618-252 Analysis

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Prerequisite/s:

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Objectives:

On completion of this subject, students should:

Comprehend:

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

• the fundamentals of continuity, differentiability of functions of several real variables;

• the concepts of an analytic function of a complex variable; complex derivative; power and Laurent series in complex variables;

• basic topological concepts in the complex plane;

• Cauchy's theorem and its applications;

Have developed:

• 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:

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

• the role of complex analytic methods in solving important problems in science and engineering.

Content:

Sequences and Series: standard sequences and series, Cauchy convergence, ratio and n-th root tests, absolute and conditional convergence, re-arrangements, power series. Continuity: continuity and differentiability of functions of several real variables. Functions of a complex variable: elementary functions of a complex variable, branches; differentiation, analytic functions, Cauchy-Riemann equations. Integration: line and contour integrals, Cauchy's integral theorem; Laurent series; singularities, poles, Liouville's theorem; residue theorem, limiting contours, evaluation of integrals using contour integration.

Assessment: