MATH 353 COMPLEX CALCULUS / FALL 2009

 

Prerequisite: Math 252 (or equivalent)

 

Instructor: Mustafa Korkmaz, M-135

 

Assistant: Ali Hakan Tor

 

Schedule: Tuesday and Thursday  8:40-10:30 M13

 

Website: http://arf.math.metu.edu.tr/~korkmaz

 

Office Hours: Tuesday     10:40-11:30

                        Thursday   10:40-11:30.

 

Textbook: Complex Calculus, 2nd ed., J. Bak and D.J. Newman, Springer, 1997.

Other complex analysis books can be used as a reference.

 

Assingments:  There will be 6-7 homework assignments. Selected problems will be graded. You will have one week to return the homework. The homework assigments will be collected at the beginning of the class on the due date. NO LATE HOMEWORK will be accepted! You are allowed to work on the problems together. But you must write your own solution. Otherwise, you won’t get any points. The affect of homework grade to the your grade in this course will be 10%.     

 

Exams and Grading: There will be two midterm exams and a final exam. The contribution of each exam to the course grade will be 30%. The final exam covers all topics. There will be no Make-up Exam!

 

 

SYLLABUS

 

Week  Dates                          Sections to be covered from Bak and Newman

 

1         Sept 28-Oct 2                1.1 The Field of Complex Numbers

                                          1.2 The complex Plane

 

2    Oct 5-9                         1.3 Topological Aspects of the Complex Plane

                                          1.4 Stereographical Projection, The Point at Infinity

 

3    Oct 12-16                     2.1 Analytic Polynomials

                                                2.2 Power Series

 

4    Oct 19-23                     2.3 Differentiability and Uniqueness of Power Series

                                          3.1 Analyticity and the Cauchy Riemann Equations

 

5    Oct 26-30                     3.2 The Functions ez , sin z , cos z

 

      6    Nov 2-7                        4.1 Properties of the Line Integral

                                          4.2 The Closed Curve Theorem for Entire Functions

                    

   Midterm 1                        NOVEMBER 10, 17:40                                     

                             

7    Nov 9-13                      5.1 The Cauchy Integral Formula and Taylor Expension for Entire Functions

                                         

8    Nov 16-20                     5.2 Liouville Theorems and The Fundamental Theorem of Algebra

                                          6.1 The Power Series Representations for Functions Analytic in a Disc

 

9    Nov 23-27                     6.2 Analyticity in an Arbitrary Open Set

                                          7.2 Morera’s Theorem

                                          6.3 The Uniqueness, Mean-Value and Maximum-Modulus Theorems

 

10  Nov 30-Dec 4               8.1 The General Cauchy Closed Curve Theorem

                                          8.2 The Analytic Function Log z

 

11  Dec 7-11                      13.1 Conformal Equivalence

                                          13.2 Special Mappings

 

12  Dec 14-18                     9.1 Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem

                                          9.2  Laurent Expansions

 

13  Dec 21-25                     10.1 Winding Numbers and The Cauchy Residue Theorem

                                                10.2 Applications of the Residue Theorem

 

    Midterm 2                      DECEMBER 23, 17:40         

 

14  Dec 28-Jan 1                11.1 Evaluation of the Definite Integrals by Contour Integral Techniques

                                          11.2 Applications of Contour Integral Methods to Evaluation and Estimation of Sums

 

15  Jan 4-8                         14.1 Conformal Mapping and Hydrodynamics

                                          14.2 The Riemann Mapping Theorem