Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 227
Note that
I exp (isAe i6 ) I = exp ( - As sin (J), (5)
and that this is < 1 and tends to 0 as A-a) if s and sin (J have the same
sign. The dominated convergence theorem shows therefore that the integral
in (3) tends to 0 if s < 0, and the one in (4) tends to 0 if s > O. Thus
lim ({J .. is) = {~
.4-+00
if s > 0,
if s < 0,
(6)
and if we apply (6) to s = t + 1 and to s = t -
1· fA sin x itx d {n
1m --e X=
A-+oo -A X 0
1, we get
if -1 < t < 1,
ifltl>1.
(7)
Since ({J A(O) = n12, the limit in (7) is nl2 when t = ± 1.
IIII
Note that (7) gives the Fourier transform of (sin x)/x. We leave it as an exercise
to check the result against the inversion theorem.
Exercises
1 The following fact was tacitly used in this chapter: If A and B are disjoint subsets of the plane, if A
is compact, and if B is closed, then there exists a b > 0 such that 1 IX - P 1 ~ b for all IX E A and P E B.
Prove this, with an arbitrary metric space in place of the plane.
2 Suppose thatJis an entire function, and that in every power series
""
J(z) = L c.(z - a)·
n=O
at least one coefficient is O. Prove thatJis a polynomial.
Hint: n! c. = p·)(a).
3 Suppose J and g are entire functions, and 1 J(z) 1 S; 1 g(z) 1 for every z. What conclusion can you
draw?
4 SupposeJis an entire function, and
1 J(z) 1 S; A + Biz Ik
for all z, where A, B, and k are positive numbers. Prove thatJmust be a polynomial.
S Suppose {f.} is a uniformly bounded sequence of holomo'rphic functions in Q such that {f.(z)}
converges for every z E Q. Prove that the convergence is uniform on every compact subset ofQ.
Hint: Apply the dominated convergence theorem to the Cauchy formula for f. - Jm'
6 There is it region Q that exp (Q) = D(1; 1). Show that exp is one-to-one in Q, but that there are
many such Q. Fix one, and define log z, for 1 z - 11 < 1, to be that W E Q for which e = W z. Prove that
log' (z) = liz. Find the coefficients a. in
1 ""
- = L a.(z - 1)·
Z n=O