WEBVTT
Kind: captions
Language: en
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Okay, so what we need to do now as I was telling
you in the last lecture is to work with domains
00:00:46.230 --> 00:00:54.250
which probably include the finite infinity,
okay so you know this is a important turning
00:00:54.250 --> 00:01:04.080
point little technical but it is very easily
understandable. So all this time you know
00:01:04.080 --> 00:01:10.120
so let me say few things when you probably
do a first course in complex analysis you
00:01:10.120 --> 00:01:14.550
are working with a domain you are working
with a function which is defined on a domain
00:01:14.550 --> 00:01:18.500
in the complex plane, okay which means it
is defined on an open connected set ofcourse
00:01:18.500 --> 00:01:22.250
non-empty, okay and the function is taking
complex values okay.
00:01:22.250 --> 00:01:30.580
But then we want to work not just with analytic
functions we want to work with meromorphic
00:01:30.580 --> 00:01:34.900
functions and the problem with a meromorphic
function is at a pole the function goes to
00:01:34.900 --> 00:01:41.140
infinity and therefore at a pole you do not
get continuity, okay so you are forced to
00:01:41.140 --> 00:01:49.040
include the point at infinity in the co-domain
of the function okay and that leads you to
00:01:49.040 --> 00:01:52.659
look at functions at values in the extended
complex plane, okay so that is what we are
00:01:52.659 --> 00:01:57.100
being doing so far, what we are doing so far
is we have been looking at functions which
00:01:57.100 --> 00:02:02.259
are defined on a domain in the complex plane,
ok but taking values in the extended complex
00:02:02.259 --> 00:02:07.200
plane, alright and for such functions we have
proved lot of theorems.
00:02:07.200 --> 00:02:13.290
So for example we have proved that you know
we have proved that if you have a family of
00:02:13.290 --> 00:02:21.870
functions which converges normally, okay then
if all the functions are analytic that is
00:02:21.870 --> 00:02:26.600
holomorphic then the limit function is either
analytic or it is identically infinity. Similarly
00:02:26.600 --> 00:02:32.410
if all the functions are meromorphic, okay
then the limit function is either meromorphic
00:02:32.410 --> 00:02:37.760
or it is identically infinity and then we
have proved ofcourse Montel’s Theorem that
00:02:37.760 --> 00:02:46.920
if the functions are all analytic if you have
a family of analytic functions okay then every
00:02:46.920 --> 00:02:54.690
sequence admits a normally convergent subsequence
if and only if the family that family is normally
00:02:54.690 --> 00:03:00.380
uniformly bounded, okay that is uniformly
bounded on compact subsets that was Montel's
00:03:00.380 --> 00:03:01.380
Theorem.
00:03:01.380 --> 00:03:07.020
And then we also proved Marty’s Theorem
which is much more stronger namely you take
00:03:07.020 --> 00:03:16.730
a family of meromorphic functions and assume
that then at the condition that any any sequence
00:03:16.730 --> 00:03:23.590
in that admits a normally convergent subsequence
is equivalent to a spherical derivatives being
00:03:23.590 --> 00:03:28.870
normally bounded and we the spherical derivatives
because for meromorphic functions usual derivatives
00:03:28.870 --> 00:03:31.190
will not be defined at the poles, okay.
00:03:31.190 --> 00:03:37.190
So in all these statements the domain of the
functions was always a subset of the complex
00:03:37.190 --> 00:03:44.190
plane and but ofcourse the co-domain we took
it to be the extended complex plane and mind
00:03:44.190 --> 00:03:47.550
you therefore what is happening is in the
domain the metric you are looking at is the
00:03:47.550 --> 00:03:51.230
usual eucledian domain because after all it
is a subset of the complex plane where you
00:03:51.230 --> 00:03:52.290
have the eucledian metric.
00:03:52.290 --> 00:03:56.550
Whereas in the co-domain the metric you are
looking at is the spherical metric because
00:03:56.550 --> 00:03:59.970
the co-domain in the extended complex plane
is identified by the stereographic projection
00:03:59.970 --> 00:04:03.890
with the Riemann sphere and you actually take
the spherical metric on the Riemann sphere,
00:04:03.890 --> 00:04:09.810
alright which is the distance between two
points being given by the length of the minor
00:04:09.810 --> 00:04:13.390
arch of the greater circle on the Riemann
sphere passing through those two points, okay.
00:04:13.390 --> 00:04:18.160
So this is the setup of things now what we
want to do is we want to extend all these
00:04:18.160 --> 00:04:24.500
theorems, okay to the domain not only a domain
in the complex plane but you want to extend
00:04:24.500 --> 00:04:28.300
it to a domain in the extended complex plane,
okay that is the next step that means you
00:04:28.300 --> 00:04:35.090
are now allowing infinity as a value of the
variable you are allowing infinity in the
00:04:35.090 --> 00:04:40.830
domain, okay and you want to write out the
same you want the same results again and the
00:04:40.830 --> 00:04:45.440
fact is it is true it will work all these
results will be true for a domain even in
00:04:45.440 --> 00:04:49.729
the even in the extended complex plane but
then we need to fix it, okay.
00:04:49.729 --> 00:04:55.110
So I will tell you where the problem lies,
the problem lies in let us try to be knife
00:04:55.110 --> 00:05:02.580
and usually see the advantage of being knife
is that you will think naturally, okay and
00:05:02.580 --> 00:05:06.380
the disadvantage is that it may not work but
the advantage is that you will know where
00:05:06.380 --> 00:05:10.070
you go wrong and then you can correct yourself,
okay so there is always an advantage to being
00:05:10.070 --> 00:05:15.680
knife in the first place. So suppose D is
a domain in the extended complex plane, okay
00:05:15.680 --> 00:05:21.320
and certainly I am looking at a domain which
contains the point at infinity, okay because
00:05:21.320 --> 00:05:25.310
if it does not contain a point infinity then
it is a usual domain I have all the theorems
00:05:25.310 --> 00:05:29.229
for usual domains in the complex plane I have
already proved, okay. So let me look at the
00:05:29.229 --> 00:05:34.759
domain in the extended complex plane which
contains a point at infinity, okay.
