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

Adding a series of similar expressions, we find

_ x_n
/
| udx = h{1/2u_m + u_m+1 + u_m+2 + ... + u_n-1 + 1/2u_n} + [phi](x_n) - [phi](x_m).
_/x_m

The function [phi](x) can be expressed in terms either of differential
coefficients of u or of advancing or central differences; thus there
are three formulae.

(i.) The Euler-Maclaurin formula, properly so called, (due
independently to Euler and Maclaurin) is

_ x_n
/ 1 du_n 1 d^3u_n 1 d^5 u_n
| udx = h.[mu][sigma]u_n - -- h^2 ---- + --- h^4 ------ - ----- h^6 ------- + ...
_/x_m 12 dx 720 dx^3 30240 dx^5

B1 du_n B2 d^3u_n B3 d^5u_n
= h.[mu][sigma]u_n - -- h2 ---- + -- h^4 ------ - -- h^6 ------ + ...,
2! dx 4! dx^3 6! dx^5

where B1, B2, B3