8.7 Analytic Functions 527function f to be analytic at cits Taylor series must converge to f(x) in some
neighborhood of c. Thus, a function f that is infinitely differentiable at c can
fail to be analytic at c in either of the following circumstances:
- Its Taylor series converges in some neighborhood of c, but not to f(x) in
any neighborhood of c. [See Exercise 13 .] - Its Taylor series diverges everywhere except at c; that is, its radius of
convergence is 0. [See Exercise 14.]
For an example of a function whose Taylor series converges on an interval
that is strictly smaller than the interval on which it is infinitely differentiable,
see Exercise 15. For an example of a function whose Maclaurin series converges
everywhere, but to f(x) only on (-oo, O], see Exercise 6.6.15.
In contrast to these negative examples, Boas [16] proves a very interesting
theorem of S. Bernstein: "If f and all its derivatives are nonnegative in an
interval, then f is analytic in that interval." Further examples and interesting
discussions of these concerns can be found in [16] pages 179-183, [49] pages
68-70, and [61] page 318.
Finally, it is important to point out that analyticity does not occur at
isolated points; if a function is analytic at c, it is analytic on an entire neigh-
borhood of c. The following theorem makes this explicit.
*Theorem 8.7.13 Suppose f has a power series representation f(x) =
00
I: ak(x - c)k, with radius of convergence p > 0. Then, V d in the interior
k=O
of the interval of convergence, f has a power series representation about d with
radius of convergence at least p - le -di. In fact,
d
t
c-P c c+pFigure 8.300
Proof. Suppose f(x) = I: ak(x - c)k, with radius of convergence p > 0,
k=O
and let d E ( c - p, c + p). Then