COMPLETING THE FUNCTIONAL CALCULUS
3
sacrificed and in this case, as in many similar cases, it is simplicity. A
comparison of Theorem 2.1 here with the clean characterization of cl {
Ap
:
A e (B(9{) } from [13] (see Corollary 2.4 of this paper for the statement) will
certainly confirm this. It is not generally the case that these complications
arise from allowing analytic functions to be constant on some of the
components of their domain. To be sure, this extra generality does
introduce some complexity, but the principal source of the difficulty is the
complex nature of an analytic function, especially if it is defined on a non-
connected open set whose components are not simply connected. But
complications may arise even if the function is a polynomial (see Example
3.8).
In Theorem 2.2, the characterization of cl[ f(5(E)) ] for a completely
non-constant analytic function (one that is not constant on any component of
its domain) is stated, though the set E is not assumed to be open. The
statement may be simplified by requiring that E be open, since f(E) can then
have no isolated points. This eliminates condition (e); conditions (a)
through (d) may not, however, be simplified.
In contrasting the results here with those from [13] for the functions
TP and
ez
, two facts emerge. First, both z
p
and
ez
have uniform valence.
Second, and more important, is that a beautiful geometric condition may be
imposed on an open subset Q of C to ensure that there exists an analytic
inverse g : Q. - C for these functions. A necessary and sufficient condition
for this is that Q does not separate 0 from oo . Nothing like this is is possible
in general, even for an analytic function defined on a disk.
Because of the nature of this material, a leisurely style ha s been
adopted. We include several details that might be eliminated in another
presentation. Many corollaries of special cases have been included. In
particular, we have included statements of these results for the special
functions zP and
ez
defined on all of C as well as the function f(z) = z defined
in a neighborhood of an arbitrary set E . This last special case allows u s to
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