Page 30 - Foundations Of Differential Calculus

1. On Finite Differences 13
If these formulas are expressed as series in the usual way, we will obtain
the expressions found above.
19. In this same way, differences of functions, either rational or irrational,
can be found. If, for example, we wish to find the first difference of the
2 2 2 2
fraction 1/ a + x , then we let y =1/ a + x , and since
I 1
y = ,
2
2
a + x +2ωx + ω 2
we have
1 1 1
∆y =∆ = − ,
2
2
2
2
a + x 2 a + x +2ωx + ω 2 a + x 2
and this expression can be converted into an infinite series.
2
2
2
We let a + x = P and 2ωx + ω = Q. Then
1 1 Q Q 2 Q 3
= − + − + ···
P + Q P P 2 P 3 P 4
and
Q Q 2 Q 3
∆y = − + − + ··· .
P 2 P 3 P 4
When we substitute the values of P and Q we obtain
1
∆y =∆ 2 2
a + x
3
2 2
2ωx + ω 2 4ω x +4ω x + ω 4
= − 2 + 3
2
2
2
2
(a + x ) (a + x )
5
4 2
3 3
8ω x +12ω x +6ω x + ω 6
− 4 + ··· .
2
2
(a + x )
If these terms are ordered by the powers of ω, we obtain
2 2 2 3 3 2
1 2ωx ω 3x − a 4ω x − a x
∆. = − 2 + 3 − 4 + ··· .
2
2
2
2
2
2
2
x + a 2 (a + x ) (a + x ) (a + x )
20. Differences of irrational functions can be expressed by similar series.
√
2
2
If we let y = a + x , and since
I 2 2 2
y = a + x +2ωx + ω ,
2
2
2
we let a + x = P and 2ωx + ω = Q, then
2 3
√
Q Q Q
∆y = P + Q − P = √ − √ + √ − ··· ,
2 P 8P P 16P 2 P