Riassunto
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.
Contenuti
1. 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 Order Homogeneous; Constant Coefficient.- n-th Order Non-Homogeneous Case; Constant Coefficients.- Techniques of Solution.- Simultaneous Difference Equations.- Hints and Answers.- References.- 13. Complex Variables.- Basic Definitions.- Modulus.- Addition, Multiplication, Division.- Conjugate.- Polar Form.- Function of a Complex Variable.- Limit at a Point.- Properties of Limit.- Differentiability at a Point.- Cauchy-Riemann Equations.- Sufficient Condition for Differentiability.- Regularity.- Derivatives of Regular Functions.- Complex Power Series.- Contours.- Integral of a Complex Function.- Properties of the Integral.- Cauchy’s Theorem and Goursat’s Lemma.- Evaluation of Certain Integrals by Cauchy’s Theorem.- Cauchy’s Integral Formula.- Integral Formula for Derivatives of Regular Functions.- Morera’s Theorem: a Converse of Cauchy’s Theorem.- Taylor’s Theorem: General Form.- Comparing Complex and Real Variables.- Integral Inequality due to Cauchy.- Liouville’s Theorem.- Fundamental Theorem of Algebra.- Zeros of a Function.- Isolated Zeros.- Poles and Singularities.- Laurent’s Theorem and Expansion.- Types of Singularities.- Residues.- Evaluation of Residues.- Method A.- Method B.- Fundamental Residue Theorem.- Applications: Evaluation of Contour Integrals.- Hints and Answers.- References.- 14. Matrices and Determinants.- Definitions.- Addition of Matrices.- Multiplication of Matrices.- Transpose of a Matrix.- Conjugate of a Matrix.- Determinant of a Square Matrix.- Submatrix, Minor, Principal Minor, Cofactor.- Evaluation of a Determinant.- Properties of a Determinant.- Inverse: Existence and Uniqueness.- Special Types of Square Matrices and Their Properties.- Singular, Non-Singular.- Symmetric.- Hermetian.- Skew-Symmetric.- Unitary.- Normal.- Orthogonal.- Hints and Answers.- References.- 15. Vectors and Vector Spaces.- Row Vectors of Complex Numbers.- Independence, Dependence of Sets of Row Vectors.- Vector Space and Vector Subspace.- Subspace Generated by Rows of a Matrix.- Basis of a Subspace.- Row Operations on a Matrix.- Existence of a Basis.- Unique Representation in Terms of a Fixed Basis.- Transformations of Basis Vectors.- Ranks of Subspaces.- Inner (dot) Product of Vectors.- Length of a Vector.- Orthogonality of Vectors.- Orthogonal Subspaces.- Ortho-Normal Basis: the Gram-Schmidt Procedure.- Conjugate Subspaces.- Hints and Answers.- References.- 16. Systems of Linear Equations and Generalized Inverse.- m Homogeneous Linear Equations in n(? m) Unknowns.- General Solution.- “Sweep Out” Technique for Finding General Solution.- Vector-Space Interpretation of General Solution.- Rank of Matrix and Rank of Subspace.- Properties of Rank.- m Non-Homogeneous Equations in n(? m) Unknowns.- Consistency and Inconsistency.- “Sweep Out” Technique for Finding General Solution.- Vector-Space Interpretation of General Solution.- Generalized Inverse of a Matrix.- Special Case of Non-Singular Matrix.- Technique for Obtaining a Generalized Inverse.- Hints and Answers.- References.- 17. Characteristic Roots and Related Topics.- Characteristic Root.- Characteristic Vector.- Characteristic Polynomial.- Characteristic Equation.- Determinant and Characteristic Roots.- Characteristic Roots of:.- Markov Matrices.- Transpose and Conjugate.- Similar Matrices.- Inverse.- Scalar Multiple.- Triangular Matrices.- Real, Symmetric Matrices.- Trace.- Characteristic Roots and Trace.- Characteristic Roots of AB and BA.- Characteristic Roots of Powers of a Matrix.- Ortho-Normal Characteristic Vectors.- Characteristic Roots of Orthogonal and Unitary Matrices.- Representation Theorem: Real, Symmetric Matrices.- Rank and Characteristic Roots.- Real Quadratic Forms.- Positive Definite, Positive Semi-Definite Forms.- Principal Axes Theorem.- Inverse of a P.D. Matrix.- Characteristic Roots and P.D., P.S.D. Matrices.- Submatrices of a P.D., P.S.D. Matrix.- Test for P.D. or P.S.D. Matrix.- Characterization of P.D. Matrices.- Largest Characteristic Root.- Applications of Largest Characteristic Root.- Hints and Answers.- References.- 18. Convex Sets and Convex Functions.- Definition: Convex Set in En.- Convex Linear Combination of Points in En.- Convex Hull.- Inner Product of Points in En.- Hyperplane in En.- Separating Hyperplanes.- Supporting Hyperplanes.- Characterizations for Convex Subsets of En:.- Separation Theorem.- Support Theorem.- Representation Theorem.- Convex Functions of n=1 Variable.- Characterization: Case When Second Derivative Exists.- Properties.- Convex Functions of n>1 Variables.- Sections of a Convex Subset of En.- Restrictions of Convex Functions n.- Jensen’s Inequality.- Concave Functions.- Arithmetic-Geometric-Harmonic Mean Inequality.- Hints and Answers.- References.- 19. Max-Min Problems.- Statement of Problem.- Relative vs. Global Max-Min.- Critical Points.- Unconstrained Max-Min: n=1 Variable.- Unconstrained Max-Min: n>1 Variables.- Constrained Max-Min.- Rationale of Lagrange Multipliers.- Lagrange Function.- Locating Critical Points.- Testing Critical Points.- Limitations.- Generalizations: Linear and Non-Linear Programming.- Hints and Answers.- References.- 20. Some Basic Inequalities.- Cauchy Inequality:.- Finite Series Version.- Infinite Series Version.- Complex Series Version.- Riemann-Stieltjes Integral Version.- p-q Inequality.- Hölder Inequality:.- Finite Series Version.- Infinite Series Version.- Complex Series Version.- Riemann-Stieltjes Integral Version.- Triangle Inequality: Finite Series Version.- Minkowski Inequality:.- Finite Series Version.- Infinite Series Version.- Riemann-Stieltjes Integral Version.- cr Inequality.- Hints and Answers.- References.
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