Fall 2008 - Academic Year 5269

(Course number 120511).

**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 midterm exam took place on Monday, 15th of December. Here are the solutions.

- Sheet number 01.
- Sheet number 02.
- Sheet number 03.
- Sheet number 04.
- Sheet number 05.
- Sheet number 06.
- Sheet number 07.
**Important notice:**Next lecture (in about one week and a half) will rely on previous knowledge on Series and Taylor series. Please have this opportunity to have a look on that topic that you learnt last year. You may use the appropriate chapters of the webbook for Infi1: series and power series. - Sheet number 08.
- Sheet number 09.
- Sheet number 10.
- Sheet number 11.

- Sheet number 1.
- Sheet number 2:a, b, c, d.
- Sheet number 6.
- A trigonometric integral.
- MacLaurin series.

- 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.

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.Computer Algebra Systems: **Useful links:**- 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 course.
- If you are interested in various mathematical topics (biographies of famous Mathematicians, famous curves, development of special topics, ...), have a look at the The MacTutor History of Mathematics archive .
- SOS Mathematics .