Mathematical Analysis
Functions, Limits, Series, Continued Fractions
Published on 15. July 2014
418 pages
978-1-4831-9436-3 (ISBN)
Description
Mathematical Analysis: Functions, Limits, Series, Continued Fractions provides an introduction to the differential and integral calculus. This book presents the general problems of the theory of continuous functions of one and several variables, as well as the theory of limiting values for sequences of numbers and vectors. Organized into six chapters, this book begins with an overview of real numbers, the arithmetic linear continuum, limiting values, and functions of one variable. This text then presents the theory of series and practical methods of summation. Other chapters consider the theory of numerical series and series of functions and other analogous processes, particularly infinite continued fractions. This book discusses as well the general problems of the reduction of functions to orthogonal series. The final chapter deals with constants and the most important systems of numbers, including Bernoulli and Euler numbers. This book is a valuable resource for mathematicians, engineers, and research workers.
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Content
ForewordChapter I The Arithmetical Linear Continuum and Functions Defined There § 1. Real Numbers and their Representation 1. Real Numbers 2. The Numerical Straight Line 3. p-Adic Systems 4. Sets of Real Numbers 5. Bounded Sets, Upper and Lower Bounds 6. The Theory of Irrational Numbers § 2. Functions. Sequences 1. Functions of One Variable 2. Upper and Lower Bounds of a Function 3. Even and Odd Functions 4. Inverse Functions 5. Periodic Functions 6. Functional Equations 7. Numerical Sequences 8. Upper and Lower Bounds of a Sequence 9. Maximum Term of a Sequence 10. Monotonic Sequences 11. Double Sequences § 3. Passage to the Limit 1. The Limit Point of a Set 2. The Limit Point and Limit of a Sequence 3. Fundamental Theorems Concerning Limits 4. Some Propositions on Limits 5. Upper and Lower Limits of a Sequence 6. Uniformly Distributed Sequences 7. Recurrent Sequences 8. The Symbols o(an) and O(an) 9. Limit of a Function 10. Right and Left Continuity of a Function 11. Continuous and Discontinuous Functions 12. Functional Sequences 13. Uniform Convergence of Functions 14. Convergence in the Mean 15. The Symbols o(x) and O(x) 16. Monotonie Functions 17. Convex FunctionsChapter II n-Dimensional Spaces and Functions Defined There Introduction § 1, n-dimensional Spaces 1. n-Dimensional Coordinate Space 2. n-dimensional Vector Space 3. Scalar Product 4. A Linear System and its Basis 5. Linear Functions 6. Linear Envelope 7. Orthogonal Systems of Vectors 8. Biorthogonal Systems of Vectors 9. The Projection of a Vector on to a Manifold § 2. Passage to the Limit, Continuous Functions and Operators 1. Passage to the Limit in n-dimensional Space 2. Series of Vectors 3. Continuous Functions of n Variables 4. Periodic Functions of n Variables. Manifolds of Constancy 5. Passage to the Limit for Linear Envelopes 6. Operators From En into Em 7. Iterative Sequences 8. The Principle of Contraction Mappings § 3. Convex Bodies in n-dimensional Space 1. Fundamental Definitions 2. Convex Functions 3. Convex Bodies and the Norm of a Vector 4. Support Hyperplanes 5. Support Functions and Conjugate Spaces 6. Fundamental Theorems on Support Hyperplanes 7. The Connection Between Reciprocal Convex Bodies 8. The Cone. The Tangent Cone 9. Helly's Theorem 10. Linear Operations on SetsChapter III Series Introduction 1. Basic Concepts 2. Some Convergence Tests for Series § 1. Numerical Series 1. Alternating Series and Series of Constant Sign 2. Properties of Convergent Series. The Associative Property 3. General Tests for the Convergence of Series of Positive Terms 4. Remainder Term Estimates Corresponding to the Various Convergence Tests 5. Special Tests for the Convergence of Series of Positive Terms. Estimates of the Remainder Term 6. The Convergence of Alternating Series 7. Infinite Products and their Convergence 8. Double Series. Fundamental Concepts and Definitions 9. Some Properties of Double Series 10. Some Convergence Tests for Double Series of Positive Terms. Estimates of Remainder Term § 2. Series of Functions 1. Fundamental Properties and Convergence Tests 2. Power Series 3. Operations on Power Series. Taylor Series. Integration and Differentiation of Power Series 4. Complex Series 5.
