*Instructor:* Dr. Szilárd RÉVÉSZ

e-mail: revesz@renyi.hu

Phone /only in case of emergency, please/: +36-30-3590489

*Text:* John B. Conway, Functions of a complex variable I, Second ed., Graduate Texts in Mathematics, Springer, 1978.

*Prerequisite:* Calculus

*Course description:* Complex numbers. The complex plane. Stereorgaphic projection. Linear and fractional linear transformations of the complex plane and the Riemann sphere. Mappings of the plane, preservation of angles. Complex differentiability. The Cauchy-Riemann differential equations. Power series, analytic functions. Complex line integrals, the "trivial estimate". Primitives and integrals. The Cauchy-Goursat integral theorems. Cauchy integral formula. Power series expansion of an analytic function. Cauchy's estimates, Liouville's theorem. Morera's Theorem. Uniqueness theorems. Open mapping theorem. Maximum modulus principle. The residuum theorem. Zeroes of an analytic function. The argument prinicple. Clasification of singularities. Schwarz Lemma, Hadamard Three Circle Theorem, Phragmen-Lindelöf Theorem. The Riemann mapping theorem. Weierstarss factorization theorem. The Gamma function. The Riemann zeta function. Topics from analytic continuation. Some applications to number theory.

Grading: Homeworks - 40 % , Midterm exam (with "take home" problems) - 30%, Final exam (written in class) - 40%.