# Theory of Functions of a Complex Variable by Markushevich

• Analysis
• micromass
In summary, A.I. Markushevich's "Theory of Functions of a Complex Variable" is a two-volume set that covers the basic concepts, sets and functions, limits and continuity, connectedness, curves and domains, and infinity and stereographic projections in volume I. Volume II discusses Laurent series, calculus of residues, inverse and implicit functions, univalent functions, and subharmonic and meromorphic functions, with applications to fluid dynamics and infinite products. The books also include a bibliography and index for further reference. These volumes are geared towards undergraduates with a background in real analysis.

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Code:
[LIST]
[*] Basic Concepts
[LIST]
[*] Introduction
[LIST]
[*] Analytic Functions of a Real Variable
[*] Infinitel y DifferentiabJe Functions
[*] Motivation for Introducing the Complex Numbers. A Preview of Analytic Functions of a Complex VariabIe
[*] Problems
[/LIST]
[*] Complex Numbers
[LIST]
[*] Geometric Representation of Complex Numbers
[*] Compiex Algebra
[*] Powers and Roots of Complex Numbers
[*] Problems
[/LIST]
[*] Sets and Functions, Limits and Continuity
[LIST]
[*] Some Basic Definitions
[*] Sequences al Complex Numbers. Limit Points and Limits of Sequences
[*] Convergence of the Real and Imaginary Parts, Moduli and Arguments of a Complex Sequence
[*] Series with Complex Terms
[*] Limit Points of Sets Bounded Sets
[*] The Lìmit of a FunctÌon of a Complex Variabie
[*] Continuous Functions. More Set Theory
[*] The Distance between Two Sets
[*] Problems
[/LIST]
[*] Connectedness, Curves and Domains
[LIST]
[*] Connected Sets. Continuous Curves and Continua
[*] Domains. Interior. Exterior and Boundary Points
[*] Simply and Multiply Connected Domains
[*] The Jordan Curve Theorem
[*] Some Further Results
[*] Problems
[/LIST]
[*] Infinity and Stereographic Projections
[LIST]
[*] Proper and Improper Complex Numbers
[*] Stereographic projection. Sets of Points on the Riemann Sphere
[*] The Extended Complex Plane, The Point at Infinity
[*] Conformality of Stereographic Projeçtion. Continuous Curves in the Extended Plane
[*] The Transformation \zeta = 1/z
[*] Another Definition of an Angle wiih its Vertex at Infinity
[*] Problems
[/LIST]
[*] Homeomorphisms
[LIST]
[*] The One-to-One Continuous Image of a Domain
[*] Some Further Results
[*] Problems
[/LIST]
[/LIST]
[*] Differentiation. Elementary Functions
[LIST]
[*] Differentiation and the Cauchy-Riemann Equations
[LIST]
[*] Derivatives and Diffentials
[*] Rules for Differentiating Functions of a Complex Variabie
[*] The Cauchy-Riemann Equations. Analytic Functions
[*] Problems
[/LIST]
[*] Geometric Interpretation of the Derivative. Conformal Mapping
[LIST]
[*] Geometric Interpretation of Arg f'(z)
[*] Geometric Interpretation of |f'(z)|
[*] The Mapping w=\frac{az+b}{cz+d}
[*] Conformal Mapping of the Extended Plane
[*] Problems
[/LIST]
[*] Elementary Entire Functions
[LIST]
[*] Polynomials
[*] The Mapping w = P_n(z)
[*] The Mapping w = (z - a)^n
[*] The Exponential
[*] The Mapping w = e^z
[*] Some Functions Related to the Exponential
[*] The Mapping w = cos Z
[*] The Image of a Half-Strip under w = cos z
[*] Problem
[/LIST]
[*] Elementary Meromorphic Functions
[LIST]
[*]
[*] Rationa] Functions
[*] The Group Property of Mobius Transformations
[*] The Circle-Preserving Property of Mobius Transforrnations
[*] Fixed Points of a Mobius Transformation. Invariance of the Cross Ratio
[*] Mapping of a Cirde onto a Circle
[*] Symmetry Transformations
[*] Examples
[*] Lobachevskìan Geometry
[*] The Mapping w = \frac{1}{2} ( z + \frac{1}{z} )
[*] Transcendental Meromorphic Functions. Trigonometric Functions
[*] Probems
[/LIST]
[*] Elementary Multiple-Valued Functions
[LIST]
[*] Sing]e-Valued Branches. Univalent Functions
[*] The Mapping w = \sqrt[n]{z}
[*] The Mapping w = \sqrt[n]{P(z)}
[*] The Logarithm,
[*] The Function z^a. Exponentials and Logarithms to an Arbitrary Base
[*] The Mapping w = Arc cos z
[*] The Mapping w = l + ln z
[*] Problems
[/LIST]
[/LIST]
[*] Integration, Power Series
[LIST]
[*] Rectifiable Curves. Complex Integrals
[LIST]
[*] Some Basic Detinitions
[*] Integra]s of Complex Functions
[*] Properties of Complex lntegrals
[*] Problems
[/LIST]
[*] Cauchy's Integral Theorem
[LIST]
[*] A Preliminary Result
[*] The Key Lemma
[*] Proof of Cauchy's IntegraI Theorem
[*] Application to the Evaluation of Definite Integrals
[*] Cauchy's Integral Theorem for a System of Contours
[*] Path-Independent Integrals. Primitives
[*] The Integra] as a Function of Its Upper Limit in a Multiply Connected Domain
[*] Problems
[/LIST]
[*] Cauchy's Integral and Related Topics
[LIST]
[*] Cauchy's lntegral Formula
[*] Some Consequences of Theorem 14.1
[*] Integrals of the Cauchy Type. Cauchy's Inequalities
[*] Boundary Values of Integrals of the Cauchy Type
[*] The Plemelj FormuJas
[*] Problems
[/LIST]
[*] Uniform Convergence. Infinite Products
[LIST]
[*] Uniformly Convergent Series
[*] Uniformly Convergent Sequences. Improper Integrals of the Cauchy Type
[*] Infinite Products
[*] Problems
[/LIST]
[*] Power Series: Rudiments
[LIST]
[*] The Cauchy-Hadamard Formula
[*] Taylor's Series. Tbe Uniqueness Property
[*] The Relation between Power Series and Fourier Series
[*] Expansion of an Analytic Function in Power Serìes
[*] Problems
[/LIST]
[*] Power Series: Ramifications
[LIST]
[*] The Interior Uniqueness Theorem. A-Points of Analytic Functions
[*] The Maximum Modulus Principle and Some of Its Consequences. Lemniscates.
[*] Circular Elements. Regular and Singular Points
[*] Behavior of a Power Series on lts Cirele of Convergence
[*] Compact Families of Analytic Functions
[*] Vitali's Theorem. Analytic Functions Defined by Integrals
[*] Problems
[/LIST]
[*] Methods for Expanding Functions in Taylor Series
[LIST]
[*] The Taylor Series of the Sum of a Series of Analytic Functions
[*] The Taylor Series of a Composite Function
[*] Division of Power Series
[*] Prob1ems
[/LIST]
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

