ABSTRACT
In this book we shall be concerned with both pure asymptotics and error analysis. In deriving implied constants frequent use will be made of the variational operator V , which we now proceed to define and discuss. 11.2 In the theory of real variables the variation, or more fully total variation, of a function f(x) over a finite or infinite interval (a, b), is the supremum of
for unbounded n and all possible modes of subdivision
with x, and x, in the closure of (a, 6). When this supremum is finite f(x) is said to be of bounded variation in (a, b), and we denote the supremum by VX=,,,{ f(x)), K,bCf), or even V(f), when there is no ambiguity. 11.3 In the case of a compact interval [a, b] one possible mode of subdivision is given by n = 1, x, = a, and x , = 6. Hence
The last relation affords a simple method for calculating the variation of a
continuous function with a finite number of maxima and minima: we subdivide [a, b] at the maxima and minima and apply (1 1.01) to each subrange. For example, in the case of the function depicted in Fig. 1 1.1, we see that
When f (x) is continuously diyerentiable in [a, b] application of the mean-value theorem gives
Continuity off '(x) implies that of (f'(x)(. Hence from Riemann's definition of an integral
11.4 Suppose now that the interval (a,b) is finite or infinite, f(x) is continuous in the closure of (a, b), f '(x) is continuous within (a, b), and (f'(x)( is integrable over (a, 6). Using the subdivision points of $1 1.2 and the result of $1 1.3, we have
Since x , and x,-, are arbitrary points in (a, b) this result implies
We also have
implying that (I 1.03) holds with the 2 sign reversed. Therefore (I 1.02) again applies. 11.5 So far it has been assumed that f(x) is real. If f(x) is a complex function of the real variable .r, then its variation is defined by (1 1.02) whenever this integral converges.
