Calculus of differences - Encyclopedia

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CALCULUS OF DIFFERENCES (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.


1st Diff.

2nd Diff.

3rd Diff.

4th Diff.



b - a






d b

e - d


a _ - 3d+3 _ b - b

e -4d+6c-4b+a

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: - Algebra of Differences and Sums. 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 a, /3, y, S, commencing with the same term a, 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 FIG. I. 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 a+a, a+2/3+- y , a---30+37+5,. 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. 13+y is the difference of a+213+y and a+a, 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 /3+2y+5 is the sum of (+y and 7+6. Hence if we take li, -y, S,. .. 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 {- .2 Ic-? (- I)n-lnb-r(- I)na, 1.2 where 1, k.. . are the (n+I)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+i)th term of the series, i.e. the nth term after a, is a+na + n.n - Iy+ ... 1.2 where /3, y, . 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 (N+na+'ß+ ... ) - N =na?-n I.2 1 ? - I - 'n. -2.n' 3 2 y+ ... 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 isna+zn(n - I)p= 1n{2a+(n -1)p }.

6. As another example, take the series 1, 8, 27,. .. the terms of which are the cubes of I, 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, n.n-1+12n.n - I.n-2 +6 n.n-1.n-2.n -3 - 1.2 1.2.3 4124+2n3+4n2={2n(n+I)}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 down wards 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 uo, u 11 u 2, u 3 .... If we merely have the terms of the series, un may be regarded as meaning the (n+I)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 xo and u 0 are a pair of corresponding values of x and u, and if any other value xo -f-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,,_ 2, u, _1, un, un+i, un+2 ..., corresponding to values of x denoted by ... xn-2, x„_ 1, x n, x n+i, x n+2 .. .



1st Diff.

2nd Diff.

3rd Diff.

4th Diff.

xn -2

un -2

A2un - 3

04un - 4

Dun -2



un -1

02un -2

0474n - 3

Dun- 1




A2un. -1

04un - 2 ...



xn +1



0421n -1...

Dun +i


xn +2

un +2

A2un +1


9. In the advancing-difference notation un+i - un is denoted by Dun. The differences Auo, Du2 ... may then be regarded as values of a function Au corresponding to values of x proceeding by constant difference h; and therefore Dun+i - Dun is denoted by dAun, or, more briefly, 0 2 un; 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 The terms of the ser i es of which. .. un_ 1, un, un+1,. .. are the first differences are denoted by Zu, with proper suffixes, so that this series is. .. Eu n _ l , Eu n, n+1. ... The suffixes are chosen so that we may have DEun=un, whatever n may be; and therefore (§ 4) Eu n may be regarded as being the sum of the terms of the series up to and including u n _ l. Thus if we write Eun-1 = C+un-2, where C is any constant, we shall have Eun = Eu n _ l +DEu n _ 1 = C -2 -1 Eun+ 1 = +un-2+un - 1+un, and so on. This is true whatever C may be, so that the knowledge of. .. u n _ l, u n,. .. gives us no knowledge of the exact value of Eu 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 Eu 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 un+1, so that Eu n = un+Aun.

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 un+1, and may be denoted by 0'u n+11 and similarly n+1n -u n _ 1 may be denoted by O' u n+1. This notation is only required for certain special purposes, and the usage is not settled (§ 9 (ii.)).

II. The central-difference notation depends on treating un - un1 as the second dffference 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 un+2-2un+l+un is a function of xn +1, the first difference un+l-un must be regarded as a function of xn +#, i.e. of a (xn+xn+1). We therefore write u„+1-un = Sun+#, 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 µ22n+2 = 2 (un+un+1), / Lou„ = 2 (Sun++bun+), &c.



1st Diff.

2nd Diff.

3rd Diff.

4th Diff.



02un_ 2


64un_2 .




x n





52un- 1

,.n 62


b3un 2


64un.. .

