Given a function which is analytic on some set, can you extend it to an analytic function
defined on a larger set? Sometimes you can do this. It was done in the proof of the
Cauchy integral formula. There are also reflection theorems like those discussed in the
exercises starting with Problem 10 on Page 5616. Here I will give a systematic way of
extending an analytic function to a larger set. I will emphasize simply connected
regions. The subject of analytic continuation is much larger than the introduction
given here. A good source for much more on this is found in Alfors [?]. The
approach given here is suggested by Rudin [?] and avoids many of the standard
technicalities.
Definition 54.4.1Let f be analytic on B
(a,r)
and let β ∈ ∂B
(a,r)
. Then β iscalled a regular point of f if there exists some δ > 0 and a function, g analytic onB
(β,δ)
such that g = f on B
(β,δ)
∩ B
(a,r)
. Those points of ∂B
(a,r)
which arenot regular are called singular.
PICT
Theorem 54.4.2Suppose f is analytic on B
(a,r)
and the power series
∑∞
f (z) = ak (z − a)k
k=0
has radius of convergence r. Then there exists a singular point on ∂B
(a,r)
.
Proof: If not, then for every z ∈ ∂B
(a,r)
there exists δz> 0 and gz analytic on
B
(z,δz)
such that gz = f on B
(z,δz)
∩ B
(a,r)
. Since ∂B
(a,r)
is compact, there exist
z1,
⋅⋅⋅
,zn, points in ∂B
(a,r)
such that
{B (zk,δzk)}
k=1n covers ∂B
(a,r)
. Now
define
{
g(z) ≡ f (z) if z ∈ B (a,r)
gzk (z) if z ∈ B (zk,δzk)
Is this well defined? If z ∈ B
(zi,δzi)
∩ B
(zj,δzj)
, is gzi
(z)
= gzj
(z)
? Consider the
following picture representing this situation.
PICT
You see that if z ∈ B
(zi,δzi)
∩B
( )
zj,δzj
then I ≡ B
(zi,δzi)
∩B
( )
zj,δzj
∩B
(a,r)
is
a nonempty open set. Both gzi and gzj equal f on I. Therefore, they must be
equal on B
(zi,δzi)
∩ B
( )
zj,δzj
because I has a limit point. Therefore, g is well
defined and analytic on an open set containing B
(a,r)
. Since g agrees with f
on B
(a,r)
, the power series for g is the same as the power series for f and
converges on a ball which is larger than B
(a,r)
contrary to the assumption that
the radius of convergence of the above power series equals r. This proves the
theorem.