An Introduction to Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
Published on 6. June 2016
624 pages
978-1-4831-3730-8 (ISBN)
Description
An Introduction to Mathematical Analysis is an introductory text to mathematical analysis, with emphasis on functions of a single real variable. Topics covered include limits and continuity, differentiability, integration, and convergence of infinite series, along with double series and infinite products.
This book is comprised of seven chapters and begins with an overview of fundamental ideas and assumptions relating to the field operations and the ordering of the real numbers, together with mathematical induction and upper and lower bounds of sets of real numbers. The following chapters deal with limits of real functions; differentiability and maxima, minima, and convexity; elementary properties of infinite series; and functions defined by power series. Integration is also considered, paying particular attention to the indefinite integral; interval functions and functions of bounded variation; the Riemann-Stieltjes integral; the Riemann integral; and area and curves. The final chapter is devoted to convergence and uniformity.
This monograph is intended for mathematics students.
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Content
PrefaceList of Symbols and NotationsI. Fundamental Ideas and Assumptions 1. Introduction 2. Assumptions Relating to the Field Operations 3. Assumptions Relating to the Ordering of the Real Numbers 4. Mathematical Induction 5. Upper and Lower Bounds of Sets of Real Numbers 6. FunctionsII. Limits and Continuity 7. Limits of Real Functions on the Positive Integers 8. Limits of Real Functions of a Real Variable x as x Tends to Infinity 9. Elementary Topological Ideas 10. Limits of Real Functions at Finite Points 11. Continuity 12. Inverse Functions and Fractional IndicesIII. Differentiability 13. Derivatives 14. General Theorems Concerning Real Functions 15. Maxima, Minima and Convexity 16. Complex Numbers and FunctionsIV. Infinite Series 17. Elementary Properties of Infinite Series 18. Series with Non-Negative Terms 19. Absolute and Conditional Convergence 20. The Decimal Notation for Real NumbersV. Functions Defined by Power Series 21. General Theory of Power Series 22. Real Power Series 23. The Exponential and Logarithmic Functions 24. The Trigonometric Functions 25. The Hyperbolic Functions 26. Complex IndicesVI. Integration 27. The Indefinite Integral 28. Interval Functions and Functions of Bounded Variation 29. The Riemann-Stieltjes Integral 30. The Riemann Integral 31. Curves 32. AreaVII. Convergence and Uniformity 33. Upper and Lower Limits and Their Applications 34. Further Convergence Tests for Infinite Series 35. Uniform Convergence 36. Improper Integrals 37. Double Series 38. Infinite ProductsHints for Solutions of ExercisesIndex
