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CHAPTER II.

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DIRECT THEOREMS OF FINITE DIFFERENCES.

1. THE operation denoted by A is capable of repetition. For the difference of a function of x, being itself a function of

is subject to operations of the same kind.

In accordance with the algebraic notation of indices, the difference of the difference of a function of x, usually called the second difference, is expressed by attaching the index 2 to the symbol A. Thus

AAU, = Aug
In like manner

AA’uz = A%ux,
and generally
ΔΔ"-'u, Ξ Δ"u,

(1), the last member being termed the nth difference of the function Uz:

If we suppose un = x", the successive values of u, with their successive differences of the first, second, and third orders will be represented in the following scheme: Values of a 1 2 3 4

6 ... Ux

1 8 27 64 125 216 ... Au, 7 19 37 61 91 ... Aug 12

18 24 A'ux 6 : 6 It may be observed that each set of differences may either be formed from the preceding set by successive subtractions in accordance with the definition of the symbol A, or calculated from the general expressions for Au, A’u, &c. by assign

30...

6 ...

ing to a the successive values 1, 2, 3, &c. Since u, = 20°, we shall have

Au, = (x+1)* — ** = 3x + 3x +1,
Aur=A (3x + 3x + 1) = 6x + 6,

Δu = 6. It may also be noted that the third differences are here constant. And generally if Uz be a rational and integral function of x of the nth degree, its nth differences will be constant. For let

u = ax" + bx*-++ &c., then

Au, = a (x + 1)" +b(x + 1)*-1 + &c.

az" bwn-1 - &c.

anx"*1 + 5,249 + b2a"-* + &c., 64, ba, &c.,,, being constant coefficients. Hence Au, is a rational and integral function of ac of the degree n – 1. Repeating the process, we have

A'un = an (n − 1) **-2 +0,ach-9 +0,21+4 + &c., a rational and integral function of the degree n–2; and so on. Finally we shall have

A"u= an (n − 1) (n − 2) ... 1,
a constant quantity.
Hence also we have
A"x"=1.2...n

(2).

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2. While the operation or series of operations denoted by.A, A5, ... A" are always possible when the subject-function Uz is given, there are certain elementary cases in which the forms of the results are deserving of particular attention, and these we shall next consider.

Differences of Elementary Functions. 1st. Let U = x (0 - 1) (2 – 2)... (x m + 1).

Then by definition, Auz=(+1) 2 (3-1)...(cc-m+2) — < (2-1) (-2) ...(x—m+1)

=mx (20 – 1) (2 — 2) ... (x – m+2). When the factors of a continued product increase or decrease by a constant difference, or when they are similar functions of a variable which, in passing from one to the other, increases or decreases by a constant difference, as in the expression

sin x sin (x-+ h) sin (x + 2h) ... sin {x+(m – 1) h}, the factors are usually called factorials, and the term in which they are involved is called a factorial term. For the particular kind of factorials illustrated in the above example it is common to employ the notation

2 (x - 1) ... (x – m+ 1) = xrm). .(1), doing which, we have Ax(m) = mcm-1

(2). Hence, acm-1) being also a factorial term,

Accm) = m (m – 1) dcfm->}, and generally = m (m - 1) (m— n+1) am-n)

(3).

1 2ndly. Let Uz =

30 (x + 1) ... (ox + m - 1)* Then by definition, 1

1 Au,=

(x + 1) (x + 2) ... (x + m) (x+1)... (x + m - 1)
1

1
Fatm
x) (0+ 1) (+2) (2c + m - 1)

.(4) 20 (x + 1)... (x + m)

(in)

2) (+1)(x+2

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xcl-m)

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(-m-1)

Hence, adopting the notation

1

+ 3 (x + 1) (x + m - 1) we have Δα-m) =- - malam

(5). Hence by successive repetitions of the operation A, Auxcm) = - m (-m - 1) ... (- m – n +1) acol==*) = (-1)" m (m +1) ... (m + n - 1) x

(6), and this may be regarded as an extension of (3).

3rdly. Employing the most general form of factorials,

1-m-n)

we find

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In like manner we have ·
A log Ug = log Uz+1 - log Uz = log

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4thly. To find the successive differences of a".

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= (a 1) a*

..(13). Hence

A’a* = (a – 1)* a*, and generally, A"a* = (a – 1)” a*

(14). Hence also, since am* = (a")*, we have A"qm* = (a" – 1)" a"

..(15). 5thly. To deduce the successive differences of sin (ax + b) and cos (ac+b).

A sin (ax +b) = sin (ax + b + a) — sin (ax +b)

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"=2

- 2 sin sin (ax +8+ , ") By inspection of the form of this result we see that

A’ sin (ax +b) = (2 sin a) sin (ax+b+a+T)...... (16). And generally,

n (a +7) A" sin

2 sin

+

2

In the same way it will be found that

17) A" cos (ax + b) = (2 sin ) cos fax +6+ *(4,7")}..(18).

These results might also be deduced by substituting for the sines and cosines their exponential values and applying (15).

3. The above are the most important forms. The following are added merely for the sake of exercise.

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