Pure Mathematics
Total Marks – 200
Paper II
Marks – 100
Candidates will be asked to attempt any three questions from Section A and two questions from Section B
Section A Calculus and Real Analysis
1. Real numbers, limits, continuity, differentiability, indefinite integration, mean value theorems. Taylor’s theorems, indeterminate form. Asymptotes, curve tracing, definite integrals, functions of several variables. Partial derivatives. Maxima and minima. Jacobeans, double and triple integration (Techniques only). Applications of Beta and Gamma functions. Areas and volumes. Riemann Stieltjes integral. Improper integrals and their conditions of existence. Implicit function theorem. Absolute and conditional convergence of series of real terms. Rearrangement of series, uniform convergence of series
2. Metric spaces. Open and closed spheres. Closure, interior and exterior of a set
3. Sequence in metric space. Cauchy sequence, convergence of sequences, examples, complete metric spaces, continuity in metric spaces. Properties of continuous functions
Section B Complex analysis
Function of a complex variable, Demoivre’s theorem and its applications. Analytic functions, Cauchy’s theorem. Cauchy’s integral formula, Taylor’s and Laurent’s series. Singularities. Cauchy residue theorem and contour integration. Fourier series and Fourier transforms. Analytic continuation