COMPLEX ANALYSIS
MA537 SPRING 2014
Instructor: Dr. Vasiliy Prokhorov, ILB 422
prokhoro@southalabama.edu
http://www.southalabama.edu/mathstat/personal pages/prokhorov/class handouts/
Phone: 460-6758
Lectures : T R 6:00 p.m. – 7:15 p.m. in ILB 345
Office Hours:
M 5:00 p.m. – 6:00 p.m., ILB 422
T R 1:30 p.m. – 3:30 p.m., ILB 422
T R 7:15 p.m. – 7:45 p.m., ILB 422
(or by appointment)
Objective: The main objective of Complex Analysis is to study the development of functions of one complex variable. Students will perform a thorough investigation of the major
theorems of complex analysis the Cauchy-Riemann Equations, Cauchy Theorem, Cauchy Integral Formula, the Maximum Modulus Principle, Liouville Theorem, the Residue Theorem,
Rouche Theorem, including their proofs. They will also apply these ideas to a wide range of
problems that include the evaluation of both complex line integrals and real integrals.
Bulletin description: Arithmetic of complex numbers; regions in the complex plane; limits,
continuity, and derivatives of complex functions; elementary complex functions; mappings by
elementary functions; contour integration; power series; Taylor series; Laurent series; calculus
of residues; conformal representation; applications.
Textbook: Complex Variables and Applications, 8th edition, by James Ward Brown and
Ruel V. Churchill; McGraw-Hill, Inc. (New York).
Coverage: Most of chapters 1 through 6, and parts of chapters 7, as time permits.
Attendance Policy:
Students are expected to attend all classes and participate in discussions.
Grading Policy:
60% Best 2 out of 3 tests to be given
February 6, March 13, April 10
No make up test will be given
30% Final (comprehensive) exam
5% Homework and class participation
5 % Project
Homework will be assigned in class. Not all of the homework will be collected and graded.
That which is collected will be called take-home quizzes. Graduate students will be given
more advanced problems in addition to the regular assignments for ungraduates.
If the student accumulates:
at least 90% =A
at least 80% =B
at least 70% =C
at least 60% =D
less than 60% =F
Final Exam: Tuesday, May 6, 6:00 p.m. – 8:00 p.m.
Projects ( Due April 22, 2014)
1. Conformal Mappings.
Elementary conformal mappings (linear functions, linear fractional transformations, w = ez ).
The Schwarz-Christoffel Transformations. Examples.
2. Harmonic Functions.
Definition and Basic Properties. The mean-value property. Harmonic conjugate. The maximum and minimum principles for harmonic functions. Poisson’s formula. The Dirichlet
problem.
3. Entire Functions.
General properties of entire functions. Measures of rate of growth. Entire functions of finite
order. Order in terms of the coefficients.
4. The Maximum Modulus Principle.
The maximum modulus principle. The Schwarz Lemma. The Three-Circle Theorem.
5. The Argument Principle.
The Argument Principle. Rouche’s Theorem. An alternative proof of the Fundamental
Theorem of Algebra.
6. Pade Approximation.
Power series. Pade approximations. Definition and basic properties. Pade approximants for
ez . The number π is irrational.
7. Laplace Transforms.
Definition and Basic Properties. Inversion of Laplace transforms. The method of Laplace
transforms in the solution of differential equations.
8. The Riemann Zeta Function ζ(s).
The number-theoretic properties. The product development. Extension of ζ(s) to the complex plane. The zeros of the Zeta Function. The Riemann conjecture.
Tutoring Lab:
The department offers a tutoring lab to all students taking mathematics and statistics classes.
There is no lab fee. Please check the bulletin board outside ILB 325 for details.
Any student with a qualified handicap requiring accommodations should talk to me after
class so that arrangements can be made. A student must verify that he/she has a qualified
handicap through the Special Student Services Office.
Students will be notified in writing in case of changes to the syllabus.
Note: In the Department of Mathematics and Statistics secretaries do not give grades over
the phone.
Students who are thinking about dropping a course please speak first with the instructor, or
department chair (M. Mulekar), or assistant to the chair (S. Brick) before dropping.