# Complex Variable and Applications (math. 3160).

 The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being. --Gottfried Whilhem Leibniz The shortest path between two truths in the real domain passes through the complex domain. -- Jacques Hadamard The complexity of the complex variable course is more imaginary than real. -- An encouraging observation "The number you have dialed is imaginary. Please rotate your phone 90 degrees and try again." -- A math joke

Instructor:

#### Andrej V. Cherkaev, professor Department of Mathematics Office: JWB 225 Email: cherk@math.utah.edu Tel : +1 801 - 581 6822 Fax : +1 801 - 581 4148

Class meets: T, Th, 9:40 - 10:30, NS 205
Office Hours: T, 4:00 - 5:00

Text : Churchill and Brown `Complex Variable and Applications', 6-th ed. chapters 1-10.

Exams
 Exam firstst midterm second midterm Final Sections 1 - 29 30 - 64 1 - 92

Policy:
Exercises: Problems are typical of what you'll be asked on exams.

Calculator: A calculator is not required for the course, although it may be useful for some homework problems. Calculators will be forbidden in exams.

Final exam: Half of the final will be devoted to material covered after the second midquarter exam. The rest will be comprehensive. Students must take the final to pass the course.

Grades: The final grade will be based on the following score: Two midquarter exams (50%) + homework(10%) + final score (40%).

ADA: The Americans with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning and psychiatric disabilities. Please contact me at the beginning of the quarter to discuss any such accommodations for the course.

Course outline and homework problems
The course discusses the algebra of complex numbers, elementary functions of complex arguments, theory of analytical functions with apllications, especially the conformal mappings.

 Section Topic Page [Problems] 1- 2 Complex Numbers 5 [2, 4, 5, 9, 12] 3- 5 Geometric Prop. 11 [3, 10, 14], 17 [1, 4, 5, 8, 13] 6- 8 Roots 22 [1, 4, 6, 7, 8], 25 [1-5] 9-14 Mappings 32 [3, 4, 5, 7] 15-18 Derivatives 42 [2, 4, 9], 47 [1, 3, 8], 54 [1, 2d, 7] 19-22 CR Eqns. 62 [1, 6, 7b, 10, 12] 23-25 Exponentials 68 [3, 6, 8, 10], 71 [2, 7, 9], 74 [2, 5] 26-29 Logs 79 [3, 6, 14], 84 [2, 6, 9, 10, 11] 30-33 Integrals 92 [3, 4, 11], 102 [1, 5, 7, 11, 13, 15] 34-35 Contour Integrals 119 [2, 3, 7] 36-38 Cauchy Goursat 128 [1, 3, 5] 39-40 Liouville's Th. 136 [1, 2, 3, 4, 5, 6] 43-45 Algebra & Series 156 [4, 6, 8] 46-51 Laurent Series 172 [1, 3, 10, 15] 53-55 Residues 188 [1, 2, 4, 7] 56-58 Poles 197 [1, 2, 3, 4, 10] 60-61 Evaluation of integrals 208 [1, 2, 3, 4, 6, 9], 214 [1, 2, 6, 13)] 62-64 Improper Integrals 218 [1, 3, 6], 226 66-67 Other applications [1, 2, 3, 7], 242 [1, 6] 68-72 Linear fractions 250 [2, 8, 11, 17], 258 [2, 5, 6, 13] 79-81 Exp-sin-log-root 266 [4, 5, 6, 9], 275 [3, 4, 8] 82-83 Harmonic Mapping 289 [1, 4, 6], 297 [1, 3, 9] 84-85 Steady Temp. 307 [1, 2, 5, 8] 86-87 Electrost. potential 312 [2, 3, 7, 10] 88-92 Stream Function 322 [3, 4, 5, 7, 11]
Acknowledgement. The selection of the homework problems has been done by Professor Andrejs Treibergs.

Internet sourses

Quadratic, cubic and quartic equations; introduction of complex numbers
Irrotational flows by conformal mappings Interactive page, a lot of pictures.
Lectures on Advanced Topics in Theory of Functions of a Complex Variable by Norm Bleistein
COMPLEX ANALYSIS: Mathematica 4.0 Notebooks by John H. Mathews, and Russell W. Howell. Lecture notes and Mathematica-based examples, including Julia & Mandelbrot sets, conformal mappings, etc.
68 lessons by Dr. John H. Mathews. A part of the book, Complex Analysis for Mathematics & Engineering, 4th Ed, 2001, by John H. Mathews and Russell W. Howell
Please let me know if you find interesting web sites related to the course
Thank you.