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The exhaustive list of topics in Modern Analysis in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Sets: Sets of numbers, set operations, family of subsets, constructing
  • new sets from old ones, De Morgan’s laws.
  • Maps: Surjection, injection, invertible maps, direct and inverse images
  • and their relations, inverse of a map, composition of maps, restriction
  • of a map.
  • Finite and infinite sets, countable and uncountable sets, Results related
  • to countable sets.
  • Countability of rational numbers.
  • Absolute value of a real number, epsilon neibourhood of a point,
  • supremum and infimum and properties. Completeness property of the
  • real numbers. Archimedean property, Density of rational numbers.
  • Nested intervals property, uncountability of reals..
  • Sequences, convergence, examples.
  • Conditions of convergence, monotone sequence, examples.
  • Tests of convergence of sequences of real numbers.
  • Subsequences, Bolzano-Weierstrass theorem, Cauchy’s criterion.
  • Infinite series, tests of convergence.
  • Definition and examples of metric spaces.
  • Open ball, neighborhood of a point, open set, closed set, and their
  • properties.
  • Interior, exterior and their properties.
  • Closure of a set and their properties.
  • Limit point of a subset and limit of a sequence in a metric space,
  • Boundary of a set and properties.
  • Equivalence of metric spaces. Product spaces.
  • Continuity of a map between two metric spaces and examples.
  • Various characterizations continuous maps.
  • Cauchy sequence, Complete metric spaces. Example of complete and
  • incomplete metric spaces.
  • Completeness of the space of real numbers.
  • Compactness, examples. Continuous image of a compact space.
  • Heine-Borel theorem.
  • Separation. Connectedness, connected subsets of the real metric space.
  • Properties of connected spaces. Examples. characterization of
  • connectedness. Intermediate value theorem.
  • Differentiable functions, higher derivatives, Leibnitz Rule.
  • Rolle’s theorem, Mean value theorems.
  • L’ Hospital rule. Examples.
  • Taylor’s formula and Taylor series.
  • Monotone functions, maxima and minima.
  • Riemann Integral, example of functions which are not Riemann
  • integrable.
  • Criterion of Riemann integrability, Properties of Riemann integral.
  • Fundamental theorems of integral calculus.
  • Integration by parts.
  • Trapezoidal rule and Simpsion’s rules of approximation of a Riemann
  • integrals.
  • Improper integrals.
  • Sequence and Series of functions, uniform convergence of functions.
  • Weierstrass M-test.
  • Power Series, radius of convergence.
  • Properties of functions represented by real power series.
  • Fourier Series.
  • More on Fourier Series.