複變函數論

Description: The aim of this course is to introduce the key mathematical ideas in complex analysis, which are used in modern mathematics and many fields of science and engineering. While the choice of topics is motivated by their use in mathematics and various applied areas, the course will emphasize the theoretical and conceptual underpinnings of this subject, just as in other mathematics course. • Syllabus: Chapter 1 Preliminaries 1.1 Complex Numbers and Complex Plane 1.2 Topology of the Complex Plane 1.3 Functions on the Complex Plane: Continuous Functions, Holomorphic Functions, Power Series 1.4 Harmonic Functions* 1.5 The Exponential Function and Trigonometric Functions 1.6 Complex Integrations Chapter 2 Cauchy’s Theorem and Its Applications 2.1 Goursat’s Theorem 2.2 Existence of a primitive in a Star Domain 2.3 Cauchy’s Theorem 2.4 Evaluation of Some Integrals 2.5 Cauchy’s Integral Formula 2.6 Further Applications 2.7 Application to Harmonic Functions* Chapter 3 Meromorphic Functions 3.1 Laurent Series 3.2 Zeros and Isolated Singularity 3.3 Meromorphic Functions 3.4 The Residue Formula 3.5 Evaluation of Definite Integrals 3.6 The Argument Principle and Rouche’s Theorem Chapter 4 Conformal Mapping 4.1 Conformal Equivalence and Examples 4.2 Mobius Transformations 4.3 The Riemann Mapping Theorem 4.4 The Schwarz-Christoffel Transformation* 4.5 Applications in Electrostatics, Heat Flow, and Fluid Mechanics* 4.6 Further Physical Applications of Conformal Mapping* Chapter 5 The Transforms of Applied Mathematics 5.1 Fourier Series (The Finite Fourier Transform) 5.2 The Fourier Transform 5.3 The Laplace Transform* 5.4 The z-Transform* 5.5 Cauchy Integrals and the Hilbert Transform*

Prerequisite: Advanced calculus

Course assistant and office hours: 黃鈺誠(Email:yuchenghuang0608@gmail.com ) Office hours : Tuesday 18:00~20:00, SC 207 Practice session: Tuesday 12:10~13:10, Room 223 Exams: • Quiz I (60 points): Tuesday , March 11, 12:10~13:10, Room 223 • Quiz II (60 points): Tuesday , March 25, 12:10~13:10, Room 223 • Midterm (180 points): Friday, April 11, 1:20-3:20 p.m., Room 223 • Quiz III (60 points): Tuesday, April 22, 12:10~13:10, Room 223 • Quiz IV (60 points): Tuesday, May 13, 12:10~13:10, Room 223 • Final (180 points): Friday, June 6, 1:20-3:20 p.m., Room 223. Grading: (60+60+180+60+60+180)/6 + (12+24)/6 (extra points/6)

References: 1. Complex Analysis by Ian Stewart and David Tall, Second Edition,Cambridge University Press 2. Fundamentals of Complex Analysis : Engineering, Science, and Mathematics by Edward B.Saff and Arthur David Snider, Third Edition 3. Complex Analysis by Joseph Bak and Donald. J. Newman, Third Edition 4. Complex Analysis by Elias M.Stein and Rami Shakarchi, Princeton University Press