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 D^6 + ...)u_1/2 \
|
[delta]u_1/2 = (hD + 1/24 h^3D^3 + 1/1920 h^5 D^5 + ...)u_1/2 |
|
[mu][delta]^2u_1/2 = (h^2D^2 + 5/24 h^4 D^4 + 91/5760 h^6 D^6 + ...)u_1/2 >
|
[delta]^3u_1/2 = (h^3D^3 + 1/8 h^5 D^5 + ...)u_1/2 |
|
[mu][delta]^4 u_1/2 = (h^4 D^4 + 7/24 h^6 D^6 + ...)u_1/2 /
. . .
. . .
. . .
u0 = u0 \
|
hDu0 = ([mu][delta] - 1/6 [mu][delta]^3 + 1/30 [mu][delta]^5 - ...)u0 |
|
h^2D^2u0 = ([delta]^2 - 1/12 [delta]^4 + 1/90 [delta]^6 - ...)u0 >
|
h^3D^3u0 = ([mu][delta]^3 1/4 [mu][delta]^5 + ...)u0 |
|
h^4 D^4 u_0 = ([delta]^4 - 1/6 [delta]^6 + ...)u0 /
. . .
. . .
. . .
u_1/2 = ([mu] - 1/8 [mu][delta]^2 + 3/128 [mu][delta]^4 - 5/1024 [mu][delta]^6 + ...)u_1/2 \
|
hDu_1/2 = ([delta] - 1/24 [delta]^3 + 3/640 [delta]^5 - ...)u_1/2 |
|
h^2D^2u_1/2 = ([mu][delta]^2 - 5/24 [mu][delta]^ + 259/5760 [mu][delta]^6 - ...)u_1/2 >
|
h^3D^3u_1/2 = ([delta]^3 - 1/8 [delta]^5 + ...)u_1/2 |
|
h^4 D^4 u_1/2 = ([mu][delta]^4 - 7/24 [mu][delta]^6 + ...)u_1/2 /
. . .
. . .
. . .
When u is a rational integral function of x, each of the above series
is a terminating series. In other cases the series will be an infinite
one, and may be divergent; but it may be used for purposes of
approximation up to a certain point, and there will be a "remainder,"
the limits of whose magnitude will be determinate.
19. _Sums and Integrals._--The relation between a sum and an integral
is usually expressed by the _Euler-Maclaurin formula_. The principle
of this formula is that, if um and um+1, are ordinates of a curve,
distant h from one another, then for a first approximation to the area
of the curve between um and um+1 we have 1/2h(u_m + u_m+1), and the
difference between this and the true value of the area can be
expressed as the difference of two expressions, one of which is a
function of x_m, and the other is the same function of x_m+1. Denoting
these by [phi](x_m) and [phi](xm+1), we have
_ x_m+1
/
| udx = 1/2h(u_m + u_m+1) + [phi](x_m+1 ) - [phi](x_m).
_/x_m