Author Description

Unit-1 Riemann integral, Integrability of continuous and monotonic functions, The fundamental theorem of integral calculus, Mean value theorems of integral calculus. Partial derivatives and differentiability of real-valued functions of two variables. Schwarz's and Young's theorem, Implicit function theorem. Unit-2 Improper integrals and their convergence, Comparison tests, Abel's and Dirichlet's tests, Frullani's integral as a function of a parameter. Continuity, derivability and integrability of an integral of a function of a parameter. Fourier series of half and full intervals. Unit -3 Definition and examples of metric spaces, Neighbourhoods, Limit points, Interior points, Open and closed sets, Closure and interior, Boundary points, Subspace of a metric space. Cauchy sequences,Completeness, Cantor's intersection theorem. Contraction principle, Real number as a complete ordered field. Dense subsets, Baire Category theorem, Separable, second countable and first countable spaces. Unit-4 Continuous functions, Extension theorem, Uniform continuity. Compactness, Sequential compactness, Totally bounded spaces, Finite intersection property. Continuous functions and compact sets, Connectedness. Unit-5 Complex numbers as ordered pairs, Geometric representation of complex numbers. Continuity and Differentiability of Complex functions. Analytic functions, Cauchy-Riemann equations, Harmonic functions. Mobius transformations, Fixed points, Cross ratio, Inverse points. Conformal Mappings.