An Introduction to Complex Analysis

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.

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

Inhalt

Preface.-Complex Numbers.-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.- Mappings by Functions II.- Curves, Contours, and Simply Connected Domains.- Complex Integration.- Independence of Path.- CauchyGoursat Theorem.- Deformation Theorem.- Cauchy's Integral Formula.- Cauchy's Integral Formula for Derivatives.- 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 Multivalued Functions .- Summation of Series. Argument Principle and Rouch´e and Hurwitz Theorems.- Behavior of Analytic Mappings.- Conformal Mappings.- Harmonic Functions.- The SchwarzChristoffel Transformation.- Infinite Products.- Weierstrass's Factorization Theorem.- MittagLeffler's Theorem.- Periodic Functions.- The Riemann Zeta Function.- Bieberbach's Conjecture.- The Riemann Surface.- Julia and Mandelbrot Sets.- History of Complex Numbers.- References for Further Reading.- Index.