# Theory of Functions of a Complex Variable by Markushevich

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## Main Question or Discussion Point

Code:
[LIST]
[*] Basic Concepts
[LIST]
[*] Introduction
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[*] 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
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[*] Complex Numbers
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[*] Geometric Representation of Complex Numbers
[*] Compiex Algebra
[*] Powers and Roots of Complex Numbers
[*] Problems
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[*] Sets and Functions, Limits and Continuity
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[*] 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
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[*] Connectedness, Curves and Domains
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[*] 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
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[*] Infinity and Stereographic Projections
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[*] 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
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[*] Homeomorphisms
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[*] The One-to-One Continuous Image of a Domain
[*] Some Further Results
[*] Problems
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[*] Differentiation. Elementary Functions
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[*] Differentiation and the Cauchy-Riemann Equations
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[*] Derivatives and Diffentials
[*] Rules for Differentiating Functions of a Complex Variabie
[*] The Cauchy-Riemann Equations. Analytic Functions
[*] Problems
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[*] Geometric Interpretation of the Derivative. Conformal Mapping
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[*] 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
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[*] Elementary Entire Functions
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[*] 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
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[*] Elementary Meromorphic Functions
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[*]
[*] 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
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[*] Elementary Multiple-Valued Functions
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[*] 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
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[*] Integration, Power Series
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[*] Rectifiable Curves. Complex Integrals
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[*] Some Basic Detinitions
[*] Integra]s of Complex Functions
[*] Properties of Complex lntegrals
[*] Problems
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[*] 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
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[*] Cauchy's Integral and Related Topics
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[*] 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
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[*] Uniform Convergence. Infinite Products
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[*] Uniformly Convergent Series
[*] Uniformly Convergent Sequences. Improper Integrals of the Cauchy Type
[*] Infinite Products
[*] Problems
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[*] Power Series: Rudiments
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[*] Taylor's Series. Tbe Uniqueness Property
[*] The Relation between Power Series and Fourier Series
[*] Expansion of an Analytic Function in Power Serìes
[*] Problems
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[*] Power Series: Ramifications
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[*] 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
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[*] 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
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[/LIST]
[*] Bibliography
[*] Index
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Code:
[LIST]
[*] Laurent Series. Calculus of Residues
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[*] Laurent's Series. Isolated Singular Points
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[*] 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
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[*] 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
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[*] 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
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[*] 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
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[*] Harmonic and Subharmonic Functions
[LIST]
[*] Basic Properties of Harmonic Functions
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[*] 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
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[*] 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
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[*] 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
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[*] 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
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[/LIST]
[*] Entire and Meromorphic Functions
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[*] Basic Properties of Entire Functions
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[*] 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
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[*] Infinite Products and Partial Fraction Expansions
[LIST]
[*] Weierstrass' Theorem
[*] The Exponent of Convergence
[*] 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
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[/LIST]
[*] Bibliography
[*] Index
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Code:
[LIST]
[*] Conformal Mapping. Approximation Theory
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[*] Conformal Mapping: Rudiments
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[*] Conformal Mapping of Annular Domains
[*] Conformal Mapping of Simply Connected Domains
[*] Basic Properties of Univalent Functions
[*] Problems
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[*] 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
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[/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
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[/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
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[*] 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
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[/LIST]
[*] Bibliography
[*] Index
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