Math 417 - Complex Analysis with Applications

Summer 2008

Welcome to Math 417! This course is an introduction to Complex Analysis. Complex Analysis is one of the great subjects of modern mathematics and an invaluable tool in physics and engineering. In this course we will explore the basic properties of complex analytic functions and conformal maps.

Lecturer: David Cabrera, dcabrera@uic.edu

Office hours: by appointment in SEO 619

Classroom: BH 316

Textbook: Complex variables and applications by J.W. Brown and R.V.Churchill, McGraw Hill, 2004, Seventh Edition. All page numbers below refer to this book.

Prerequisites: A grade of C or better in Math 210 (multivariable calculus). Some background in analysis and writing proofs is helpful but not essential.

Homework: Homework will be assigned and due daily. Late homework will not be accepted without prior permission from the instructor. You are allowed to discuss problems with each other. However, the write-up must be your own and should reflect your own understanding of the problem.

Grading: There will be a midterm exam and a final exam. The midterm and the final will each count for 40% of your grade. Homework will count for 20%.

Links to other complex analysis webpages: Caution I have not checked the material in these pages. Some seem to have beautiful pictures and applications.

Homework Assignments:


Week Date Topics Problems Due Date
1 Mon 6/16 Complex Numbers
Exponential Form
p. 4-5: 1, 2    sol
p. 11: 1ac, 4ab   
p. 13-14: 1, 3, 16    sol
p. 21: 1, 2    sol
Wed 6/18
Wed 6/20 Roots
Functions of a Complex Variable
p. 28-29: 1b, 3a    sol
p. 31: 1, 2, 3    sol
p. 35: 1ad, 2    sol
p. 42: 1    sol
Fri 6/20
Fri 6/20 Limits p. 42: 6    sol
p. 53: 2b, 5, 10ab    sol
Mon 6/23
2 Mon 6/23 Derivative
Cauchy-Riemann Eqns.
p. 59: 1bc, 8b, 9    sol
p. 68-69: 1bc, 2c, 3b, 5    sol
Wed 6/25
Wed 6/25 Analytic Functions
Harmonic Functions
p. 68: 4c    sol
p. 73-74: 1d, 2a, 4b    sol
p. 78-79: 1bd, 3    sol
Fri 6/27
Fri 6/27 Exponentials, Logarithms
Complex Exponents
p. 89-90: 1bc, 3, 5    sol
p. 94-95: 1b, 2c, 3a    sol
p. 96-97: 1    sol
p. 99: 1a, 2c    sol
Mon 6/30
3 Mon 6/30 Trigonometric, Hyperbolic Functions
Inverse Trig. Functions, Integrals
p. 103-104: 1, 2, 16    sol
p. 107: 1, 14, 15a    sol
p. 110: 1b    sol
p. 115-116: 2b    sol
Wed 7/2
Wed 6/30 Contours, Contour Integrals
ML-Bound
p. 120-121: 2    sol
p. 129-130: 1, 2, 3, 8    sol
p. 133-134: 1, 2, 3    sol
Mon 7/7
Fri 7/4 No Class Independence Day  
4 Mon 7/7 Cauchy-Goursat Theorem, Path Independence
Path Deformation
p. 141-142: 2abc    sol
p. 153-155: 1ace, 2abc, 3    sol
Wed 7/9
Wed 7/9 Cauchy Integral Formula p. 162-164: 1abc, 2, 4    sol Mon 7/14
Fri 7/11 Midterm Exam    
5 Mon 7/14 Liouville's Theorem, Maximum Modulus Principle
Sequences
p. 172: 6    sol
p. 181: 1    sol
Wed 7/16
Wed 7/16 Series, Radius of Convergence
Taylor Series
p. 188-190: 1, 3, 6    sol
these problems    sol
Fri 7/18
Fri 7/18 Laurent Series
Series Properties
p. 198-199: 1, 2, 4, 5    sol
p. 212-215: 1, 3    sol
p. 218-219: 1, 3, 5    sol
Mon 7/21
6 Mon 7/21 Residues
Classification of Singularities
p. 230: 1abc, 2ab, 3a    sol
p. 233: 1abd, 2b    sol
Wed 7/23
Wed 7/23 Residues, Zeros, Poles p. 238: 1ab, 2a, 3, 4    sol
p. 245: 1, 2ab, 4a    sol
Fri 7/25
Fri 7/25 Improper Integrals p. 257: 3, 4, 6    sol
p. 265: 3, 7    sol
Mon 7/28
7 Mon 7/28 Improper Integrals p. 265: 4, 6    sol
p. 276: 2, 4    sol
Wed 7/30
Wed 7/30 Improper Integrals
Trig Integrals
p. 276: 5, 6    sol
p. 280: 1, 2    sol
Fri 8/1
Fri 8/1 Inverse Laplace Transform
Argument Principle
Rouche's Theorem
p. 285-287: 1bc, 6, 7    sol
p. 296: 1, 2, 3    sol
Mon 8/4
8 Mon 8/4 Linear Fractional Transformations p. 312: 1, 3, 4, 7 Wed 8/6