| 1/h | ∫ | xn | udx = μσ(un − u0) + 1⁄12(Δu0 − ½Δ²u0 + 19⁄60Δ³u0 − ...) + 1⁄12(Δ′ un − ½Δ′ ²un + 19⁄60Δ′ ³un − ...), |
| x0 |
where accented differences denote that the values of u are read backwards from un; i.e. Δ′un denotes un-1 − un, not (as in § 10) un − un-1.
(iii.) Expressed in terms of central differences this becomes
| 1/h | ∫ | xn | udx = μσ(un − u0) − 1⁄12μδun + 11⁄720 μδ³un − ... + 1⁄12μδu0 − 11⁄720 μδ³u0 + ... = μ(σ − 1⁄12δ + 11⁄720δ³ − 191⁄60480δ5 + 2497⁄3628800δ7 − ...)(un − u0). |
| x0 |
(iv.) There are variants of these formulae, due to taking hum+1/2 as the first approximation to the area of the curve between um and um+1; the formulae involve the sum u1/2 + u3/2 + ... + un-1/2 ≡ σ(un − u0) (see [Mensuration]).
20. The formulae in the last section can be obtained by symbolical methods from the relation
| 1/h | ∫ | udx = 1/h D-1u = 1/hD · u. |
Thus for central differences, if we write θ ≡ ½hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ-1, and the result in (iii.) corresponds to the formula
sinh θ = θ cosh θ/(1 + 1⁄3 sinh² θ − 2⁄3·5 sinh4 θ + 2·4⁄3·5·7 sinh6 θ − ...).
References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’s Finite Differences (1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’s Differenzenrechnung (German trans., 1896) contains general formulae. (Both these works ignore central differences.) Encycl. der math. Wiss. vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books, e.g. G. Chrystal’s Algebra (pt. 2, ch. xxxi.). A. W. Sunderland, Notes on Finite Differences (1885), is intended for actuarial students. Various central-difference formulae with references are given in Proc. Lond. Math. Soc. xxxi. pp. 449-488. For other references see [Interpolation].