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Complex Variables deals with complex variables and covers topics ranging from Cauchy's theorem to entire functions, families of analytic functions, and the prime number theorem. Major applications of the basic principles, such as residue theory, the Poisson integral, and analytic continuation are given.
Comprised of seven chapters, this book begins with an introduction to the basic definitions and concepts in complex variables such as the extended plane, analytic and elementary functions, and Cauchy-Riemann equations. The first chapter defines the integral of a complex function on a path in the complex plane and develops the machinery to prove an elementary version of Cauchy's theorem. Some applications, including the basic properties of power series, are then presented. Subsequent chapters focus on the general Cauchy theorem and its applications; entire functions; families of analytic functions; and the prime number theorem. The geometric intuition underlying the concept of winding number is emphasized. The linear space viewpoint is also discussed, along with analytic number theory, residue theory, and the Poisson integral.
This book is intended primarily for students who are just beginning their professional training in mathematics.
0.1 Basic Definitions
0.2 The Extended Plane
0.3 Analytic Functions
0.4 Cauchy-Riemann Equations and Applications
0.5 The Elementary Functions
0.6 Logarithms and Roots
Remarks on Notation
1 The Elementary Theory
1.1 Integration on Paths
1.2 Power Series
2 The General Cauchy Theorem
2.1 Logarithms and Arguments
2.2 The Index of a Point with Respect to a Closed Curve
2.3 Cauchy’s Theorem
3 Applications of the Cauchy Theory
3.2 Residue Theory
3.3 Inverse Functions
3.4 Analytic Mappings of One Disk into Another
3.5 Extension of Cauchy’s Theorem and Integral Formula
3.6 The Poisson Integral Formula and Its Applications
3.7 The Jensen and Poisson-Jensen Formulas
3.8 Analytic Continuation
4 Entire Functions
4.1 Infinite Products
4.2 Canonical Products and the Weierstrass Factorization Theorem
4.3 Order of an Entire Function
4.4 The Hadamard Factorization Theorem
5 Families of Analytic Functions
5.1 The Spaces A(U) and C(U)
5.2 Riemann Mapping Theorem
5.3 The Homotopic Version of Cauchy’s Theorem
6 The Prime Number Theorem
6.1 The Gamma Function
6.2 The Riemann Zeta Function
6.3 Proof of the Prime Number Theorem
Solutions to Problems
- No. of pages:
- © Academic Press 1971
- 1st January 1971
- Academic Press
- eBook ISBN:
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