| up/q = u0 + p∂u0 + | p·p − 1 | ∂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 Δ, E, D, ... as if they were algebraical quantities. Thus the relation E = ehD (§ 17) gives
hD = loge (1 + Δ) = Δ − ½Δ² + 1⁄3Δ³ ...
or
h(du/dx)0 = Δu0 − ½Δ²u0 + 1⁄3Δ³u0 ....
The formulae connecting central differences with differential coefficients are based on the relations μ = cosh ½hD = ½(e1/2hD + e-1/2hD), δ = 2 sinh ½hD − e1/2hD − e-1/2hD, and may be grouped as follows:—
| u0 | = u0 | } |
| μδu0 | = (hD + 1⁄6 h³D³ + 1⁄120 h5D5 + ...)u0 | |
| δ²u0 | = (h²D² + 1⁄12 h4D4 + 1⁄360 h6D6 + ...)u0 | |
| μδ³u0 | = (h³D³ + 1⁄4 h5D5 + ...)u0 | |
| δ4u0 | = (h4D4 + 1⁄6 h6D6 + ...)u0 | |
| · · · | ||
| · · · | ||
| · · · | ||
| μu1/2 | = (1 + 1⁄8 h²D² + 1⁄384 h4D4 + 1⁄46080 h6D6 + ...)u1/2 | } |
| δu1/2 | = (hD + 1⁄24 h³D³ + 1⁄1920 h5D5 + ...)u1/2 | |
| μδ²u1/2 | = (h²D² + 5⁄24 h4D4 + 91⁄5760 h6D6 + ...)u1/2 | |
| δ³u1/2 | = (h³D³ + 1⁄8 h5D5 + ...)u1/2 | |
| μδ4 u1/2 | = (h4D4 + 7⁄24 h6D6 + ...)u1/2 | |
| · · · | ||
| · · · | ||
| · · · | ||
| u0 | = u0 | } |
| hDu0 | = (μδ − 1⁄6 μδ³ + 1⁄30 μδ5 − ...)u0 | |
| h²D²u0 | = (δ² − 1⁄12 δ4 + 1⁄90 δ6 − ...)u0 | |
| h³D³u0 | = (μδ³ − 1⁄4 μδ5 + ...)u0 | |
| h4D4u0 | = (δ4 − 1⁄6 δ6 + ...)u0 | |
| · · · | ||
| · · · | ||
| · · · | ||
| u1/2 | = (μ − 1⁄8 μδ² + 3⁄128 μδ4 − 5⁄1024 μδ6 + ...)u1/2 | } |
| hDu1/2 | = (δ − 1⁄24 δ³ + 3⁄640 δ5 − ...)u1/2 | |
| h²D²u1/2 | = (μδ² − 5⁄24 μδ4 + 259⁄5760 μδ6 − ...)u1/2 | |
| h³D³u1/2 | = (δ³ − 1⁄8 δ5 + ...)u1/2 | |
| h4D4 u1/2 | = (μδ4 − 7⁄24 μδ6 + ...)u1/2 | |
| · · · | ||
| · · · | ||
| · · · | ||
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 ½h(um + um+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 xm, and the other is the same function of xm+1. Denoting these by φ(xm) and φ(xm+1), we have