16. The cases of greatest practical importance are those in which u is a continuous function of x. The terms u1, u2 ... of the series then represent the successive values of u corresponding to x = x1, x2.... The important applications of the theory in these cases are to (i.) relations between differences and differential coefficients, (ii.) interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.
17. Starting from any pair of values x0 and u0, we may suppose the interval h from x0 to x1 to be divided into q equal portions. If we suppose the corresponding values of u to be obtained, and their differences taken, the successive advancing differences of u0 being denoted by dPu0, dP²u0 ..., we have (§ 3 (ii.))
q·q - 1 u1 = u0 + qdPu0 + ------- dP²u0 + .... 1·2
When q is made indefinitely great, this (writing f(x) for u) becomes Taylor's Theorem (INFINITESIMAL CALCULUS)
h² f(x + h) = f(x) + hf'(x) + --- f"(x) + ..., 1·2
which, expressed in terms of operators, is
h² h³ E = 1 + hD + ---D² + ----- D³ + ... = e^(hD). 1·2 1·2·3
This gives the relation between [Delta] and D. Also we have
2q·2q - 1 u2 = u0 + 2qdPu0 + --------- dP²u0 + ... 1·2
3q·3q - 1 u3 = u0 + 3qdPu0 + --------- dP²u0 + ... 1·2 . . . . . .
p·p - 1 u_(p/q) = u0 + pdPu0 + ------- dP²u0 + .... 1·2
From these equations up/q could be expressed in terms of u0, u1, u2, ...; this is a particular case of interpolation (q.v.).
18. _Differences and Differential Coefficients._--The various formulae are most quickly obtained by symbolical methods; i.e. by dealing with the operators [Delta], E, D, ... as if they were algebraical quantities. Thus the relation E = e^(hD) (§ 17) gives
hD = log_e (1 + [Delta]) = [Delta] - ½[Delta]² + 1/3 [Delta]³ ...
/du\ or h( -- ) = [Delta]u0 - ½[Delta]²u0 + 1/3 [Delta]³u0 .... \dx/0
The formulae connecting central differences with differential coefficients are based on the relations [mu] = cosh ½hD = ½(e^ ½hD + e^ -½hD), [delta] = 2 sinh ½hD - e^ ½hD - e^ -½hD, and may be grouped as follows:--
u0 = u0 \ | [mu][delta]u0 = (hD + 1/6 h³D³ + 1/120 h^5 D^5 + ...)u0 | | [delta]²u0 = (h²D² + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0 > | [mu][delta]³u0 = (h³D³ + 1/4 h^5 D^5 + ...)u0 | | [delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0 /