524, + 4




(/ - tSun+l)


(µs3un +1)

b4un +1

n +3



un +2

(µbun +2)


(µb3un +2)

54u4, +2

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: - 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 _ 1r x n _ 1 to x n,.. The terms of the series of which the values of u are the first differences are denoted by au, with suffixes on the same principle; the suffixes being chosen so that bau n shall be equal to u n . Thus, if aun_3 = C+un-2, then aun - '? C+un-2+un-1, n+'s C+un-2+un-1+un, &C., and also µnun-1 = C+un-2+ 2un-1, µrun = C +un-2+un-1+2un, &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 0, E, b, a, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.

(i.) Instead of treating Du as a function of x, so that ztu n means (Au) n, we may regard A as denoting an operation performed on u, and take Du n as meaning A.u n. This applies to the other symbols E, S, &c., whether taken simply or in combination. Thus DEun means that we first replace u n by un+1, and then replace this by n+2 n+1.

(ii.) The operations 0, E, b, 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. 0(u n +v n) =Dun-f-Ovn, ADu n = Dou 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 Eu n = u n l = u n +Au n =I un 4'Dun, so that we may write E = I +0, or A =E-i. 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 E = (I +0), O = (E-1) and expand by the binomial theorem and then operate on uo, we get the general formulae un=uo+nAuo n I .2 1A2uo+ -I-Anuo, n.n - uo=unnun-1+ .2 n-2+

+(- I) nuo, 1.

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 E and a, and to the operation which represents integration. The latter is sometimes denoted by D-; and, since Ant ?, = un, and oau n = u n , we might similarly replace E and a by 0 - and S-. These symbols can be combined with 0, 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, 0 -1, - Applications to Algebraical Series. 13. Summation of Series. - If u r denotes the (r+ i)th term of a series, and if y r is a function of r such that Ov r = u r for all integral values of r, then the sum of the terms u m, u,n + 1,. .. . u n is n+l 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 Dur is a rational integral function of r of degree p - I .

(ii.) A particular case is that of a factorial, i.e. a product of the form (r+a+I) (r+a+2).. (r+b), each factor exceeding the preceding factor by I. We have A. (r+a+I) (r+a+2). .. (r+b) = (b-a).(r+a+2). .. (r+b), whence, changing a into a-I, E(r+a+I) (r+a+2). .. (r+b) =const.+(r+a)(r+a+1). .. (r+b)/(b-a-1-1). A similar method can be applied to the series whose (r + I) th term is of the form I/(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 the coefficient of n being I /(p+ I).

'15.' Difference-equations. - The summation of the series. .. +un+2+un -1 +un is a solution of the difference equation Ovn =un which may also be written (E -I)v n = n +1This 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 coeficients- vn+m + l n+m-1 + 2 n+'rn-2 +

+ mvn = N, where al, a2, ... a m are constants, and N is a given function of n. This may be written (E7"-}.a1Em -1+. .. +am)vn = N or (E-p1) (-2) ... (E-p m) vn = N.

The solution, if p1, p2,.. pm are all different, is v n = Clp1n+ C 2 p 2 +

+Cmpm' + n, where C 2, C2.. are constants, and 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 C 1, 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 is a continuous function of x. The terms u l, u 2. of the series then represent the successive values of u corresponding to x= x 1 , x2 The important applications of the theory in these cases are to (i.) relations between differences and differential coefficients, (ii.) 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, n, and the other is the same function of x,,,+l. Denoting these by ¢(x„,,) and ¢(x m+1), we have Xm+i interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.

17. Starting from any pair of values xo and uo, we may suppose the interval h from xo to x 1 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 uo being denoted by Ouo, 0 2 uo ..., we have (§ 3 (ii.)) u1= uo+gauo+q o I.2 I a o u When q is made indefinitely great, this (writing f(x) for u) becomes Taylor's Theorem (Infinitesimal Calculus) f(x+ h) =f(x)+hf'(x)+ Ih2 f" (x) ..., which, expressed in terms of operators, is E = I +hD -{- h2 D 2 + h3 D 3 I- ... = ehD.

