Real and Complex Analysis (Rudin)

APPROXIMA TIO~S BY RATIONAL FUNCTIONS 273
The Mittag-Leffler Theorem
Runge's theorem will now be used to prove that meromorphic functions can be
constructed with arbitrarily preassigned poles.
13.10 Theorem Suppose n is an open set in the plane, A c n, A has no limit
point in n, and to each ex E A there are associated a positive integer m(ex) and a
rational Junction
m("1
P,,(Z) = L cj,,,(z-ex)-j.
j= 1
Then there exists a meromiJrphic Junction J in n, whose principal part at each
ex E A is P" and which has no other poles in n.
PROOF We choose a sequence {Kn} of compact sets in n, as in Theorem 13.3:
For n = 1, 2, 3, ..., Kn lies in the interior of Kn+ h every compact subset of n
lies in some K n , and every component of S2 - Kn contains a component of
S2 - n. Put Al = A () K 1, and An = A () (Kn - K no1) for n = 2, 3, 4, ....
Since An C Kn and A has no limit point in n (hence none in K n), each An is a
finite set. Put
Qn(Z) = L P..(z) (n = 1, 2, 3, ...). (1)
lIeAn
Since each An is finite, each Qn is a rational function. The poles of Qn lie in
Kn - K n- 1 , for n:2= 2. In particular, Qn is holomorphic in an open set containing
Kn _ l' It now follows from Theorem 13.6 that there exist rational
functions R n , all of whose poles are in S2 - n, such that
We claim that
has the desired properties.
Fix N. On K N , we have
I Rn(z) - Qn(z) I < 2- n (2)
co
J(z) = Ql(Z) + L (Qn(z) - Rn(z)) (Z E n) (3)
n=2
N
co
J = Ql + L (Qn - Rn) + L (Qn - Rn)· (4)
n=2 N+l
By (2), each term in the last sum in (4) is less than 2- n on K N ; hence this last
series converges uniformly on K N , to a function which is holomorphic in the
interior of K N • Since the poles of each Rn are outside n,