A First Course in Complex Analysis with Applications

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The new Second Edition of A First Course in Complex Analysis with Applications is a truly accessible introduction to the fundamental principles and applications of complex analysis. Designed for the undergraduate student with a calculus background but no prior experience with complex variables, this text discusses theory of the most relevant mathematical topics in a student-friendly manner. With Zill’s clear and straightforward writing style, concepts are introduced through numerous examples and clear illustrations. Students are guided and supported through numerous proofs providing them with a higher level of mathematical insight and maturity. Each chapter contains a separate section on the applications of complex variables, providing students with the opportunity to develop a practical and clear understanding of complex analysis. Key features Portions of the text and examples have been revised or rewritten to clarify or expand upon the topics at hand. Each chapter opens with an overview of the material to be covered in the chapter. End-of-chapter materials include Computer Lab Assignments and a Chapter Review Quiz. Numerous Exercises are included at the end of each section. Remarks conclude most sections and discuss the relationships and differences between important concepts. Proof Problems offer an additional learning experience for students, who are supported through each step of the process by hints and guides. Table of Contents Chapter 1: Complex Numbers and the Complex Plane Complex numbers and their properties Complex plane Polar form of complex numbers Power and roots Sets of points in the complex plane Applications Chapter 1 Review quiz Chapter 2: Complex Functions and Mappings Complex functions Complex functions as mappings Linear mappings Special power functions Reciprocal function Limits and continuity Applications Chapter 2: Review quiz Chapter 3: Analytic Functions Differentiability and analyticity Cauchy- Reimann equations Harmonic functions Applications Chapter 3 Review quiz Chapter 4: Elementary Functions Exponential and logarithmic functions Complex powers Trigonometric and hyperbolic functions Inverse trigonometric and hyperbolic functions Applications Chapter 4 Review Quiz Chapter 5: Integration in the Complex Plane Real integrals Complex integrals Cauchy-Goursat Theorem Independence of Path Cauchy’s Integral formulas and their consequences Applications Chapter 5 Review quiz Chapter 6: Series and Residues Sequences and series Taylor series Laurent series Zeros and poles Residue and residue theorem Some consequences of the residue theorem Applications Chapter 6 Review quiz Chapter 7: Conformal Mappings Conformal mapping Linear fractional transformations Schwartz- Christoffel Transformations Poisson integral formulas Applications 1 Chapter 7 Review quiz Appendices Proof of Theorem 2.6.1 APP-1 Proof of the Cauchy- Goursat Theorem Table of conformal mappings