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 |