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.

Inhalt

Sets, Sequences, Series, and Functions.- Basic Set Definitions.- Unions, Intersections (multiple).- Lim Inf, Lim Sup, Limit, Convergence of Set Sequences.- Sup, Inf, Max, Min of Real Sets.- Limit Point of Real Sets; Closure, Boundary.- Open, Closed Real Sets.- Bolzano-Weierstrass Theorem.- Limit Point of Real Sequences.- Lim Inf, Lim Sup, Limit, Convergence of Real Sequences.- Cauchy Criterion for Convergence.- Sup, Inf, Max, Min of Functions over Sets.- General Principle of Convergence for Real Series.- Properties of Convergent Series.- Bracketing and Reordering.- Non-negative Series, Absolute Convergence.- Tests for Convergence of Real Series.- Hints and Answers.- References.- 2. Doubly Infinite Sequences and Series.- Definitions and Notation: Sequences.- Lim Inf, Lim Sup, Limit, Convergence: Sequences.- Cauchy Criterion: Sequences.- Iterated Limits: Sequences.- Definition and Notation: Series.- Iterated Sums: Series.- Convergence: Series.- Non-Negative Series.- Absolute Convergence: Series.- Tests for Convergence: Series.- Interchange of Summation Order: Series.- Hints and Answers.- References.- 3. Sequences and Series of Functions.- Real Function Sequences: Definition, Notation.- Lim Inf, Lim Sup, Pointwise Convergence, Limit.- Pointwise Convergence: Shortcomings.- Uniform Convergence: Real Function Sequences.- Continuity of Limit Under Uniform Convergence.- Real Function Sequences: Monotone, Continuous.- Term-by-Term Integration.- Term-by-Term Differentiation.- Real Function Series: Definition, Notation.- Sum Function and Pointwide Convergence.- Interchanging Limit Operations: Dominated Convergence.- Interchanging Limit Operations: Fatou's Lemma.- Uniform Convergence: Real Function Series.- Real Function Series: Uniform Convergence Tests.- Continuity of Sum Function.- Term-by-Term Integration.- Term-by-Term Differentiation.- Multiply Infinite Case.- Hints and Answers.- References.- 4. Real Power Series.- Real Power Series about a Point.- Radius of Convergence.- Convergence.- Uniform Convergence of Real Power Series.- Interval of Convergence.- Continuity of Sum Function.- Term-by-Term Integration.- Term-by-Term Differentiation.- Taylor Series: Definition.- Real Geometric Series.- Hints and Answers.- References.- 5. Behavior of a Function Near a Point: Various Types of Limits.- Notation: Types of Limits.- Two-Sided Limit.- Continuity.- Left-Hand Limit.- Left-Continuity.- Right-Hand Limit.- Right-Continuity.- Extensions.- Operations With Limits.- L'Hospital's Rules.- Limit Infimum: Definition, Properties.- Limit Supremum: Definition, Properties.- Limit Infimum and Supremum: Combined Properties.- Applications: Generalized Inequalities.- Hints and Answers.- References.- 6. Orders of Magnitude: The 0, o, ~ Notation.- Comparing Asymptotic Magnitudes.- Same Order of Magnitude: the ~ Relation.- At Most Order of Magnitude: the 0 Relation.- Smaller Order of Magnitude: the o Relation.- Hints and Answers.- References.- 7. Some Abelian and Tauberian Theorems.- The Laplace Transform of a Function.- Nature of Abelian and Tauberian Theorems.- Classical Results.- Reformulations.- Functions of Slow and Regular Variation.- A General Abelian-Tauberian Theorem.- Infinite Series Version.- Hints and Answers.- References.- 8. 1-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- 1-C.D.F.: Definition, Properties.- 1-C.D.F.: Riemann-Continuous Case.- Functions of 1-C.F.F.'s.- Sequences of 1-C.D.F.'s: Complete, Weak Convergence.- Convergence Properties.- 1-B.V.F.'s: Definition, Relation to 1-C.D.F.'s.- 1-B.V.F.'s: Properties.- 1-B.V.F.'s: Alternate Definition by Variation Sums.- Combinations of 1-B.V.F.'s.- Sequences and Convergences of 1-B.V.F.'s.- Hints and Answers.- References.- 9. 1-Dimensional Riemann-Stieltjes Integral.- Approximating Sums: Partition of Bounded Interval [a,b).- Definition and Notation: Integral with 1-C.D.F. Integrator.- Sufficient Conditions for Existence.- Integration Over a Single Point.- Physical Interpretation of the Integral.- Extension of Definition: all Bounded Intervals.- Properties of the Integral.- Integral Inequalities.- A Mean Value Theorem.- Extension of Definition: Unbounded Intervals.- Limit Properties:.- Varying Integrand.- Varying Integrator.- Varying Integration Interval.- Integration-by-Parts:.- Version A.- Version B.- Extension of the Integral to Case of:.- 1-B.V.F. Integrators.- Complex-Valued Integrands.- Discontinuous Integrands.- Change-of-Variables Formula.- Differentiation of the Indefinite Integral.- Hints and Answers.- References.- 10. n-Dimensional Cumulative Distribution Functions and Bounded Variation Functions.- n-Monotonicity: Definition.- n-Monotonicity: Characterization in Differentiable Case.- n-C.D.F.'s: Definition, Properties.- Functions of n-C.D.F.'s.- n-C.D.F.'s: Riemann Continuous Case.- Sequences of n-C.D.F.'s: Complete, Weak Convergence.- n-B.V.F.'s: Definition, Relation to n-C.D.F.'s.- n-B.V.F.'s: Case of Differentiability.- n-B.V.F.'s: Alternate Characterization; Variation Sums.- Hints and Answers.- References.- 11. n-Dimensional Riemann-Stieltjes Integral.- Approximating Sums: Partition of n-Rectangle [a',b').- Definition and Notation: n-C.D.F. Integrator.- Sufficient Conditions for Existence.- Extensions of Definition to:.- Various Bounded Rectangles.- Unbounded Rectangles.- Unions of Rectangles.- Properties of the Integral.- A Mean Value Theorem.- Factoring, Iterated Integrals.- Limit Properties:.- Varying Integrand.- Varying Integrator.- Varying Integration Set.- An Integration-by-Parts Formula.- Extension of the Integral to Case of:.- n-B.V.F. Integrators.- Complex Valued Integrands.- Discontinuous Integrands.- Hints and Answers.- References.- 12. Finite Differences and Difference Equations.- Definition: ? and E Operators.- Definition: First Unit Differences.- Definition: n-th Unit Difference (n > 1).- Simple Ascending and Descending Factorials.- Stirling Numbers: First Kind.- Stirling Numbers: Second Kind.- ? Operator: Properties.- General Ascending and Descending Factorials.- Definition: ?-Inverse Operator.- Anti-Differences and Properties of ?-Inverse Operators.- Anti-Differences: Techniques for Obtaining.- Application: Summation of Series.- Definition: Difference Equations.- n-th Order Linear Difference Equations.- General Solution: n-th Ord…