h^2
f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,
1.2

which, expressed in terms of operators, is

h^2 h^3
E = 1 + hD + ---D^2 + ----- D^3 + ... = e^(hD).
1.2 1.2.3

This gives the relation between [Delta] and D. Also we have

2q.2q - 1
u2 = u0 + 2qdPu0 + --------- dP^2u0 + ...
1.2

3q.3q - 1
u3 = u0 + 3qdPu0 + --------- dP^2u0 + ...
1.2
. .
. .
. .

and, if p is any integer,

p.p - 1
u_(p/q) = u0 + pdPu0 + ------- dP^2u0 + ....
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) (S 17) gives

hD = log_e (1 + [Delta]) = [Delta] - 1/2[Delta]^2 + 1/3 [Delta]^3 ...

/du\
or h( -- ) = [Delta]u0 - 1/2[Delta]^2u0 + 1/3 [Delta]^3u0 ....
\dx/0

The formulae connecting central differences with differential
coefficients are based on the relations [mu] = cosh 1/2hD = 1/2(e^1/2hD
+ e^ -1/2hD), [delta] = 2 sinh 1/2hD - e^ 1/2hD - e^ -1/2hD, and
may be grouped as follows:--

u0 = u0 \
|
[mu][delta]u0 = (hD + 1/6 h^3D^3 + 1/120 h^5 D^5 + ...)u0 |
|
[delta]^2u0 = (h^2D^2 + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0 >
|
[mu][delta]^3u0 = (h^3D^3 + 1/4 h^5 D^5 + ...)u0 |
|
[delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0 /

. . .
. . .
. . .

[mu]u_1/2 = (1 + 1/8 h^2D^2 + 1/384 h^4 D^4 + 1/46080 h^6