| = u0 + xΔu0 + | x(x-1) | Δ2u0 + ... |
| 1×2 |
a formula much used by calculators, and known as Newton's interpolation formula.
The above symbolic method of proof only applies when x is a positive integer, but the result is used in practice even for fractional values of x, as in most cases the high differences become negligible.
If n is a positive integer, it is easy to prove that
| Δnux = ux+n - nux+n-1 + | n(n-1) | ux+n-2 - ... |
| 1×2 |
If the nth differences vanish, or are negligible, this gives
| 0 = ux+n - nux+n-1 + | n(n-1) | ux+n-2 - ... +(-1)nux, |
| 1×2 |
another useful interpolation formula, by which we can calculate any missing term of a series.—Bibliography: G. Boole, Finite Differences; Textbook of the Institute of Actuaries.
Differential Equation, an algebraical relation involving derivatives or differentials. Examples:
| d2z | = g: ydx + xdy + zdz = 0. |
| dt2 |