620-221 Real and Complex Analysis

Students may only gain credit for one of 620-221 and 620-252.

One of 620-122, [05]620-194, 620-211 together with one of 620-113 or 620-123; a grade of H3 or better in each of the prerequisites is recommended. Students with a grade of H1 in 620-142 or [05]620-192 together with a grade of H1 in 620-143 or [05]620-193 will be permitted to enrol on completion of additional summer reading as prescribed by the coordinator.

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; and to 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, and absolute and conditional convergence; tests for convergence; power series of complex numbers and radius of convergence; basic topological concepts in the complex plane; continuous functions and their properties; homomorphic function and 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 and Weierstrass M-test; equivalence of complex differentiability to the local power series expansion; Laurent series, singularities and poles; and residue theorem, evaluation of integrals and summation of series.

Up to 24 pages of written assignments during the semester and a 50-minute written test held mid semester (equally weighted, with a total of either 0% or 20%); a 3-hour written examination in the examination period (80% or 100%). The relative weighting of the examination and the total assignment plus test mark will be chosen so as to maximise the student's final mark.