Real and Complex Analysis (Rudin)

332 REAL AND COMPLEX ANALYSIS
value O. The latter theorem is actually true in a local situation: Iff has an isolated
Singularity at a point Zo and iff omits two values in some neighborhood of zo, then
Zo is a removable singularity or a pole of! This so-called "big Picard theorem" is
a remarkable strengthening of the theorem of Weierstrass (Theorem 10.21) which
merely asserts that the image of every neighborhood of Zo is dense in the plane iff
has an essential singularity at Zo. We shall not prove it here.
16.22 Theorem Iff is an entire function and if there are two distinct complex
numbers IX and fJ which are not in the range off, thenfis constant.
PROOF Without loss of generality we assume that IX = 0 and fJ = 1; if not,
replace f by (f -
1X)/(fJ - IX). Then f maps the plane into the region n
described in Theorem 16.20.
With each disc Dl en there is associated a region VI c rr+ (in fact,
there are infinitely many such Vto one for each ((J E n such that A is one-toone
on VI and A(V 1) = D 1 ; each such VI intersects at most two of the
domains ((J(Q). Corresponding to each choice of VI there is a function "'1 E
H(D1) such that "'1(A(Z» = z for all z E VI'
If D2 is another disc in n and if Dl () D2 "# 0, we can choose a corresponding
V2 so that VI () V2 "# 0. The function elements ("'to D 1) and
("'2, D2) will then be direct analytic continuations of each other. Note that
"'i(DJ c rr+.
Since the range off is in n, there is a disc Ao with center at 0 so that
f(Ao) lies in a disc Do in n. Choose "'0 E H(Do), as above, put g(z) = "'o(f(z»
for z E Ao, and let y be any curve in the plane which starts at O. The range of
f 0 y is a compact subset of n. Hence y can be covered by a chain of discs,
A o, ... , All' so that each f(AJ lies in a disc Di in n, and we can choose
"'i E H(Di) so that ("'i' Di) is a direct analytic continuation of ("'i-I' Di- 1), for
i = 1, ... n. This gives an analytic continuation of the function element
(g, Ao) along the chain {Ao, ... , All}; note that "'" 0 f has positive imaginary
part.
Since (g, Ao) can be analytically continued along every curve in the plane
and since the plane is simply connected, the monodromy theorem implies
that g extends to an entire function. Also, the range of g is in rr +, hence
(g - i)/(g + i) is bounded, hence constant, by Liouville's theorem. This shows
that g is constant, and since "'0 was one-to-one onf(Ao) and Ao was a nonempty
open set, we conclude thatfis constant.
IIII
Exercises
1 Suppose f(z) = :Ea. z·, a. ~ 0, and the radius of convergence of the series is 1. Prove that f has a
singularity at z = 1. Hint: Expandfin powers of z - t. If 1 were a regular point off, the new series
would converge at some x > 1. What would this imply about the original series?
2 Suppose (f, D) and (g, D) are function elements, P is a polynomial in two variables, and P(f, g) = 0
in D. Suppose f and g can be analytically continued along a curve y, to fl and gl' Prove that