Analysis

SYLLABUS- ANALYSIS, Unit-I
Definition and examples of metric spaces, Neighbourhoods, Interior points, Limit
points, Open and closed sets, Subspaces, Convergent and Cauchy sequences,
Completeness, Cantor's intersection theorem. Riemann integral, Integrability of
continuous and monotonic functions, Fundamental theorem of integral calculus, Mean
value theorems of integral calculus, Improper integrals and their convergence,
Comparison test, m-test, Abel's test, Dirichlet's test, Integral as a function of a parameter
and its differentiability and integrability.
Unit-II
Series of arbitrary terms, Convergence divergence and oscillation, Uniform convergence
of sequences and series of functions, Uniform convergence and continuity, Uniform
convergence and integration, Uniform convergence and differentiation, Power series,
Partial derivation and differentiability of real valued functions of two variables, Schwarz
and Young's theorem, Implicit function theorem, Fourier series, Fourier expansion of
piece wise monotonic functions.
Unit-III
Complex numbers as ordered pairs, geometric representation of complex numbers,
Stereographic projection, Continuity and Differentiability of complex functions,
Analytic functions, Cauchy Riemann equations, Harmonic functions, complex
integration, Cauchy-Goursat theorem, Cauchy's Integral formula, Formulae for first,
second and nth derivatives, Cauchy's Inequality, Maximum Moduli theorem, Liouville's
Theorem, Elementary functions, Mapping by elementary functions.
Unit-IV
Series, Taylor and Laurent Series, Absolute and uniform convergence of Power series,
Residues and Poles, Residue theorem, Zeros and Poles of order m, Evaluation of improper
real integrals, Improper Integrals and definite integrals involving sines and cosines,
conformal mapping, Analytic continuation.