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Mathematics - Analysis | An Introduction to Complex Analysis

An Introduction to Complex Analysis

Agarwal, Ravi P., Perera, Kanishka, Pinelas, Sandra

2011, XIV, 331 p.

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  • Provides a rigorous introduction to complex analysis
  • Arranges the material effectively in 50 class-tested lectures
  • Uses ample illustrations and examples to explain the subject
  • Provides problems for practice

This textbook introduces the subject of complex analysis to advanced undergraduate and graduate students in a clear and concise manner.

 

Key features of this textbook:

-Effectively organizes the subject into easily manageable sections in the form of 50 class-tested lectures

- Uses detailed examples to drive the presentation

-Includes numerous exercise sets that encourage pursuing extensions of the material, each with an “Answers or Hints” section

-covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics

-Provides a concise history of complex numbers

 

 

An Introduction to Complex Analysis will be valuable to students in mathematics, engineering and other applied sciences. Prerequisites include a course in calculus.

Content Level » Upper undergraduate

Keywords » analytic function - complex function - complex variables - series

Related subjects » Analysis

Table of contents 

-Preface. -Complex Numbers I. - Complex Numbers II. - Complex Numbers III. - Set Theory in the Complex Plane. - Complex Functions. -Analytic Functions I. - Analytic Functions II. - Elementary Functions I. - Elementary Functions II. - Mappings by Functions I. - Mappings by Functions II. - Curves, Contours, and Simply Connected Domains. - Complex Integration. -Independence of Path. - Cauchy-Goursat Theorem. - Deformation Theorem. - Cauchy’s Integral Formula. - Cauchy’s Integral Formula for Derivatives. - The Fundamental Theorem of Algebra. - Maximum Modulus Principle. - Sequences and Series of Numbers. - Sequences and Series of Functions. - Power Series. -Taylor’s Series. -Laurent’s Series. - Zeros of Analytic Functions. -Analytic Continuation. -Symmetry and Reflection. -Singularities and Poles I. -Singularities and Poles II. - Cauchy’s Residue Theorem. - Evaluation of Real Integrals by Contour Integration I. - Evaluation of Real Integrals by Contour Integration II. -Indented Contour Integrals. -Contour Integrals Involving Multi-valued Functions. -Summation of Series. -Argument Principle and Rouch´e and Hurwitz Theorems. -Behavior of Analytic Mappings. - Conformal Mappings. -Harmonic Functions. -The Schwarz-Christoffel Transformation. -Infinite Products. - Weierstrass’s Factorization Theorem. - Mittag-Leffler Theorem. -Periodic Functions. -The Riemann Zeta Function. -Bieberbach’s Conjecture. -Riemann Surfaces. -Julia and Mandelbrot Sets. -History of Complex Numbers. -References for Further Reading. -Index

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