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1. Purpose The purpose of this work is to provide, in one volume, a wide spectrum of essential (non-measure theoretic) Mathematics for use by workers in the variety of applied fields. To obtain the background developed here in one volume would require studying a prohibitive number of separate Mathematics courses (assuming they were available). Before, much of the material now covered was (a) unavailable, (b) too widely scattered, or (c) too advanced as presented, to be of use to those who need it. Here, we present a sound basis requiring only Calculus through however, Differential Equations. It provides the needed flexibility to cope, in a rigorous manner, with the every-day, non-standard and new situations that present themselves. There is no substitute for this. 2. Arrangement The volume consists of twenty Sections, falling into several natural units: Basic Real Analysis 1. Sets, Sequences, Series, and Functions 2. Doubly Infinite Sequences and Series 3. Sequences and Series of Functions 4. Real Power Series 5. Behavior of a Function Near a Point: Various Types of Limits 6. Orders of Magnitude: the D, 0, ~ Notation 7. Some Abelian and Tauberian Theorems v Riemann-Stieltjes Integration 8. I-Dimensional Cumulative Distribution Functions and Bounded Variation Functions 9. I-Dimensional Riemann-Stieltjes Integral 10. n-Dimensional Cumulative Distribution Functions and Bounded Variation Functions 11. n-Dimensional Riemann-Stieltjes Integral The Finite Calculus 12. Finite Differences and Difference Equations Basic Complex Analysis 13. Complex Variables Applied Linear Algebra 14. Matrices and Determinants 15.
List of contents
1. Sets, Sequences, Series, and Functions.- 2. Doubly Infinite Sequences and Series.- 3. Sequences and Series of Functions.- 4. Real Power Series.- 5. Behavior of a Function Near a Point: Various Types of Limits.- 6. Orders of Magnitude: The 0, o, ~ Notation.- 7. Some Abelian and Tauberian Theorems.- 8. 1-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- 9. 1-Dimensional Riemann-Stieltjes Integral.- 10. n-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- 11. n-Dimensional Riemann-Stieltjes Integral.- 12. Finite Differences and Difference Equations.- 13. Complex Variables.- 14. Matrices and Determinants.- 15. Vectors and Vector Spaces.- 16. Systems of Linear Equations and Generalized Inverse.- 17. Characteristic Roots and Related Topics.- 18. Convex Sets and Convex Functions.- 19. Max-Min Problems.- 20. Some Basic Inequalities.
