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. .
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