This is a textbook in the theory of analytic functions with coverage adequate for the syllabus of physico-mathematical faculties of universities. Numerous worked-out examples of the general ideas and methods are given. A small part of the material is not obligatory and only supplements the basic course. The interested reader may turn to the bibliography on the subject provided at the end of the book.
In preparing this textbook, I have drawn extensively on my other book, The Theory of Analytic Functions [English translation: A.I. Markushevich, Theory of Functions of a Complex Variable, 3 vols., Prentice-Hall, Englewood Cliffs, N.J., 1965-71].
Translated from the Russian by Eugene Yankovsky
About the Author
Aleksei Ivanovich Markushevich (1908–1979) received his degree of Doctor of Physico-Mathematical Sciences from Moscow State University in 1944. He served as a professor at Moscow University and became a Full Member of the Academy of Pedagogical Sciences of the USSR in 1967. He was also First Deputy Minister of Education in the Russian Federation.
Professor Markushevich’s research covered a wide range of topics in pedagogy, the history of science, and mathematics. His main interest was the theory of functions of a complex variable. He authored many mathematics textbooks, popular-science booklets, and numerous articles on the history of function theory, the theory of entire functions, and the theory of conformal mapping. His comprehensive course, Theory of Functions of a Complex Variable, was published in the United States by Prentice-Hall in three volumes. Professor Markushevich was Chief Editor of the Encyclopedia of Elementary Mathematics and the Children’s Encyclopedia.
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Contents
0. Introduction
0.1. The Theory of Analytic Functions
0.2. Analytic Functions of a Complex Variable
1. Complex Numbers and Their Geometric Interpretation
1.1. The Interpretation of Complex Numbers on a Plane
1.2. Operations on Complex Numbers
1.3. The Limit of a Sequence
1.4. Infinity and the Stereographic Projection
1.5. Sets of Points in a Plane
2. Functions of a Complex Variable. The Derivative and Its Geometric and Hydromechanical Interpretations
2.1. A Function of a Complex Variable
2.2. The Limit of a Function at a Point
2.3. Continuity
2.4. A Continuous Curve
2.5. The Derivative and the Differential
2.6. Rules of Differentiation
2.7. Necessary and Sufficient Conditions for Differentiability at an Inner Point of a Domain
2.8. The Geometric Interpretation of the Argument of a Derivative
2.9. The Geometric Interpretation of the Modulus of a Derivative
2.10. The Linear and Linear-Fractional Functions
2.11. An Angle with Its Vertex at the Point at Infinity
2.12. Quasiconformal Mappings
2.13. Harmonic and Conjugate Harmonic Functions
2.14. The Hydromechanical Interpretation of an Analytic Function
2.15. Examples
3. Elementary Analytic Functions and the Corresponding Conformal Mappings
3.1. The Polynomial
3.2. The Points at Which Conformal Mapping Does Not Work
3.3. The Mapping \(w=(z-a)^n\)
3.4. The Group Properties of Linear-Fractional Transformations
3.5. The Circular Property
3.6. The Invariance of the Anharmonic Ratio
3.7. Mapping Domains Bounded by Straight Lines or Circles
3.8. Symmetry and Its Conservation
3.9. Examples
3.10. The Zhukovskii (Joukowski) Function
3.11. The Exponential Function
3.12. Mapping by the Exponential Function
3.13. Trigonometric Functions
3.14. The Geometric Behaviour of Trigonometric Functions I
3.15. The Geometric Behaviour of Trigonometric Functions II
3.16. Branches of Multiple-Valued Functions
3.17. The Function \(w=\sqrt{z}\)
3.18. The Function \(w=\sqrt[n]{P(z)}\)
3.19. The Logarithm
3.20. The Power and Exponential Functions: The General Case
3.21. Inverse Trigonometric Functions
4. Series with Complex Terms. Power Series
4.1. Convergent and Divergent Series
4.2. The Cauchy–Hadamard Theorem
4.3. Analyticity of the Sum of a Power Series
4.4. Uniformly Convergent Series
5. Integrating Functions of a Complex Variable
