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Complex Analysis
Ghiocel Groza

1. Complex numbers

1.1. The algebra of complex numbers. Cauchy’s inequality.
1.2. The geometric representation of complex numbers. The spherical representation.

2. Complex functions

2.1. Introduction to the topology.
2.2. Sequences and series of complex numbers.
2.3. Continuity of a function of a complex variable
2.4. Analytic functions. Cauchy-Riemann equations.
2.5. Polynomial and rational functions. Lucas’s theorem.
2.6. Elementary theory of power series. Abel’s limit theorem.
2.7. The exponential, logarithm and trigonometric functions.
2.8. Conlbrmal mapping. The principle of correspondence of boundaries. Riemann’s mapping theorem.
2.9. Linear transformation. The symmetry principle.
2.10. Elementary conformal mapping. The use of level curves. Zhukovsky’s function. Elernentary Riemann surfaces.

3. Complex integration

3.1. Complex line integral.
3.2. Cauchy’s theorem. Indefinite integral.
3.3. The index of a point with respect to a closed curve. Cauchy’s integral formula.
3.4. Morera’s theorem. Liouville’s theorem. The fundamental theorem of algebra.
3.5. Taylor’s theorem.
3.6. Zeros, poles, essential isolated singularities. Theorem of Weierstrass on the behavior of a function in the neighborhood of a essential isolated singularity.
3.7. Formula for the total number of zeros of an analytic function enclosed by a closed curve. The openness of an analytic function. The maximum principle.
3.8. Chains and cycles. Cauchy’s theorem for a cycle which is homologous to zero.
3.9. The residue theorem. Argument principle. Rouché’s theorem. Evaluation of definite integrals.

4. Series and product developments

4.1. The Taylor series of an analytic function. Analytic continuation.
4.2. The Laurent series.
4.3. Partial fraction. Theorem of Mittag-Leffler.
4.4. Infinite products. Canonical product. Theorem of Weierstrass on the entire function.
4.5. Euler’s gamma function.

Books

1. L. Ahifors, Complex analysis, Ch.1- Ch.5, McGraw-Hill Book Company, 1966.
2. A. Sveshnikov and A. Tikhonov, The theory qtfunctions of a complex variable, Ch.1-Ch.6, Mir Publishers, 1978.

E-mail, grozag@mail.utcb.ro
Department of Mathematics and Informatics,
Technical University of Civil Engineering, Bucharest, Romania
.