618-201 Real and Complex Analysis

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Objectives:

On completion of this subject, students should:

Comprehend:

• the structure and methods of proof;

• the concept of convergence of sequences and series; elementary topology of the real line;

• basic topological concepts in the complex plane;

• the concept of a homomorphic function of a complex variable;

• power and Laurent series in complex variables;

• Cauchy's theorem and its applications.

Have developed:

• skills in constructing rigorous and accurate arguments;

• skills in determining the convergence or otherwise of sequences and series;

• skills in differentiating functions of a complex variable;

• skills in calculating contour integrals;

• the ability to work with analytic functions in the cut plane;

• the ability to apply Cauchy's integral formula and the residue theorem.

Appreciate:

• the importance and satisfaction of rigorous arguments via proofs;

• the fundamental concepts of topology of the complex plane;

• differences between functions of a real and a complex variable.

Content:

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: