Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.

**The Riemann Integral; **Upper and lower sums, Definition of a Riemann integral, Integrability criterion, Classes of integrable functions, Properties of the Riemann integral,

**Infinite Series; **Review of sequences, Geometric series, Tests for convergence, Conditional and absolute convergence. Regrouping and rearrangement of series. Power series, radius of convergence,

**Uniform Convergence;** Uniform convergence of a sequence and a series, the M-test, properties of uniformly convergent series. Weierstrass approximation theorem,

**Improper Integrals; **Classification, tests for convergence, Absolute and conditional convergence, Convergence of cos(x) sinx dx, the Gamma function. Uniform convergence of integrals, M-text, properties of uniformly convergent integrals,

**Fourier series; **Orthogonal functions, Legendre, Hermite and Laguerre polynomials, Convergence in the mean. Fourier-Legendre and Fourier-Bessel series, Bessel inequality, Parseval equality. Convergence of the trigonometric Fourier series.

- Principles of Real Analysisby Malik, S.C.

Date Published: 2011

Pages: 386 - Multidimensional Real Analysis IIby Duistermaat, J. J.

Kolk, J. A. C.

van Braam Houckgeest, J. P. - Counterexamples in Calculusby Klymchuk, Sergiy

Date Published: 2014

Pages: 112