Functions of a complex variable

0366.2123.03

FUNCTIONS OF A COMPLEX VARIABLE

Dany Leviatan(leviatan@math.tau.ac.il)

School of Mathematical Sciences

Spring Semester 2003.

Time and place

Sunday 12.00-13.00 in Orenstein 005

Wedneday 13.00-15.00 in Orenstein 111

Grade : An examination at the end of the semester

The syllabus of the course

1. Complex numbers

Arithmetics of the complex numbers;

Absolute value, Triangle inequality;

Polar representation, De Moivre's formula;

The sphere, Spherical metric;

2. Differentation

Cauchy-Riemann equations;

Holomorphic functions;

Harmonic functions, Harmonic conjugate.

3. The complex plane

Trajectories;

Linear fraqctional transformations;

The projective plane.

4. Elemntary functions

The exponential function;

The hyperbolic functions;

Trigonometric functions;

Logaritm.

5. Power series

Convergence and absolute convergence;

Sequences and series of functions;

Differentiation of serires, in particualr power series;

Cauchy products.

6. Integration

Cauchy's integral;

Taylor series and Laurent series;

Zeros and poles, classification;

Residues and applications.

7. Further results

The maximum modulus principle;

The argument principle;

Rouche's theorem.

We recommend as bibliography the following books :

1) Agmon S. Classical analysis (In hebrew)

2) Sarason D. Notes on complex function theory

3) Ahlfors L. Complex analysis

4) Hille E. Analytic function theory, vol1.