5.1. An Integral with Respect to a Complex Variable
5.2. Basic Properties of Integrals
5.3. Reduction to Ordinary Integrals
5.4. Cauchy’s Integral Theorem
5.5. Proof of Cauchy’s Integral Theorem
5.6. Evaluation of Definite Integrals
5.7. The Integral and the Primitive
5.8. Generalisation of Cauchy’s Integral Theorem to Functions Non-analytic on the Integration Contour
5.9. The Composite Contour Theorem
5.10. Integrals in Multiply Connected Domains
6. Cauchy’s Integral Formula and Its Implications
6.1. Cauchy’s Integral Formula
6.2. Expanding an Analytic Function in a Power Series. Liouville’s Theorem
6.3. The Infinite Differentiability of Analytic and Harmonic Functions
6.4. Change of Variable Under the Integral Sign
6.5. Morera’s Theorem
6.6. Weierstrass’s Theorem on Uniformly Convergent Series of Analytic Functions
6.7. The Compactness Principle
6.8. The Uniqueness Theorem. Vitali’s Theorem
6.9. Runge’s Theorems
6.10. Integrals Dependent on a Parameter
6.11. \(A\)-points. The Maximum Modulus Principle
6.12. Power Series Expansions for Analytic Functions
6.13. Substituting a Series into a Series
6.14. Dividing Power Series
6.15. Power Series Expansions for \(\cot z\), \(\tan z\), \(\csc z\), and \(\sec z\)
6.16. Series Expansions for Harmonic Functions. The Poisson Integral and Schwarz’s Formula
6.17. Analytic Functions of Several Complex Variables
7. The Laurent Series. Isolated Singular Points of Single-Valued Analytic Functions. Entire and Meromorphic Functions
7.1. The Laurent Series
7.2. The Laurent Theorem
7.3. Isolated Singular Points of Single-Valued Analytic Functions
7.4. The Sokhotski–Casorati–Weierstrass Theorem
7.5. Singular Points of Derivatives and Rational Combinations of Analytic Functions
7.6. Singularities at Infinity
7.7. Entire and Meromorphic Functions
7.8. The Mittag-Leffler Theorem
7.9. Product Expansion of Entire Functions
7.10. The Gamma Function
7.11. An Integral Representation of the Gamma Function
7.12. The Order and Type of an Entire Function. Theorems of Hadamard and Borel
7.13. The Phragmén–Lindelöf Principle and Functions
7.14. Jensen’s Formula
7.15. Nevanlinna’s First Fundamental Theorem
8. Residues and Their Applications. The Argument Principle. Elliptic Functions
8.1. The Residue Theorem and the Evaluation of Definite Integrals
8.2. The Argument Principle and Its Implications
8.3. The Residue at the Point at Infinity
8.4. The Partial-Fraction Expansion of Meromorphic Functions by Means of the Residue Theorem
8.5. Partial-Fraction Expansions for \(\sec z\), \(\cot z\), \(\csc z\), and \(\tan z\)
8.6. Elliptic Functions
9. Analytic Continuation. The Riemann Surface. Singular Points
9.1. The Concept of Analytic Continuation
9.2. Direct Analytic Continuation
9.3. Constructing an Analytic Function from Its Elements
9.4. Constructing Riemann Surfaces
9.5. The Riemann–Schwarz Symmetry Principle
9.6. Singularities on the Boundary of the Convergence Circle of a Power Series
9.7. Criterion for Detecting Singular Points
9.8. Determining the Convergence Radius of a Power Series from the Distribution of Singularities
9.9. Branch Points
9.10. Analytic Continuation Along a Curve. The Monodromy Theorem
9.11. Analytic Continuation and the Borel Transformation
10. Mappings Performed by Analytic Functions. Riemann’s Mapping Theorem. The Schwarz–Christoffel Transformation Formula
10.1. Mapping a Domain by an Analytic Function
10.2. The Maximum Modulus Principle and Schwarz’s Lemma
10.3. Local Criterion for Univalence
10.4. Inversion of Analytic Functions
10.5. Extending the Concept of Univalence to Functions with Poles
10.6. Riemann’s Mapping Theorem. The Uniqueness of a Mapping
10.7. Correspondence Between Boundaries
10.8. Mapping the Upper Half-Plane by Means of an Elliptic Integral
10.9. Jacobi’s Elliptic Function \(\operatorname{sn} w\). Inversion of Ultraelliptic Integrals
10.10. The Schwarz–Christoffel Transformation Formula
10.11. Schwarz’s Modular Function and Picard’s First Theorem
Bibliography
Name Index
Subject Index