Code:
[LIST]
[*] Laurent Series. Calculus of Residues
[LIST]
[*] Laurent's Series. Isolated Singular Points
[LIST]
[*] Laurent's Theorem
[*] Poles and Essential Singular Points
[*] Singular Points or f(z) \pm g(z), f(z)g(z) and f(z)/g(z)
[*] Behavior at Infinity. The Poles of g(z)(d/dz) Ln [f(z) - A]
[*] Dirichlet Series
[*] Problems
[/LIST]
[*] The Calculus of Residues and Its Applications
[LIST]
[*] The Residue Theorem
[*] The Argument Principle. The Theorems of Rouché and Hurwitz
[*] Residues at Infinity
[*] Cauchy's Theorem on Partial Fraction Expansions
[*] Examples or Partial Fraction Expansions
[*] Interpolation Theory
[*] Problems
[/LIST]
[*] Inverse and Implicit Functions
[LIST]
[*] Inverse Functions: The Single-Valued Case
[*] Inverse Functions: The Multìple Valued Case
[*] Examples of Lagrange's Series
[*] Functions of Two Complex Variables
[*] Weierstrass' Preparation Theorem. The Implicit Function Theorem
[*] Problems
[/LIST]
[*] Univalent Functions
[LIST]
[*] Some Elementary Results
[*] Sufficient Conditions for Univalence
[*] Mapping of the Upper Half-Plane onto a Rectangle
[*] The Schwarz-Christoffel Transformation
[*] Sufficient Conditions for Univalent Mapping onto a Half-Plane
[*] Problems
[/LIST]
[/LIST]
[*] Harmonic and Subharmonic Functions
[LIST]
[*] Basic Properties of Harmonic Functions
[LIST]
[*] Laplace's Equation. Conjugate Harmonic Functions
[*] Poisson's Integral. Schwarz's Formula
[*] The Dirichlet Problem for a Disk
[*] Behavior of a Harmonic Function near an Isolated Singular Point
[*] Sequences of Harmonic Functions. Harnack's Theorem
[*] Generalizatìon of Poisson's Integral. The Dirichlet Problem for a Jordan Domain
[*] Problems
[/LIST]
[*] Applications to Fluid Dynamics
[LIST]
[*] Irrotational and Solenoidal Flows. The Complex Potential
[*] Examples
[*] Flow past a Circular Cylinder
[*] Flow past an Arbitrary Cy1indrical Object. The Kutta-Joukowski Theorem
[*] Problems
[/LIST]
[*] Subharmonic Functions
[LIST]
[*] The Key Lemma. The Converse of Theorem 5.6
[*] The Generalized Maximum Modulus Principle and Its Application
[*] The Phragmén-LindelOf Theorems
[*] Problems
[/LIST]
[*] The Poisson-Jensen Formula and Related Topics
[LIST]
[*] Various Versions of the Poisson-Jensen Formula
[*] Jensen's Inequality, Blaschke Products
[*] Functions of Bounded Characteristic
[*] Nevanlinna's Theorem
[*] Problems
[/LIST]
[/LIST]
[*] Entire and Meromorphic Functions
[LIST]
[*] Basic Properties of Entire Functions
[LIST]
[*] Growth of an Entire Function
[*] Behavior of e^{P(z)}
[*] Order and Type in Terms of the Taylor Coefficients
[*] Distribution of Zeros
[*] A-Points of Entire Functions
[*] Picard's First Theorem
[*] The Phragmén-Lindelof Indicator Function
[*] Problems
[/LIST]
[*] Infinite Products and Partial Fraction Expansions
[LIST]
[*] Weierstrass' Theorem
[*] The Exponent of Convergence
[*] Hadamard's Factorization Theorem
[*] Borel's Theorem
[*] Meromorphic Functions
[*] Mittag-Leffler's Theorem
[*] The Gamma Function
[*] Integral Representations of \Gamma(z). Partial Fraction Expansion of \Gamma(z)
[*] Asymptotc Behavior of \Gamma(z). Stirling's Formula
[*] Problems
[/LIST]
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

