_ 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 ... 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) - ...