Department of Mathematics
Fall 2008 - Academic Year 5269

## Functions of a Complex Variable (Course number 120511).

### Syllabus:

• 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.

### Homework:

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.

### Samples of past exams:

• 5764.
• 5767 with solutions.

### Course Textbooks:

• B. Kohn: Complex Variables (in Hebrew), Bak Editions, Haifa.
• T. Dana-Picard: Complex Numbers (in Hebrew), JCT.

### Recommended Texts:

• 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.

### Help material:

• 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.