Code:
[LIST]
[*] Conformal Mapping. Approximation Theory
[LIST]
[*] Conformal Mapping: Rudiments
[LIST]
[*] Conformal Mapping of Annular Domains
[*] Conformal Mapping of Simply Connected Domains
[*] Basic Properties of Univalent Functions
[*] Problems
[/LIST]
[*] Conformal Mapping: Ramifications
[LIST]
[*] Conformal Mapping of Sequences of Domains
[*] Curvilinear Ha1f-Intervals
[*] Accessible Boundary Points
[*] Prime Ends
[*] Boundary Behavior of Conformal Mappings
[*] Problems
[/LIST]
[*] Approximation by Rational Functions and Polynomials
[LIST]
[*] Locally Analytic Functions
[*] Functions Meromorphic on a Domain
[*] Runge's Theorem and Related Results
[*] ApproximatÌon on Closed Domains
[*] Approximation on Continua
[*] Faber Polynomials
[*] Bernstein's Theorem
[*] Approximation in the Mean
[*] Polynomials Orthogonal on a Domain
[*] Problems
[/LIST]
[/LIST]
[*] Periodic and Elliptic Functions
[LIST]
[*] Periodic Meromorphic Functions
[LIST]
[*] Preliminaries
[*] Periodic Entire Functions. Trigonometric Polynomials
[*] Elliptic Functions
[*] Problems
[/LIST]
[*] Elliptic Functions: Weierstrass Theory
[LIST]
[*] Weierstrass' Elliptic Functions
[*] The Functions P(z | a, ib) and P(z | a - ib, a + ib)
[*] The Differential Equation for P(z)
[*] Inversion of Elliptic Integrals
[*] The Functions \xi(z) and \sigma(z)
[*] The Addition Theorem for P(z)
[*] The Spherical Pendulum
[*] Problems
[/LIST]
[*] Elliptic Functions: Jacobi's Theory
[LIST]
[*] Jacobi's Elliptic Functions
[*] Theta Functions and Their Relation to Elliptic Functions
[*] Infinite Product Expansions of Theta Functions
[*] Problems
[/LIST]
[/LIST]
[*] Riemann Surfaces, Analytic Continuation
[LIST]
[*] Riemann Surfaces
[LIST]
[*] Topological Preliminaries
[*] Abstract Riernann Surfaces
[*] Triangulations
[*] Interior Mappings
[*] Riemann Covering Surfaces
[*] Regular Analytic Curves
[*] The Riemann Surface of a Meromorphic Function
[*] Examples
[*] Problem
[/LIST]
[*] Analytic Continuation
[LIST]
[*] Elements. The Complete Analytic Function
[*] Circular Elements. The Monodromy Theorem.
[*] Analytic Continuation in a Star
[*] Singular Points. Generalized Elements and the Analytic Configuration
[*] The Ana]ytic Configuration as a Topological Surface
[*] The Analytic Configuration as a Riemann Surface
[*] Algebraic Functions
[*] Problems
[/LIST]
[*] The Symmetry Principle and Its Applications
[LIST]
[*] The Symmetry Principle
[*] More on the Schwarz-Christoffel Transformation
[*] Examples
[*] The Modular Function. Picard's First Theorem
[*] Normal Families of Analytic Functions
[*] Picard's Second Theorem. Julia Directions
[*] Problems
[/LIST]
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

