620-221 Real and Complex Analysis

One of 620-112, 620-122, [99]620-200 or 620-211; a grade of H3 or better in the prerequisite is recommended. Students with a grade of H1 in 620-142 will be permitted to enrol on completion of additional summer reading.

One of 620-113, 620-123, 620-143.

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; and 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 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.

Topics include 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; and residue theorem, evaluation of integrals, summation of series.

Up to 24 pages of written assignments and a 3-hour end-of-semester written examination and class tests totalling not more than 1.5 hours.