Analysis Real and Complex/R.Y. Denis, S.K.D. Dubey, M.U. Khan and R.P.
Singh. New Delhi, Dominant, 2005, x, 378 p., ISBN 81-7888-297-3.
Analysis Real and Complex/R.Y. Denis, S.K.D. Dubey, M.U. Khan and R.P.
Singh. New Delhi, Dominant, 2005, x, 378 p., ISBN 81-7888-297-3.
Contents: Real analysis: I. Sequences of real numbers: 1. Basic concepts of bounds, intervals and limiting points. 2. Limit of a sequence, bounded sequence, convergent and divergent, sequences. 3. Monotonic sequences, operation on convergent sequences. 4. Cauchy's sequence, Cauchy's theorem on limits and Cauchy's principle on convergence of a real sequence. 5. Limit point of a sequence, upper and lower limits of sequence in R. 6. Limit superior and limit inferior. II. Sequences of rational numbers: 1. Axiomatic study of real numbers as the limit of the sequence of rational numbers. 2. Countability. 3. Neighbourhoods, open sets, closed sets, derived sets and their related theorems. 4. Bolzano-Weierstrass theorem and closures. 5. Dense, non-dense perfect and isolated sets and their theorems. 6. Definition of metric spaces and some theorems. III. Limits and continuity: 1. Limits of real valued functions. 2. Right hand and left hand limits. 3. Continuity of a function and their properties. 4. Discontinuity of functions and its kinds and simple problems based on them. Complex Analysis: IV. Analytic functions and power series: 1. Basic concepts of complex numbers. 2. Analytic functions of complex variable. 3. Cauchy-Riemann equations. 4. Orthogonal system. 5. Convergence, circle of convergence and radius of convergence of the power series. V. Conformal mappings and complex integration: 1. Basic concepts and necessary and sufficient conditions for conformal mappings. 2. Linear; Bilinear transformations and invariance of cross ratio. 3. The transformation w = Zn; w= Z. 4. The inverse transformation Z = W. 5. Real line integral and complex integration. 6. Cauchy's theorem. 7. Cauchy's integral formula. 8. Morera's theorem. 9. Taylor's series. 10. Laurent's theorem (series). 11. Singularities and poles. 12. Zeros and poles of an analytic function. VI. Calculus of residues: 1. Residues, Cauchy's residue theorem. 2. Certain theorems and Jordan's inequality. 3. Examples of real integrals by contour integral: infiniteintegrals. 4. Evaluation of contour integrals round the unit circle.