Concrete mathematics : a foundation for computer science

2.6 FINITE AND INFINITE CALCULUS 49
example, C might be the periodic function a + b sin2nx; such functions get
washed out when we take differences, just as constants get washed out when
we take derivatives. At integer values of x, the function C is constant.
Now we’re almost ready for the punch line. Infinite calculus also has
definite integrals: If g(x) = Df(x), then
/‘g(x)dx = f(x)11 = f(b) -f(a).
a
Therefore finite calculus-ever mimicking its more famous cousin- has definite
Sims: If g(x) = Af(x), then
Lb g(x) 6x = f(x)i’ = f(b) -f(a). (2.47)
a a
This formula gives a meaning to the notation x.“, g(x) 6x, just as the previous
formula defines Jl g(x) dx.
But what does xi g(x) 6x really mean, intuitively? We’ve defined it by
analogy, not by necessity. We want the analogy to hold, so that we can easily
remember the rules of finite calculus; but the notation will be useless if we
don’t understand its significance. Let’s try to deduce its meaning by looking
first at some special cases, assuming that g(x) = Af(x) = f(x + 1) -f(x). If
b = a, we have
tIg(x)bx = f(a)-f(a) = 0.
Next, if b = a + 1, the result is
xl+’ g(x) dx = f(a+ 1) -f(a) = g(a).
More generally, if b increases by 1, we have
- x: g(x) 6x = (f(b + 1) -f(a)) - (f(b) -f(a))
= f(b+ 1) -f(b) = g(b).
These observations, and mathematical induction, allow us to deduce exactly
what x.“, g(x) 6x means in general, when a and b are integers with b > a:
~-$xi~x = ~g&, = x g(k),
k=a a