620-221 Real and Complex Analysis

Note

Students may only gain credit for one of 620-221, 620-252, (1997 Handbook 618-201, 618-251;1996 Handbook 618-202).

Credit Points

Coordinator

Prerequisites

One of 620-112, 620-122, 620-200 or 620-211 (1997 Handbook one of 618-112, 618-122, 618-211 or 618-200); a grade of H3 or better in the prerequisite is recommended.

Semester

Contact

Subject Description

This subject introduces the structure and methods of proof; the concept of convergence of sequences and series; basic topological concepts in the real line and complex plane; the basic concepts of functions of a complex variable.

Students completing this subject develop an ability to construct rigorous and accurate arguments; determine convergence or otherwise of sequences and series; differentiate functions of a complex variable; calculate contour integrals; work with analytic functions in the cut plane and apply Cauchy's integral formula and the residue theorem.

The subject demonstrates the importance and of rigorous arguments via proofs; the fundamental concepts of topology of the complex plane; and the differences between functions of a real and a complex variable.

Sequences of real and complex numbers and their properties. Rigorous definition of the limit, Cauchy sequences. Series of real or complex numbers, absolute and conditional convergence; tests for convergence. Power series of complex numbers, radius of convergence. Basic topological concepts in the complex plane. Continuous functions and their properties. Homomorphic functions, Cauchy-Riemann conditions. Exponential and logarithm of the complex variable; other elementary functions. Contour integration, Cauchy's theorem and Cauchy's integral formula. Uniform convergence, Weierstrass M-test. Equivalence of complex differentiability to the local power series expansion. Laurent series, singularities, poles. Residue theorem, evaluation of integrals, summation of series.

Assessment

Up to 24 pages of written assignments and a three-hour end-of-semester written examination.