Richardson C.H. An Introduction to the Calculus of Finite Differences

New York: D. Van Nostrand Company, Inc., 1954. — 149 p.Definitions and Notation
Tables of Differences
Difference Formulas
Symbolic Operators
Finite Integration and Applications
Finite Integration
Summation of Series
More Advanced Methods of Integration
A Integration by Parts
B Undetermined Coefficients and Functions
Stirling Numbers
Bernoulli and Euler Polynomials
Bernoulli Functions, P_n(x)
Properties of the Polynomials, P_n(x)
Bernoulli Polynomials, B_n(x), and Numbers
Other Developments of the Bernoulli Functions
Bernoulli Polynomials of the Second Kind
Euler Polynomials
Interpolation Approximate IntegrationNewton's Interpolation Formulas
Formulas of Gauss, Stirling, and Bessel
Equidistant Terms with Terms Missing
Lagrange's Interpolation Formula
Concluding Remarks on Interpolation
Approximate Integration
(a) The Trapezoidal Rule
(b) Simpson's One-Third Rule
(c) Simpson's Three-Eighths Rule
(d) Weddle's Rule
(e) The Euler-Maclaurin Sum Formula
Beta and Gamma FunctionsThe Gamma Function
The Beta Function
Illustrative Examples
Difference EquationsSolution of a Difference Equation
Derivation of a Difference Equation
Solution of Simple Difference Equations
Linear Equations of Order One
Linear Equations of Order n
Linear Equations with Constant Coefficients
Some Helpful Theorems
Applications of Theorems Illustrative Examples
Seliwanoff's Treatment of the Homogeneous Equation
Multiple Roots in Auxiliary Equation by Operators
Simultaneous Equations
Rational Fractions The Psi Function
Miscellaneous Equations
The Linear Equation of Order Two
Concluding Remarks
Appendix I Mathematical Induction
Appendix II Hyperbolic Functions