COMPLETING THE FUNCTIONAL CALCULUS 11

Q = {X: \X\ r, \X\ e, and IX,- 3el e } ,

where r » 1 » e 0 are chosen so that BQ consists of three circles: yr of

radius r and center 0, ye of radius e and center 0, and ye' of radius e and

center 3e . If r is sufficiently large and e is sufficiently small, then D = f~l{Q]

is an analytic Cauchy region and 3D = g

r

u g

e l

u - - - u g

g s

u g'

8 1

u • • • u

g'

e

, where these sets in the union are pairwise disjoint analytic Jordan

curves, g

r

is the boundary of the unbounded component of C \ D and f is a p-

to-one map of gr onto yr ; g

e k

is the boundary of some neighborhood of X^

and f is a d

k

-to-one map of g

e k

onto y

e

(1 k s); and f is a bijection of

each ge^ j ' onto y

£

(1 j p).

By combining this polynomial with the Ahlfors function mapping Q

strictly 3 -to-one onto D , even more pathology can be obtained.

1.3 Proposition If f: G - C is an analytic function that is completely non-

constant , A e 5(G) , and T = f(A) , then the following statements are true.

(a) If Cl is an analytic Cauchy region with dQ n f(Z(f')) = 0 and c l ^ c

p±oo(T) , then there is an analytic Cauchy region H such that c l H c P±00(A) ,

f(H) = Cl , f(9H) = 3Q , and there is a natural number p such that f is a strictly

p-valent map of H onto Cl .

(b) If Cl is an analytic Cauchy region such that cl Cl e P±(T) \ P±eo(T)

and dCl n f(Z(f')) = 0 , then there are analytic Cauchy regions Hl , . . . , H

d

such that for i j d , cl Hj c P±(A) \ P±00(A) , f(Hj) = ft , f(3Hj) = dQ , and

there are positive integers p

x

, . . . , p

d

such that if nij = ind (a - A) for a in

Hj , then :

(i) f is a strictly pj-valent map of Hj onto Q ;

(ii) ind (X -T) = ]\ Dj mj for all X in Cl ;

(iii) nul (X - T) ]T {

P j m j

: mj 0 ) for all X in Q. .