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Instructor: Oleksandr Voskoboynikov 霍斯科

Prof. Alex

Phone: 5712121 ext. 54174

Office: 646 ED bld.4

E-mail: vam@ faculty.nctu.edu.tw

Web: http:// web2.cc.nctu.edu.tw/~vam/

Office hours: by appointment

Complex Variables

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Complex Variables

Web: http://web2.cc.nctu.edu.tw/~vam/

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Credits: 3 (Hours for Weekly Study: 3)

Time & Rooms: Tuesday EDB06 CD 10:10 ~ 12:00 AM

Thursday ED101 G 15:30 ~ 16:20 AM

Pre-requisite Courses: Calculus, Linear Algebra, and … English

Grade:

Home Works: 20%

Midterm: 40%

Final: 40%

Complex Variables

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Text Book:

A.David Wunsch,

Complex Variables with Applications, 3rd Edition,

Pearson Education, Inc., 2005.

Reference Books:

1. D. G. Zill and P. D. Shanahan,

A first Course in Complex Analysis with Applications,

2nd Edition, Jones and Bartlett Publishers, 2009

2. J. W, Brown and R. V. Churchill, Complex Variables and

Applications, 7th Edition, McGraw Hill, 2004.

3. Lecture Notes: http://web2.cc.nctu.edu.tw/~vam/

Complex Variables

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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.

Complex Variables

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Complex Variables

Why and when do we need functions in complex variables?

- Not every algebraic (e.g. quadratic) equation

has a solution in real numbers

after Gerolamo Cardano (1501-1576)

- The term “imaginary” was used fist by

after René Descartes (1596 – 1650)

- The “i” notation and Euler's identity

after Leonhard Euler (1707 – 1783)

- rigorous theory

after Johann Carl Friedrich Gauss (1777 – 1855)

William Rowan Hamilton (1805–1865), and…..

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Complex Variables

Analytic functions those have extremely elegant and useful properties

– derivatives of all orders

– many useful transformations (Fourier, Laplace, ...)

– their series expansions are commonly used to calculate solutions

to multi-dimensional differential equations

Analytic functions are widely used in

- Heat conduction

- Fluid flows

- Electrostatics

- Electromagnetics

- Quantum mechanics

- e.t.c.

We use functions in complex variables to formulate:

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Complex Variables

“Function in Complex World”

“Function in Real World”

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iivuw

iyxz

2 real numbers{x,u}

2D presentation 2D plane {x,u}

4 real numbers{x,y,u,v}

4D presentation 4D space {x,y,u,v}

2)( xxu

),(),(2)( 222yxivyxuxyiyxiyxzw

x y

u,v u

x y x y

v

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Complex Variables

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1 (Chapter 1) Complex Numbers

1.1 Introduction

1.2 More Properties of Complex Numbers

1.3 Complex Numbers and the Argand Plane

1.4 Integer and Fractional Powers of Complex Numbers

1.5 Points, Sets, Loci, and Regions in the Complex Plane

2 (Chapter 2) The Complex Function and Its Derivative

2.1 Introduction

2.2 Limits and Continuity

2.3 The Complex Derivative

2.4 The Derivative and Analyticity

2.5 Harmonic Functions

2.6 Some Physical applications of Harmonic Functions

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3 (Chapter 3) The Basic Transcendental Functions

3.1 The Exponential Function

3.2 Trigonometric Functions

3.3 Hyperbolic Function

3.4 The Logarithmic Function

3.5 Analycity of the Logarithmic Function

3.6 Complex Exponentials

3.7 Inverse Trigonometric and Hyperbolic Functions

3.8 Branch Cuts and Branch Points

4 (Chapter 4) Integration in the Complex Plane

4.1 Introduction to Line Integration

4.2 Complex Line Integration

4.3 Contour Integration and Green’s Theorem

4.4 Path Independence, Indefinite Integrals, Fundamental Theorem

of Calculus in the Complex Plane

4.5 The Cauchy Integral Formula and Its Extension

4.6 Some applications of the Cauchy Integral Formula

4.7 Introduction to Dirichlet Problems –

The Poisson Integral Formula for the Circle and Half Plane

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5 (Chapter 5) Infinite Series Involving a Complex Variable

5.1 Introduction and Review of Real Series

5.2 Complex Sequences and Convergence of Complex Series

5.3 Uniform Convergence of Series

5.4 Power Series and Taylor Series

5.5 Techniques for Obtaining Taylor Series Expansions

5.6 Laurent Series

5.7 Properties of Analytic Functions Related to Taylor Series:

Isolation of Zeros, Analytic Continuation, Zeta Function, Reflection

5.8 The z Transformation

6 (Chapter 6) Residues and Their Use in Integration

6.1 Introduction and Definition of the Residue

6.2 Isolated Singularities

6.3 Finding the Residue

6.4-5-6 Evaluation of Real Integrals with Residue Calculus

6.7 Integrals Involving Indented Contours

6.8 Contour Integrations Involving Branch Points and Branch Cuts

6.9 Residue Calculus Applied to Fourier Transforms

6.10 The Hilbert Transform

Uniform Convergence of Integrals and the Gamma Function

Complex Variables