1911 Encyclopædia Britannica/Differences, Calculus of

by William Fleetwood Sheppard

DIFFERENCES, CALCULUS OF (Theory of Finite Differences), that branch of mathematics which deals with the successive differences of the terms of a series.

1. The most important of the cases to which mathematical methods can be applied are those in which the terms of the series are the values, taken at stated intervals (regular or irregular), of a continuously varying quantity. In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e. the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry), i.e. where the value of each term depends on the values of a pair of other quantities.

2. The first differences of a series are obtained by subtracting from each term the term immediately preceding it. If these are treated as terms of a new series, the first differences of this series are the second differences of the original series; and so on. The successive differences are also called differences of the first, second, ... order. The differences of successive orders are most conveniently arranged in successive columns of a table thus:—

Term.	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
a
	b − a
b		c − 2b + a
	c − b		d − 3c + 3b − a
c		d − 2c + b		e − 4d + 6c − 4b + a
	d − c		e − 3d + 3c − b
d		e − 2d + c
	e − d
e

Algebra of Differences and Sums.

Fig. 1.

3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms α, β, γ, δ, commencing with the same term α, and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law; i.e. the coefficients in b − a, c − 2b + a, d − 3c + 3b − a, ... and in α + β, α + 2β + γ, α + 3β + 3γ + δ, ... are respectively the same as in y − x, (y − x)², (y − x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B); e.g. β + γ is the difference of α + 2β + γ and α + β, just as c − b is the difference of c and b: and d − c is the sum of c − b and d − 2c + b, just as β + 2γ + δ is the sum of β + γ and γ + δ. Hence if we take β, γ, δ, ... of (B) as being the same as b − a, c − 2b + a, d − 3c + 3b − a, ... of (A), all corresponding terms in the two diagrams will be the same.

Thus we obtain the two principal formulae connecting terms and differences. If we provisionally describe b − a, c − 2b + a, ... as the first, second, ... differences of the particular term a (§ 7), then (i.) the nth difference of a is

$l-nk+\ldots +(-1)^{n-2}{\frac {n\cdot n-1}{1\cdot 2}}c+(-1)^{n-1}nb+(-1)^{n}a,$

where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c, ...; the coefficients being those of the terms in the expansion of (y − x)ⁿ: and (ii.) the (n + 1)th term of the series, i.e. the nth term after a, is

$a+n\beta +{\frac {n\cdot n-1}{1\cdot 2}}\gamma +\ldots$

where β, γ, ... are the first, second, ... differences of a; the coefficients being those of the terms in the expansion of (x + y)ⁿ.

4. Now suppose we treat the terms a, b, c, ... as being themselves the first differences of another series. Then, if the first term of this series is N, the subsequent terms are N + a, N + a + b, N + a + b + c, ...; i.e. the difference between the (n + 1)th term and the first term is the sum of the first n terms of the original series. The term N, in the diagram (A), will come above and to the left of a; and we see, by (ii.) of § 3, that the sum of the first n terms of the original series is

$\left(N+na+{\frac {n\cdot n-1}{1\cdot 2}}\beta +\ldots \right)-N{=}na+{\frac {n\cdot n-1}{1\cdot 2}}\beta +{\frac {n\cdot n-1\cdot n-2}{1\cdot 2\cdot 3}}\gamma +\ldots$

5. As an example, take the arithmetical series

a, a + p, a + 2p, ...

The first differences are p, p, p, ... and the differences of any higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and, by § 4, the sum of the first n terms is na + ½n(n − 1)p = ½n{2a + (n − 1)p}.

6. As another example, take the series 1, 8, 27, ... the terms of which are the cubes of 1, 2, 3, ... The first, second and third differences of the first term are 7, 12 and 6, and it may be shown (§ 14 (i.)) that all differences of a higher order are zero. Hence the sum of the first n terms is

$n+7{\frac {n\cdot n-1}{1\cdot 2}}+12{\frac {n\cdot n-1\cdot n-2}{1\cdot 2\cdot 3}}+6{\frac {n\cdot n-1\cdot n-2\cdot n-3}{1\cdot 2\cdot 3\cdot 4}}{=}{\tfrac {1}{4}}n^{4}+{\tfrac {1}{2}}n^{3}+{\tfrac {1}{4}}n^{2}{=}{{\tfrac {1}{2}}n(n+1)}^{2}$

7. In § 3 we have described b − a, c − 2b + a, ... as the first, second, ... differences of a. This ascription of the differences to particular terms of the series is quite arbitrary. If we read the differences in the table of § 2 upwards to the right instead of downwards to the right, we might describe e − d, e − 2d + c, ... as the first, second, ... differences of e. On the other hand, the term of greatest weight in c − 2b + a, i.e. the term which has the numerically greatest coefficient, is b, and therefore c − 2b + a might properly be regarded as the second difference of b, and similarly e − 4d + 6c − 4b + a might be regarded as the fourth difference of c. These three methods of regarding the differences lead to three different systems of notation, which are described in §§ 9, 10 and 11.

Notation of Differences and Sums.

8. It is convenient to denote the terms a, b, c, ... of the series by u₀, u₁, u₂, u₃, ... If we merely have the terms of the series, u_n may be regarded as meaning the (n + 1)th term. Usually, however, the terms are the values of a quantity u, which is a function of another quantity x, and the values of x, to which a, b, c, ... correspond, proceed by a constant difference h. If x₀ and u₀ are a pair of corresponding values of x and u, and if any other value x₀ + mh of x and the corresponding value of u are denoted by x_m and u_m, then the terms of the series will be ... u_n-2, u_n−1, u_n, u_n+1, u_n+2 ..., corresponding to values of x denoted by ... x_n-2, x_n−1, x_n, x_n+1, x_n+2....

9. In the advancing-difference notation u_n+1 − u_n is denoted by Δu_n. The differences Δu₀, Δu₁, Δu₂ ... may then be regarded as values of a function Δu corresponding to values of x proceeding by constant difference h; and therefore Δu_n+1 − Δu_n denoted by ΔΔu_n, or, more briefly, Δ²u_n; and so on. Hence the table of differences in § 2, with the corresponding values of x and of u placed opposite each other in the ordinary manner of mathematical tables, becomes

x	u	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·
xn-2	un-2		Δ²un-3		Δ4un-4 ...
		Δun-2		Δ³un-3
xn−1	un−1		Δ²un-2		Δ4un-3 ...
		Δun−1		Δ³un-2
xn	un		Δ²un−1		Δ4un-2 ...
		Δun		Δ³un−1
xn+1	un+1		Δ²un		Δ4un−1 ...
		Δun+1		Δ³un
xn+2	un+2		Δ²un+1		Δ4un  ...
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·

The terms of the series of which ... u_n−1, u_n, u_n+1, ... are the first differences are denoted by Σu, with proper suffixes, so that this series is ... Σu_n-1, Σu_n, Σu_n+1.... The suffixes are chosen so that we may have ΔΣu_n = u_n, whatever n may be; and therefore (§ 4) Σu_n may be regarded as being the sum of the terms of the series up to and including u_n-1. Thus if we write Σu_n-1 = C + u_n-2, where C is any constant, we shall have

Σu_n = Σu_n-1 + ΔΣu_n-1 = C + u_n-2 + u_n-1,

Σu_n+1 = C + u_n-2 + u_n-1 + u_n,

and so on. This is true whatever C may be, so that the knowledge of ... u_n-1, u_n, ... gives us no knowledge of the exact value of Σu_n; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of Σu corresponding to different suffixes.

There is another symbol E, used in conjunction with u to denote the next term in the series. Thus Eu_n means u_n+1, so that Eu_n = u_n + Δu_n.

10. Corresponding to the advancing-difference notation there is a receding-difference notation, in which u_n+1 − u_n is regarded as a difference of u_n+1, and may be denoted by Δ′u_n+1, and similarly u_n+1 − 2u_n + u_n-1 may be denoted by Δ′²u_n+1. This notation is only required for certain special purposes, and the usage is not settled (§ 19 (ii.)).

11. The central-difference notation depends on treating u_n+1 − 2u_n − u_n-1 as the second difference of u_n, and therefore as corresponding to the value x_n; but there is no settled system of notation. The following seems to be the most convenient. Since u_n is a function of x_n, and the second difference u_n+2 − 2u_n+1 + u_n is a function of x_n+1, the first difference u_n+1 − u_n must be regarded as a function of x_n+1/2, i.e. of ½(x_n + x_n+1). We therefore write u_n+1 − u_n = δu_n+1/2, and each difference in the table in § 9 will have the same suffix as the value of x in the same horizontal line; or, if the difference is of an odd order, its suffix will be the means of those of the two nearest values of x. This is shown in the table below.

In this notation, instead of using the symbol E, we use a symbol μ to denote the mean of two consecutive values of u, or of two consecutive differences of the same order, the suffixes being assigned on the same principle as in the case of the differences. Thus

μu_n+1/2 = ½(u_n + u_n+1, μδu_n = ½(δu_n-1/2 + δu_n+1/2, &c.

If we take the means of the differences of odd order immediately above and below the horizontal line through any value of x, these means, with the differences of even order in that line, constitute the central differences of the corresponding value of u. Thus the table of central differences is as follows, the values obtained as means being placed in brackets to distinguish them from the actual differences:—

x	u	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·
xn-2	un-2	(μδun-2)	δ²un-2	(μδ³un-2)	δ4un-2 ...
		δun-3/2		δ³un-3/2
xn-1	un-1	(μδun-1)	δ²un-1	(μδ³un-1)	δ4un-1 ...
		δun-1/2		δ³un-2
xn	un	(μδun)	δ²un	(μδ³un)	δ4un  ...
		δun+1/2		δ³un+1/2
xn+1	un+1	(μδun+1)	δ²un+1	(μδ³un+1)	δ4un+1 ...
		δun+3/2		δ³un+3/2
xn+2	un+2	(μδun+2)	δ²un+2	(μδ³un+2)	δ4un+2 ...
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·

Similarly, by taking the means of consecutive values of u and also of consecutive differences of even order, we should get a series of terms and differences central to the intervals x_n-2 to x_n-1, x_n-1 to x_n, ....

The terms of the series of which the values of u are the first differences are denoted by σu, with suffixes on the same principle; the suffixes being chosen so that δσu_n shall be equal to u_n. Thus, if

σu_n-3/2 = C + u_n-2,

then

σu_n-1/2 = C + u_n-2 + u_n-1, σ_n+1/2 = C + u_n-2 + u_n-1 + u_n, &c.,

and also

μσu_n-1 = C + u_n-2 + ½u_n-1, μσu_n = C + u_n-2 + u_n-1 + ½u_n, &c.,

C being an arbitrary constant which must remain the same throughout any series of operations.

Operators and Symbolic Methods.

12. There are two further stages in the use of the symbols Δ, Σ, δ, σ, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.

(i.) Instead of treating Δu as a function of x, so that Δu_n means (Δu)_n, we may regard Δ as denoting an operation performed on u, and take Δu_n as meaning Δ.u_n. This applies to the other symbols E, δ, &c., whether taken simply or in combination. Thus ΔEu_n means that we first replace u_n by u_n+1, and then replace this by u_n+2 − u_n+1.

(ii.) The operations Δ, E, δ, and μ, whether performed separately or in combination, or in combination also with numerical multipliers and with the operation of differentiation denoted by D (≡ d/dx), follow the ordinary rules of algebra: e.g. Δ(u_n + v_n) = Δu_n + Δv_n, ΔDu_n = DΔu_n, &c. Hence the symbols can be separated from the functions on which the operations are performed, and treated as if they were algebraical quantities. For instance, we have

E·u_n = u_n+1 = u_n + Δu_n = 1·u_n + Δ·u_n,

so that we may write E = 1 + Δ, or Δ = E − 1. The first of these is nothing more than a statement, in concise form, that if we take two quantities, subtract the first from the second, and add the result to the first, we get the second. This seems almost a truism. But, if we deduce Eⁿ = (1 + Δ)ⁿ, Δⁿ = (E-1)ⁿ, and expand by the binomial theorem and then operate on u₀, we get the general formulae

$u_{n}{=}u_{0}+n\Delta u_{0}+{\frac {n\cdot n-1}{1\cdot 2}}\Delta ^{2}u_{0}+\ldots +\Delta ^{n}u_{0},$

$\Delta ^{n}u_{0}{=}u_{n}-nu_{n-1}+{\frac {n\cdot n-1}{1\cdot 2}}u_{n-2}+\ldots +(-1)^{n}u_{0},$

which are identical with the formulae in (ii.) and (i.) of § 3.

(iii.) What has been said under (ii.) applies, with certain reservations, to the operations Σ and σ, and to the operation which represents integration. The latter is sometimes denoted by D^-1; and, since ΔΣu_n = u_n, and δσu_n = u_n, we might similarly replace Σ and σ by Δ^-1 and δ^-1. These symbols can be combined with Δ, E, &c. according to the ordinary laws of algebra, provided that proper account is taken of the arbitrary constants introduced by the operations D^-1, Δ^-1, δ^-1.

Applications to Algebraical Series.

13. Summation of Series.—If u_r, denotes the (r + 1)th term of a series, and if v_r is a function of r such that Δv_r = u_r for all integral values of r, then the sum of the terms u_m, u_m+1, ... u_n is v_n+1 − v_m. Thus the sum of a number of terms of a series may often be found by inspection, in the same kind of way that an integral is found.

14. Rational Integral Functions.—(i.) If u_r is a rational integral function of r of degree p, then Δu_r, is a rational integral function of r of degree p − 1.

(ii.) A particular case is that of a factorial, i.e. a product of the form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding factor by 1. We have

Δ · (r + a + 1) (r + a + 2) ... (r + b) = (b − a)·(r + a + 2) ... (r + b),

whence, changing a into a-1,

Σ(r + a + 1) (r + a + 2) ... (r + b) = const. + (r + a)(r + a + 1) ... (r + b)/(b − a + 1).

A similar method can be applied to the series whose (r + 1)th term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).

(iii.) Any rational integral function can be converted into the sum of a number of factorials; and thus the sum of a series of which such a function is the general term can be found. For example, it may be shown in this way that the sum of the pth powers of the first n natural numbers is a rational integral function of n of degree p + 1, the coefficient of n^p+1 being 1/(p + 1).

15. Difference-equations.—The summation of the series ... + u_n+2 + u_n-1 + u_n is a solution of the difference-equation Δv_n = u_n+1, which may also be written (E − 1)v_n = u_n+1. This is a simple form of difference-equation. There are several forms which have been investigated; a simple form, more general than the above, is the linear equation with constant coefficients—

v_n+m + a₁v_n+m-1 + a₂v_n+m-2 + ... + a_mv_n = N,

where a₁, a₂, ... a_m are constants, and N is a given function of n. This may be written

(E^m + a₁E^m-1 + ... + a_m)v_n = N

or

(E − p₁)(E − p₂) ... (E − p_m)v_n = N.

The solution, if p₁, p₂, ... p_m are all different, is v_n = C₁p₁ⁿ + C₂p₂ⁿ + ... + C_mp_mⁿ + V_n, where C₁, C₂ ... are constants, and v_n = V_n is any one solution of the equation. The method of finding a value for V_n depends on the form of N. Certain modifications are required when two or more of the p’s are equal.

It should be observed, in all cases of this kind, that, in describing C₁, C₂ as “constants,” it is meant that the value of any one, as C₁, is the same for all values of n occurring in the series. A “constant” may, however, be a periodic function of n.

Applications to Continuous Functions.

16. The cases of greatest practical importance are those in which u is a continuous function of x. The terms u₁, u₂ ... of the series then represent the successive values of u corresponding to x = x₁, x₂.... The important applications of the theory in these cases are to (i.) relations between differences and differential coefficients, (ii.) interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.

17. Starting from any pair of values x₀ and u₀, we may suppose the interval h from x₀ to x₁ to be divided into q equal portions. If we suppose the corresponding values of u to be obtained, and their differences taken, the successive advancing differences of u₀ being denoted by ∂u₀, ∂²u₀ ..., we have (§ 3 (ii.))

$u_{1}{=}u_{0}+q\partial u_{0}+{\frac {q\cdot q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots .$

When q is made indefinitely great, this (writing ƒ(x) for u) becomes Taylor’s Theorem (Infinitesimal Calculus)

$f(x+h){=}f(x)+hf'(x)+{\frac {h^{2}}{1\cdot 2}}f''(x)+\ldots ,$

which, expressed in terms of operators, is

$\mathrm {E} {=}1+h\mathrm {D} +{\frac {h^{2}}{1\cdot 2}}\mathrm {D} ^{2}+{\frac {h^{3}}{1\cdot 2\cdot 3}}\mathrm {D} ^{3}+\ldots {=}e^{h\mathrm {D} }.$

This gives the relation between Δ and D. Also we have

$u_{2}{=}u_{0}+2q\partial u_{0}+{\frac {2q\cdot 2q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots$

$u_{3}{=}u_{0}+3q\partial u_{0}+{\frac {3q\cdot 3q-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots$

$\vdots \qquad \qquad \quad \vdots$

and, if p is any integer,

$u_{\frac {p}{q}}{=}u_{0}+p\partial u_{0}+{\frac {p\cdot p-1}{1\cdot 2}}\partial ^{2}u_{0}+\ldots .$

From these equations u_p/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 = e^hD (§ 17) gives

hD = log_e (1 + Δ) = Δ − 1/2Δ² + ¹⁄₃Δ³ ...

or

h(du/dx)₀ = Δu₀ − 1/2Δ²u₀ + ¹⁄₃Δ³u₀ ....

