Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions (Paperback)

One Infinite Series and Infinite Sequences.- 1 Operations with Power Series.- Additive Number Theory, Combinatorial Problems, and Applications.- Binomial Coefficients and Related Problems.- Differentiation of Power Series.- Functional Equations and Power Series.- Gaussian Binomial Coefficients.- Majorant Series.- 2 Linear Transformations of Series. A Theorem of Cesaro.- Triangular Transformations of Sequences into Sequences.- More General Transformations of Sequences into Sequences.- Transformations of Sequences into Functions. Theorem of Cesaro.- 3 The Structure of Real Sequences and Series.- The Structure of Infinite Sequences.- Convergence Exponent.- The Maximum Term of a Power Series.- Subseries.- Rearrangement of the Terms.- Distribution of the Signs of the Terms.- 4 Miscellaneous Problems.- Enveloping Series.- Various Propositions on Real Series and Sequences.- Partitions of Sets, Cycles in Permutations.- Two Integration.- 1 The Integral as the Limit of a Sum of Rectangles.- The Lower and the Upper Sum.- The Degree of Approximation.- Improper Integrals Between Finite Limits.- Improper Integrals Between Infinite Limits.- Applications to Number Theory.- Mean Values and Limits of Products.- Multiple Integrals.- 2 Inequalities.- Inequalities.- Some Applications of Inequalities.- 3 Some Properties of Real Functions.- Proper Integrals.- Improper Integrals.- Continuous, Differentiate, Convex Functions.- Singular Integrals. Weierstrass Approximation Theorem.- 4 Various Types of Equidistribution.- Counting Function. Regular Sequences.- Criteria of Equidistribution.- Multiples of an Irrational Number.- Distribution of the Digits in a Table of Logarithms and Related Questions.- Other Types of Equidistribution.- 5 Functions of Large Numbers.- Laplace s Method.- Modifications of the Method.- Asymptotic Evaluation of Some Maxima.- Minimax and Maximin.- Three Functions of One Complex Variable. General Part.- 1 Complex Numbers and Number Sequences.- Regions and Curves. Working with Complex Variables.- Location of the Roots of Algebraic Equations.- Zeros of Polynomials, Continued. A Theorem of Gauss.- Sequences of Complex Numbers.- Sequences of Complex Numbers, Continued: Transformation of Sequences.- Rearrangement of Infinite Series.- 2 Mappings and Vector Fields.- The Cauchy-Riemann Differential Equations.- Some Particular Elementary Mappings.- Vector Fields.- 3 Some Geometrical Aspects of Complex Variables.- Mappings of the Circle. Curvature and Support Function.- Mean Values Along a Circle.- Mappings of the Disk. Area.- The Modular Graph. The Maximum Principle.- 4 Cauchy s Theorem The Argument Principle.- Cauchy s Formula.- Poisson s and Jensen s Formulas.- The Argument Principle.- Rouche s Theorem.- 5 Sequences of Analytic Functions.- Lagrange s Series. Applications.- The Real Part of a Power Series.- Poles on the Circle of Convergence.- Identically Vanishing Power Series.- Propagation of Convergence.- Convergence in Separated Regions.- The Order of Growth of Certain Sequences of Polynomials.- 6 The Maximum Principle.- The Maximum Principle of Analytic Functions.- Schwarz s Lemma.- Hadamard s Three Circle Theorem.- Harmonic Functions.- The Phragmen-Lindelof Method.- Author Index.