_ x_n / 1 du_n 1 d³u_n 1 d^5 u_n | udx = h·[mu][sigma]u_n - -- h² ---- + --- h^4 ----- - ----- h^6 ------- + ... _/x_m 12 dx 720 dx³ 30240 dx^5
B1 du_n B2 d³u_n B3 d^5u_n = h·[mu][sigma]u_n - -- h2 ---- + -- h^4 ----- - -- h^6 ------ + ..., 2! dx 4! dx³ 6! dx^5
where B1, B2, B3 ... are _Bernoulli's numbers_.
(ii.) If we express differential coefficients in terms of advancing differences, we get a theorem which is due to Laplace:--
_ x_n 1 / - | udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]²u_n - [Delta]²u0) h _/x0
- 19/720 ([Delta]³u_n - [Delta]³u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...
For practical calculations this may more conveniently be written
_ x_n 1 / - | udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - ½[Delta]²u0 + 19/60 [Delta]³u0 - ...) h _/x0
+ 1/12 ([Delta]'u_n - ½[Delta]'²u_n + 19/60 [Delta]'³u_n - ...),
where accented differences denote that the values of u are read backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in § 10) u_n - u_n-1.
(iii.) Expressed in terms of central differences this becomes
_ x_n 1 / - | udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]³u_n - ... h _/x0 + 1/12 [mu][delta]u0 - 11/720 [mu][delta]³u0 + ...
/ 1 11 191 2497 \ / \ = [mu]([sigma] - -- [delta] + --- [delta]³ - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0). \ 12 720 60480 3628800 / \ /
(iv.) There are variants of these formulae, due to taking hum+½ as the first approximation to the area of the curve between um and um+1; the formulae involve the sum u_½ + u_3/2 + ... + u_n-½ := [sigma](u_n - u0) (see MENSURATION).
20. The formulae in the last section can be obtained by symbolical methods from the relation
_ 1 / 1 1 - | udx = - D^1 u = --·u. h _/ h hD
Thus for central differences, if we write [theta] := ½hD, we have [mu] = cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and the result in (iii.) corresponds to the formula
/ / 1 2 2·4 \ sinh [theta] = [theta] cosh [theta]/ (1 + - sinh²[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...). / \ 3 3·5 3·5·7 /
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. (W. F. SH.)
