Author(s): C. H. Richardson
Publisher: D. Van Nostrand Company, Inc.
Year: 1954
Language: English
Pages: 149
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