1.2 1.2.3 This gives the relation between A and D. Also we have U2 = uo+2gauo 12 - '02U0+ U3 = U0+30 U0+ g - II a2u0+ and, if is any integer, up/ =uo+pauo+ p'P 2 Iaeuo+. .. .

From these equations U p /, could be expressed in terms of uo, u1, 242,. .; 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 0, E, D,. .. as if they were algebraical quantities. Thus the relation. E = e hD (§ 17) gives hD o = loge(I + 0) =A - 2 02 +3? 3. .

or h (- du) x = Auo - 1 02u o+0 3 auo

The formulae connecting central differences with differential coefficients are based on the relations µ = coshlhD = I (eihD µ u; = (1 +s h2D2 +??4 h4D4 +1s?ea h6D6 + Su; = (hD -}-z 1 4 h3 L D3 +i?a h5D5 +.. .)u; 82u = (h2D2 + ? h4D4 +5360 h6D6 + . .)u; (h 3 D 3 +gh 5 D'.. .1,8 4 u; _ (h 4 D 4 +z h 6 D 6 +.. .)u; uo =uo hDuo = (µ8 - 83 +a l oµ 85 -. ..) h 2 D 2 uo = (52 - 1.84 + 1 56 u0 h 3 D 3 u0 = (A V - 41c 8 5 +..)u0 h 4 D 4 uo = (8 4 - 18 6 +.. .,)u0 10 2 _ (µaµ 82 +rfiµ 84 iowµ 86 -f-. ..)uf hDu; -2483+64085 )u1 h 2 D 2 U3 - (µ82-254 µ84 )ul h 3 D 3 u = (8 3 - g8 5 +. ..)u3 h 4 D 4 u; = 4 6 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 7n+11 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 Ih(um+Um+l), udx = Ih (um+Um+1) +¢(xm+1)-¢(xm) Adding a series of similar expressions, we find fxn l udx=h{Ium+26m+1+um+2++ +u un-1tnJ +¢(xn)-¢ (xm)' 'xm The function ¢(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 Xfl = h. vun1 h?dx n z+7 21Y 11 4(1311 "- dx 5n+=h4-"Tun-2! B h2 d x „ `+ B? h 4 dx 3 - 6 ! L ? dx5 + where B 1, 132, 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: - fx„ h udx =jor(un-uo)11z (Dun Duo) +2 4 (z2un02uo) xo (A3u, -03uo) + T 0 (Mun-0 4 uo) -... .

For practical calculations this may more conveniently be written xn u d x = µw(unuo) + (Auo-20 2 uo+sot1 3 u0 - ...) xo +12(0'unlA'2un+6?(0f0'3un. .), where accented differences denote that the values of u are read backwards from u,; i.e. Q'u n denotes u,_ 1 -u n , not (as in § Io)un ' n-1 (iii.) Expressed in terms of central differences this becomes I x,b u d x =, U c r (Un U 0)12 b un ¦ AoO ati n -.

xo + µ8140 - 10 µ8 3 uo+. .

=t1 (a - 12 8 +7 1 A 53 - a Th 55 a-B W$OO 57 -.. )(uri u0)(iv.) There are variants of these formulae, due to taking hum+2 as the first approximation to the area of the curve between u m and u m+1; the formulae involve the sum u;+-u2 + ... +u n _3 - o (un-uo) (see M Ensuration).

20. The Formulae In The Last Section Can Be Obtained By Symbolical Methods From The Relation Udx = H D 1 U = Hd 'U. Thus For Central Differences, If We Write 0 Ihd, Wet Have A = Cosh 0, 8=2 Sinh 0, V =51 , And The Result In (Iii.) Corresponds To The Formula Sinh 0=0 Cosh 0/(1 3 Sinh 2 0 3 ? 5 Sinh 4 0 Xi Sinh 6 0 ...).


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.)

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