Page 31 - Foundations Of Differential Calculus

14 1. On Finite Differences
so that
2 2
3
2ωx + ω 2 4ω x +4ω x + ω 4
2 2
∆y =∆. a + x = √ − √ + ··· ,
2
2
2
2
2 a + x 2 8(a + x ) a + x 2
or
2 2 2 3
ωx a ω a ω x
∆y = √ + √ − 2 √ + ··· .
2
2
2
2
2
2
2
a + x 2 2(a + x ) a + x 2 2(a + x ) a + x 2
From this we gather the fact that the difference of any function of x, which
we call y, can be put into this form, so that
2 3 4
∆y = Pω + Qω + Rω + Sω + ··· ,
where P, Q, R, S,... are certain functions of x that in any case can be
defined in terms of the function y.
21. We do not exclude from this form of expression even the differences
of transcendental functions, as will clearly appear from the following ex-
amples.
Example 1. Find the first difference of the natural logarithm of x.
I
Let y =ln x. Since y =ln (x + ω) , we have
ω
I
∆y = y − y =ln (x + ω) − ln x =ln 1+ .
x
2
Elsewhere we have shown how this kind of logarithm can be expressed in
an infinite series. We use this to obtain
ω ω 2 ω 3 ω 4
∆y =∆ ln x = − + − + ··· .
x 2x 2 3x 3 4x 4
x
Example 2. Find the first difference of exponential functions a .
3
I
x
x ω
Let y = a , so that y = a a . We have also shown that
3
2
ω ln a ω (ln a) 2 ω (ln a) 3
ω
a =1 + + + + ··· .
1 1 · 2 1 · 2 · 3
From this we have
2
3
x
x
x
I
x
∆.a = y − y =∆y = a ω ln a + a ω (ln a) 2 + a ω (ln a) 3 + ··· .
1 1 · 2 1 · 2 · 3
Example 3. In a unit circle, to find the difference of the sine of the arc
x.
2 Introduction to Analysis of the Infinite, Book I, Chapter VII; see also note on page 1.
3 Introduction, Book I, Chapter VIII; see also note on page 1.