Department of Mathematics
Fall 2008 - Academic Year 5269
Functions of a Complex Variable
(Course number 120511).
Lecture hall: Lau 540 - Monday 17:00-17:45 and Thursday 14:30-16:00
Assistants: Mr Netanel Altschuler, Dr. Yaacov Itin
- Complex numbers: algebraic form and trigonometric form. Geometric applications. Roots of order n of a complex number. Solutions of polynomial equations in one complex unknown.
- Functions of one complex variable: limits, continuity, differentiability. Cauchy-Riemann equations.
- Analytic functions and harmonic functions. Analytic continuation.
- Trigonometric and hyperbolic functions.
- The logarithm of a complex number. Multi-valued functions of one complex variable. Branch points points, branch cuts.
- Inverse trigonometric and inverse hyperbolic functions.
- Line integrals, Cauchy's theorem and its consequences. Applications.
- The maximum principle and the minimum principle. Liouville's theorem. Applications.
- The fundamental theorem of Algebra.
- Taylor series and Laurent series.
- Residues and their applications (complex integrals, real integrals).
- Conformal mappings.
The grade will be determined by final exam (85%), one mid-term exam (10%) and homework assignments (5%).
The midterm exam took place on Monday, 15th of December. Here are the solutions.
Every week, a homework sheet should be posted on this site.
All the students must have it printed before next exercise session. There will be no handout.
A couple of solutions:
Samples of past exams:
- 5767 with solutions.
- B. Kohn: Complex Variables (in Hebrew), Bak Editions, Haifa.
- T. Dana-Picard: Complex Numbers (in Hebrew), JCT.
- A.D. Wunsch: Complex Variables with Applications 3/e, Addison-Wesley (2007).
- E.B. Saff and al.: Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Prentice-Hall (1993).
- J. Bak and D. Newman: Complex Analysis, UTM, Springer Verlag (1997).
- New books arrived one month ago. Pay a visit to the library and check them.
- Examples of level curves for the real part and the imaginary part of an analytic function are here. Yes, we know, the quality is not so good; we"ll try to improve it.
- A short webbook for the course, written by Prof. Dana-Picard, is available here.
Improvements to the current version will be introduced from time to time, so please check them. And most important, enjoy.
- Presentation on conformal mappings.
Computer Algebra Systems:
A set of online lessons using Maple software. This software is installed in our labs, but with a few licences only. Anyway, you are warmly invited to use it.
- A bibliography on the History of Complex Numbers.
- Have a look at some graphical examples .
- The Applied Maths Book , by Sean Mauch, is very nice; it includes a large chapter on Complex Variables, and the graphics are worth a look.
- The WIMS site at the University of Nice (France) provides many useful tools: 2D and 3D animated graphics, interactive exercises in many fields, including topics related to the present
- If you are interested in various mathematical topics (biographies of famous
Mathematicians, famous curves, development of special topics, ...), have a look
The MacTutor History of Mathematics archive .
- SOS Mathematics .