5.1.2 Sequences of meromorphic functions
5.1.3 Meromorphic function series
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 P-function
5.4 Infinite Products and Weierstrass's Theorem
5.4.1 Infinite numerical products
5.4.2 Infinite products of holomorphic functions
5.4.3 Weierstrass's Factorization Theorem
5.5 The 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 Uniformization Theorem
6.1 Conformal Equivalencies
6.1.1 Elementary notations and properties
6.1.2 Examples
6.2 Automorphisms of C and the unit disk
6.2.1 Some properties of homography
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 the simply connected open sets of C
6.3.4 Proof of Lemma 6.1
Bibliography
Index of Authors
Index