複變函數

This course will enhance students understanding of and interest on the functions of complex variables. The course is designed to provide students with a comprehensive introduction to the concepts and ideas that form the basis of applications of complex functions – a powerful tool of researchers and developers. The field of applications of the complex functions is very wide and it includes: control theory, improper integrals, fluid dynamics, electromagnetism, electrical engineering, quantum physics, fractals, etc. The course is addressed to a typical student majoring in engineering and science who is prepared in calculus and linear algebra. Some working knowledge of differential equations would be helpful. In the course we pay attention to the theory of the complex variables, but only within bars we feel are necessary in the first course. So, the course is a continuation of the calculus of functions but of a complex variable. Nevertheless, proofs of major results are presented and standard terminology is used. The course will help students to understand the complex functions those can be used to answer pertinent questions.

微積分、線性代數

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Assignments (or Quiz):10% Midterm exam:30% * 2 Final exam:30%

• Introduction

• Complex Numbers

  1.Introduction 2.More Properties of Complex Numbers 3.Complex Numbers and the Argand Plane 4.Integer and Fractional Powers of Complex Numbers 5.Points, Sets, Loci, and Regions in the Complex Plane

• Analytic Functions

  1.Introduction 2.Limits and Continuity 3.The Complex Derivative 4.The Derivative and Analyticity 5.Harmonic Functions 6.Some Physical applications of Harmonic Functions

• Elementary Functions

  1.The Exponential Function 2.Trigonometric Functions 3.Hyperbolic Function 4.The Logarithmic Function 5.Analycity of the Logarithmic Function 6.Complex Exponentials 7. Inverse Trigonometric and Hyperbolic Functions 8. Branch Cuts and Branch Points

• Complex Integration

  1.Introduction to Line Integration 2.Complex Line Integration 3.Contour Integration and Green’s Theorem 4.Path Independence, Indefinite Integrals, Fundamental Theorem of Calculus in the Complex Plane 5.The Cauchy Integral Formula and Its Extension 6. Some applications of the Cauchy Integral Formula 7.Introduction to Dirichlet Problems – The Poisson Integral Formula for the Circle and Half Plane

• Series Representations for Analytic Functions

  1.Introduction and Review of Real Series 2.Complex Sequences and Convergence of Complex Series 3.Uniform Convergence of Series 4.Power Series and Taylor Series 5.Thechniques for Obtaining Taylor Series Expansions 6. Laurent Series 7. Properties of Analytic Functions Related to Taylor Series: Isolation of Zeros, Analytic Continuation, Zeta Function, Reflection 8. The z Transformation

• Residue Theory

  1.Introduction and Definition of the Residue 2.Isolated Singularities 3.Finding the Residue 4.Evaluation of Real Integrals with Residue Calculus 5.Integrals Involving Indented Contours 6.Contour Integrations Involving Branch Points and Branch Cuts 7.Residue Calculus Applied to Fourier Transforms 8.The Hilbert Transform, Uniform Convergence of Integrals and the Gamma Function

Textbook: E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis, 3rd ed., 2014. Reference: 1. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9th ed., 2014. 2. A. David Wunsch, Complex Variables with Applications, 3rd ed., 2005.

地點

工程四館408室

時間

每週三AM10:00~12:00

聯絡方式

Tel: 03-5131289 Email: graylin@mail.nctu.edu.tw