The formulae connecting central differences with differential coefficients are based on the relations μ = cosh 1/2hD = 1/2(e^1/2hD + e^-1/2hD), δ = 2 sinh 1/2hD − e^1/2hD − e^-1/2hD, and may be grouped as follows:—

u0	= u0	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
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	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
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	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
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	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
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 u_m and u_m+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between u_m and u_m+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 φ(x_m) and φ(x_m+1), we have

$\int _{x_{m}}^{x_{m+1}}udx{=}{\tfrac {1}{2}}h(u_{m}+u_{m+1})+\phi (x_{m+1})-\phi (x_{m}).$

Adding a series of similar expressions, we find

$\int _{f}^{x_{n}}rac{x_{m}}udx{=}h{{\tfrac {1}{2}}u_{m}+u_{m+1}+u_{m+2}+\ldots +u_{n-1}+{\tfrac {1}{2}}u_{n}}+\phi (x_{n})-\phi (x_{m}).$

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

$\int ^{x_{n}}udx{=}h\cdot \mu \sigma u_{n}-{\frac {1}{12}}h^{2}{\frac {du_{n}}{dx}}{\tfrac {1}{2}}+{\tfrac {1}{720}}h^{4}{\frac {d^{3}u_{n}}{dx^{3}}}-{\tfrac {1}{30240}}h^{6}{\frac {d^{5}u_{n}}{dx^{5}}}+\ldots {=}h\cdot \mu \sigma u_{n}-{\frac {\mathrm {B} _{1}}{2!}}h_{2}{\frac {du_{n}}{dx}}+{\frac {\mathrm {B} _{2}}{4!}}h^{4}{\frac {d^{3}u_{n}}{dx^{3}}}-{\frac {\mathrm {B} _{3}}{6!}}h^{6}{\frac {d^{5}u_{n}}{dx^{5}}}+\ldots$

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:—

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})-{\tfrac {1}{12}}(\Delta u_{n}-\Delta u_{0})+{\tfrac {1}{24}}(\Delta ^{2}u_{n}-\Delta ^{2}u_{0})-{\tfrac {19}{720}}(\Delta ^{3}u_{n}-\Delta ^{3}u_{0})+{\tfrac {3}{160}}(\Delta ^{4}u_{n}-\Delta ^{4}u_{0})-\ldots$

For practical calculations this may more conveniently be written

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})+{\tfrac {1}{12}}(\Delta u_{0}-{\tfrac {1}{2}}\Delta ^{2}u_{0}+{\tfrac {19}{60}}\Delta ^{3}u_{0}-\ldots )+{\tfrac {1}{12}}(\Delta 'u_{n}-{\tfrac {1}{2}}\Delta '^{2}u_{n}+{\tfrac {19}{60}}\Delta '^{3}u_{n}-\ldots ),$

where accented differences denote that the values of u are read backwards from u_n; i.e. Δ′u_n denotes u_n-1 − u_n, not (as in § 10) u_n − u_n-1.

(iii.) Expressed in terms of central differences this becomes

${\frac {1}{h}}\int _{x_{0}}^{x_{n}}udx{=}\mu \sigma (u_{n}-u_{0})-{\tfrac {1}{12}}\mu \delta u_{n}+{\tfrac {11}{720}}\mu \delta ^{3}u_{n}-\ldots +{\tfrac {1}{12}}\mu \delta u_{0}-{\tfrac {11}{720}}\mu \delta ^{3}u_{0}+\ldots {=}\mu (\sigma -{\tfrac {1}{12}}\delta +{\tfrac {11}{720}}\delta ^{3}-{\tfrac {191}{60480}}\delta ^{5}+{\tfrac {2497}{3628800}}\delta ^{7}-\ldots )(u_{n}-u_{0}).$

(iv.) There are variants of these formulae, due to taking hu_m+1/2 as the first approximation to the area of the curve between u_m and u_m+1; the formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 ≡ σ(u_n − u₀) (see Mensuration).

20. The formulae in the last section can be obtained by symbolical methods from the relation

${\frac {1}{h}}\int udx{=}{\frac {1}{h}}\mathrm {D} ^{-1}u{=}{\frac {1}{h\mathrm {D} }}\cdot u.$

Thus for central differences, if we write θ ≡ 1/2hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ^-1, and the result in (iii.) corresponds to the formula

sinh θ = θ cosh θ/(1 + 1/3 sinh² θ − 2/3·5 sinh⁴ θ + 2·4/3·5·7 sinh⁶ θ − . . .).

References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’s Finite Differences (1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’s Differenzenrechnung (German trans., 1896) contains general formulae. (Both these works ignore central differences.) Encycl. der math. Wiss. vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books, e.g. G. Chrystal’s Algebra (pt. 2, ch. xxxi.). A. W. Sunderland, Notes on Finite Differences (1885), is intended for actuarial students. Various central-difference formulae with references are given in Proc. Lond. Math. Soc. xxxi. pp. 449-488. For other references see Interpolation.  (W. F. Sh.)