| ∫ | xm+1 | udx = ½h(um + um+1) + φ(xm+1) − φ(xm). |
| xm |
Adding a series of similar expressions, we find
| ∫ | xn | udx = h{½um + um+1 + um+2 + ... + un-1 + ½un} + φ(xn) − φ(xm). |
| xm |
The function φ(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
| ∫ | xn | udx = h·μσun − 1⁄12 h² | dun | ½ + 1⁄720 h4 | d³un | − 1⁄30240 h6 | d5un | + ... = h·μσun − | B1 | h2 | dun | + | B2 | h4 | d3un | − | B3 | h6 | d5un | + ... |
| dx | dx3 | dx5 | 2! | dx | 4! | dx3 | 6! | dx5 |
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:—
| 1/h | ∫ | xn | udx = μσ(un − u0) − 1⁄12(Δun − Δu0) + 1⁄24(Δ²un − Δ²u0) − 19⁄720(Δ³un − Δ³u0) + 3⁄160(Δ4un − Δ4u0) − ... |
| x0 |
For practical calculations this may more conveniently be written