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