Page 23 - Foundations Of Differential Calculus

6 1. On Finite Differences
and

I
q = q +∆q,
we have
I I
p q = pq + p∆q + q∆p +∆p∆q,
so that
∆y = p∆q + q∆p +∆p∆q.

Hence, if p is a constant equal to a, since ∆a = 0 and the function y = aq,
the first difference ∆y equals a∆q. In a similar way the second difference
3
3
∆∆y equals a∆∆q, the third difference ∆ y equals a∆ q, and so forth.
13. Since every polynomial is the sum of several powers of x,wecan
find all of the differences of polynomials, provided that we know how to
find the differences of these powers. For this reason we will investigate the
differences of powers of x in the following examples.
0 0 0
Since x = 1,wehave∆x = 0, because x does not change when x
changes to x + ω.
Also, since as we have seen, ∆x = ω and ∆∆x = 0, all of the following
differences vanish. Since these things are clear, we begin with the second
power of x.
2
Example 1. Find the differences of all orders of x .
2
I
2
Since here y = x ,wehave y =(x + ω) , so that
2
∆y =2ωx + ω ,
and this is the first difference. Now, since ω is a constant, we have ∆∆y =
4
3
2
2ω and ∆ y =0, ∆ y =0,... .
3
Example 2. Find the differences of all orders of x .
3 I 3
Let y = x . Since y =(x + ω) ,wehave
2 2 3
∆y =3ωx +3ω x + ω ,
2
2
2
which is the first difference. Then, since ∆x =2ωx+ω , we have ∆3ωx =
3
2
3
2
3
6ω x +3ω , ∆3ω x =3ω , and ∆ω =0. We put it all together to obtain
2 3
∆∆y =6ω x +6ω
and
3 3
∆ y =6ω .
The differences of higher order vanish.