Complex Analysis

About the Book

Complex Analysis 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 analysis, this text discusses the theory of the most relevant mathematical topics in a student-friendly manner. With a clear and straightforward writing style, concepts are introduced through numerous examples, illustrations, and applications. Each section of the text contains an extensive exercise set
containing a range of computational, conceptual, and geometric problems. In the text and exercises, 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 devoted exclusively to the applications of complex analysis to science and engineering, providing students with the opportunity to develop a practical and clear
understanding of complex analysis.

About the Author

Dennis G. Zill- Loyola Marymount University Dennis Zill received a PhD in Applied Mathematics from Iowa State University, and is a former professor of Mathematics at Loyola Marymount University in Los Angeles, Loras College in Iowa, and California Polytechnic State University. He is also the former chair of the Mathematics department at Loyola Marymount University, where he currently holds a rank as Professor Emeritus of Mathematics. Zill holds interests in astronomy, modern literature, music, golf, and good wine, while his research interests include Special Functions, Differential Equations, Integral Transformations, and Complex Analysis. Patrick D. Shanahan- Loyola Marymount University Patrick Shanahan received his Ph.D. from the University of California, Santa Barbara, 1996; his M.A. from the University of California, Santa Barbara, 1992; and his B.A. from California State University, Long Beach, 1990. Currently Patrick’s research interests include Geometric topology, knot theory, hyperbolic geometry, and representation theory.

Table of Contents:
Preface Chapter 1: Complex Numbers and the Complex Plane • Complex Numbers and Their Properties • Complex Plane • Polar Form of Complex Numbers • Powers 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 • The Power Function zn • The Power Function z1/n • Reciprocal Function • Applications • Chapter 2 Review Quiz Chapter 3: Analytic Functions • Limits and Continuity • Limits • Continuity • Differentiability and Analyticity • Cauchy-Riemann Equations • Harmonic Functions • Applications • Chapter 3 Review Quiz Chapter 4: Elementary Functions • Exponential and Logarithmic Functions • Complex Exponential Function • Complex Logarithmic Function • Complex Powers • Trigonometric and Hyperbolic Functions • Complex Trigonometric Functions • Complex 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 • Cauchy’s Two Integral Formulas • Some Consequences of the Integral Formulas • Applications • Chapter 5 Review Quiz Chapter 6: Series and Residues • Sequences and Series • Taylor Series • Laurent Series • Zeros and Poles • Residues and Residue Theorem • Some Consequences of the Residue Theorem • Evaluation of Real Trigonometric Integrals • Evaluation of Real Improper Integrals • Integration along a Branch Cut • The Argument Principle and Rouche’s Theorem • Summing Infinite Series • Applications • Chapter 6 Review Quiz Chapter 7: Conformal Mappings • Conformal Mapping • Linear Fractional Transformations • Schwarz-Christoffel Transformations • Poisson Integral Formulas • Applications • Boundary-Value Problems • Fluid Flow • Chapter 7 Review Quiz Appendixes: I Proof of Theorem 3.1.1 APP- 1 II Proof of the Cauchy-Goursat Theorem APP- 3 III Table of Conformal Mappings APP- 7 Answers to Selected Odd-Numbered Problems ANS- 1 • Symbol Index IND- 1 • Word Index IND- 50