DEPARTMENT OF MATHEMATIC

THE UNIVERSITY OF KARACHI.

KARACHI.

M. A. PREVIOUS (EXTERNAL)

There will be five papers each of 100 marks and 3 hours’ duration.

Paper 1. Analysis

Algebra of sets: partition and equivalent classes: partially ordered sets and Axiom of Choice. Canonical decomposition of function.

R___________________: Euclidean metric space, Convergence of sequences, Completeness.

Functions of several real variables: their continuity and differentiability. Implicit and Inverse function Theorems Jacobians and functional dependence. Taylor’s Theorem (several variables)

Maxima Minima; Lagrange’s method of undetermined multipliers.

Riemann and Riemann — Stieltijes integrals. Differentiation under integral sign.

Line and surface integrals. Theorems of Gauss, Green and Stokes and their applications.

Uniform and absolute convergence of sequences and series of functions. Uniform convergence and continuity. Term by term differentiation and integration. Improper integrals and their convergence; their absolute and uniform convergence.

Functions of a complex variable. Analytic functions; power series.

Cauchy’s Theorem and integral formulas. Singularities and branch points. Taylor’s and Laurent series, Residue theorem and contour integration, Conformal mapping.

Reference Books

l. Apostol, T.M., Mathematical Analysis, Addison Wesley, 1976.
2. Kaplan, W., Advanced Calculus, Addison Wesley, 1965.
3. Rudin, W., Principles of Mathematical Analysis, Mc Graw Hill, 1976: (3/e)
4. Taylor, A. E., and Mann, W. R., Advanced Calculus, John Wiley, New York, 1983.
5. Churchill, R. V., Complex Variables, and Applications. Mc Graw Hill. 3 / 3,
6. Paliouras, F. D., Complex variables, Collier McMillan, New York, 1975.
7. Pennesi L.L., Elements of Complex variables, Holt, Rinehart and Winston, New York, 1967.