An Introduction to the Calculus of Finite Differences

Preface iii

@=8
I. Introduction
...1. Definitions and Notation 1
...2. Tables of Differences  4
...3. Difference Formulas 7
...4. Symbolic Operators 16

II. Finite Integration and Applications
...1. Finite Integration 20
...2. Summation of Series 25
...3. More Advanced Methods of Integration 29
......A. Integration by Parts 29
......B. Undetermined Coefficients and Functions 31
...4. Stirling Numbers 35

III. Bernoulli and Euler Polynomials
...1. Bernoulli Functions, P_n(x) 44
...2. Properties of the Polynomials, P_n(x) 46
...3. Bernoulli Polynomials, B_n(x), and Numbers 48
...4. Other Developments of the Bernoulli Functions 51
...5. Bernoulli Polynomials of the Second Kind 55
...6. Euler Polynomials  58

IV. Interpolation. Approximate Integration
...1. Introduction 63
...2. Newton's Interpolation Formulas 64
...3. Formulas of Gauss, Stirling, and Bessel 68
...4. Equidistant Terms with Terms Missing 72
...5. Lagrange's Interpolation Formula 73
...6. Concluding Remarks on Interpolation  75
...7. Approximate Integration 76
......(a) The Trapezoidal Rule 77
......(b) Simpson's One-Third Rule 79
......(c) Simpson's Three-Eighths Rule  80
......(d) Weddle's Rule 81
......(e) The Euler-Maclaurin Sum Formula 82

V. Beta and Gamma Functions
...1. Introduction 87
...2. The Gamma Function 87
...3. The Beta Function 92
...4. Illustrative Examples 93

VI. Difference Equations
...1. Introduction 98
...2. Solution of a Difference Equation  99
...3. Derivation of a Difference Equation 101
...4. Solution of Simple Difference Equations 102
...5. Linear Equations of Order One 103
...6. Linear Equations of Order n 106
...7. Linear Equations with Constant Coefficients 107
...8. Some Helpful Theorems 111
...9. Applications of Theorems. Illustrative Examples 114
...10. Seliwanoff's Treatment of the Homogeneous Equation 117
...11. Multiple Roots in Auxiliary Equation by Operators 119
...12. Simultaneous Equations 121
...13. Rational Fractions. The Psi Function  123
...14. Miscellaneous Equations 126
...15. The Linear Equation of Order Two 127
...16. Concluding Remarks 129

Appendix I. Mathematical Induction 131
Appendix II. Hyperbolic Functions 136
Index 141