COMPLETING THE FUNCTIONAL CALCULUS 7
neighborhood of o(A). If T = f(A) , then the following statements are true.
(a) a(T) = f(a(A)) .
(b) ae(T) = f(ae(A}) .
(c) alre(T) = f [ alre(A) ] u { f(a) : a e P+00(A) and there exists a point b
in P.^A) with f(b) = f(a) } .
(d) ale(T) = f [ ale(A) ] u alre(T) .
(e) are(T) = f [ ore(A) ] u alre(T) .
(f) If A , P±(T) , then f~l{X) n a (A) is a finite subset { a
x
, . . . , a
n
} of
o(A) \ alre(A) . Moreover, if { ^ , . . . , a
n
} n Z(f') = 0 , then
(i) nul (X - T) = £ . nul (at - A) ;
(ii) nul (X - T)* = X i nul (^ - A)* ;
(iii) ind ( X - T) = ]\ ind (at - A) .
An analytic Cauchy domain is a bounded open set Q contained in C
whose boundary consists of a finite number of pairwise disjoint analytic
Jordan curves. An analytic Cauchy region is a connected analytic Cauchy
domain. Note that if K is a compact subset of an open set G contained in C ,
the n there is an analytic Cauchy domain Q with K c £ 2 c c l Q c G .
Moreover, if F is any countable set in C (for example, if, as will often be the
case in the paper, F is f(Z(f)) = the image under f of the zeros of the
derivative of f), then Q. can be chosen such that dQ n F = 0 . To see this
assume that Q. is connected and let D be a circle domain (a r e g i o n bounded
by a finite number of p a i r w i s e d i s j o i n t circles) and 0 : D - Q a conformal
equivalence. For all small e 0 let D
e
be a circle domain with cl D
e
c D and
so that D c an e- neighborhood of D
e
. Then Kc)( De) c §( D) for small e .
Since there are uncountably many e 's , one can be chosen with 3(|)( De)
disjoint from F .
If f : G - C is an analytic function and p is a natural number, say that f
is a strictly p-valent function if for every a in f(G), the equation f(z) = a has
p solutions in G counting multiplicities. Because the concept is frequently
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