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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, Continuous functions, Uniform continuity, Properties of continuous fuctions on compact sets. Unit-4 Continuity and differentiability of complex functions. Analytic functions. Cauchy-Reimann equations. Harmonic functions, Cauchy's Theorem, Cauchy's Integral formula. Unit-5 Power series representation of an analytical function, Taylor's series, Laurant's series, Singularities, Cauchy's Residue Theorem, Contour Integration.