Last edited by a moderator:
This book doesn't play around. It's got complete sections & proofs of things like the Jordan Curve theorem & big Picard, & I guess it probably should considering it's 1200 pages. Definitely not for an intro. It's actually the main companion text to the Volkovyskii book, except the sections (in my copy anyway) are out of order. I mean Volkovyskii will reference a section in this one but the numbering or content won't make any sense. Other than that I wouldn't know what other 'editorializing' the translator has done.

Nathanmaul

## What is the Theory of Functions of a Complex Variable?

The Theory of Functions of a Complex Variable is a branch of mathematics that deals with the study of functions defined on the complex numbers. It is a fundamental tool in the fields of physics, engineering, and other areas of mathematics.

## Who is Markushevich and what is his contribution to the Theory of Functions of a Complex Variable?

Markushevich was a prominent Russian mathematician who made significant contributions to the Theory of Functions of a Complex Variable. He is best known for his two-volume textbook, "Theory of Functions of a Complex Variable," which has become a standard reference for students and researchers in the field.

## What are some applications of the Theory of Functions of a Complex Variable?

The Theory of Functions of a Complex Variable has numerous applications in various fields such as physics, engineering, computer science, and economics. It is used to model and analyze complex systems, solve differential equations, and understand the behavior of physical phenomena.

## What are some important concepts in the Theory of Functions of a Complex Variable?

Some important concepts in the Theory of Functions of a Complex Variable include analytic functions, Cauchy-Riemann equations, contour integration, and the fundamental theorem of algebra. These concepts are essential for understanding the behavior and properties of complex functions.

## How does the Theory of Functions of a Complex Variable differ from the Theory of Real Functions?

The Theory of Functions of a Complex Variable is a generalization of the Theory of Real Functions. It deals with functions of a complex variable, which has two components (real and imaginary), while the Theory of Real Functions deals with functions of a real variable. The Theory of Functions of a Complex Variable has its own unique properties and techniques, such as the use of contour integration and Cauchy's integral theorem.

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