00:05:34.759 --> 00:05:40.350
Now I can just say a sequence of functions
converges normally on the domain if it converges
00:05:40.350 --> 00:05:46.690
on compact subsets this is usual definition,
okay but the problem is that this definition
00:05:46.690 --> 00:05:52.680
will not work, why it will not work is because
if you take a compact subset of a domain even
00:05:52.680 --> 00:05:58.690
if you take a domain in the extended complex
plane that compact subset cannot contain the
00:05:58.690 --> 00:06:05.510
point at infinity because any infinity is
unbounded, do you understand?
00:06:05.510 --> 00:06:13.770
So therefore the problem is that if you say
a family of functions or suppose you are looking
00:06:13.770 --> 00:06:18.810
at a sequence of functions and if you say
if you say it normally converges on a domain
00:06:18.810 --> 00:06:23.259
which contains the point at infinity actually
you are not taking care of the normal convergence
00:06:23.259 --> 00:06:28.510
at infinity because when you say it normally
converges what does it mean it means that
00:06:28.510 --> 00:06:34.300
it is converging on compact subsets uniformly
convergent on compact subsets but what are
00:06:34.300 --> 00:06:37.630
compact subsets of a domain even in the extended
complex plane?
00:06:37.630 --> 00:06:48.750
See if you look at it with respect to the
usual eucledian plane, okay no neighbourhood
00:06:48.750 --> 00:06:57.180
of infinity, okay can be compact subset of
the usual plane, okay. So what is happening
00:06:57.180 --> 00:07:06.920
is that even if you naively define that a
sequence of functions is converging normally
00:07:06.920 --> 00:07:12.509
on a domain containing the point at infinity
what you are actually defining is only that
00:07:12.509 --> 00:07:19.990
it is that this sequence of functions is converging
normally only on the domain the punctured
00:07:19.990 --> 00:07:23.850
domain with the point at infinity removed,
okay.
00:07:23.850 --> 00:07:28.120
So you are not able to take care of normal
convergence at infinity that is the problem,
00:07:28.120 --> 00:07:33.360
okay. So how do you tackle this? So the way
of tackling this is is again the same old
00:07:33.360 --> 00:07:37.850
philosophy of you know how to tackle the point
at infinity, you tackle the point at infinity
00:07:37.850 --> 00:07:44.360
by you know inviting the variable and looking
at the point at 0, okay. So you know whenever
00:07:44.360 --> 00:07:50.319
you want to look study f of z at z equal to
infinity what was original idea we studied
00:07:50.319 --> 00:07:56.960
f of 1 by w at w equal to 0 and when the moment
so the neighbourhood of infinity will translate
00:07:56.960 --> 00:08:00.870
to a neighbourhood of 0 and a neighbourhood
of 0 is again now a good old neighbourhood
00:08:00.870 --> 00:08:05.710
in the good old complex plane and you can
do work with it we have already proved theorems
00:08:05.710 --> 00:08:07.580
there, okay.
00:08:07.580 --> 00:08:12.220
So here is the definition so the definition
is suppose D is a domain in the extended complex
00:08:12.220 --> 00:08:16.919
plane containing the point infinity, when
do I see a sequence of functions converges
00:08:16.919 --> 00:08:24.360
normally on D? I say that I have to say it
in two pieces, I have to first say that the
00:08:24.360 --> 00:08:30.750
convergence is normal on D minus infinity,
okay that is throw away infinity you are throwing
00:08:30.750 --> 00:08:36.490
away one point, okay so it is still an open
set, okay mind you infinity is a closed point
00:08:36.490 --> 00:08:41.169
if you look at in the extended complex plane
which is identified by the Riemann Sphere
00:08:41.169 --> 00:08:44.430
the infinity is identified with the north
pole on the Riemann Sphere, okay.
00:08:44.430 --> 00:08:51.920
So you take D minus infinity that is an open
that is a nice domain in the complex plane,
00:08:51.920 --> 00:08:56.180
okay it is continue it is going to just by
removing the point you cannot make it disconnected,
00:08:56.180 --> 00:09:01.700
okay and because it is open okay so it is
still going to be open connected. So it is
00:09:01.700 --> 00:09:06.500
a domain in the complex plane you require
that on that domain punctured at infinity
00:09:06.500 --> 00:09:11.520
the given family converges normally, okay
which means you are requiring that that is
00:09:11.520 --> 00:09:16.560
uniform convergence on compact subsets of
on compact subsets of the plane which intersect
00:09:16.560 --> 00:09:19.810
D that is all, okay that is one condition.
00:09:19.810 --> 00:09:24.700
The second condition is you take the same
sequence of functions change the variable
00:09:24.700 --> 00:09:32.290
from z to 1 by w and say that that converges
normally in a neighbourhood of the origin,
00:09:32.290 --> 00:09:38.269
okay. So you give a definition outside infinity
and you give a definition in a neighbourhood
00:09:38.269 --> 00:09:42.700
of infinity by translating it to a neighbourhood
of 0 and this is the definition that we will
00:09:42.700 --> 00:09:47.250
make and this is the definition that works,
okay you will see that with this definition
00:09:47.250 --> 00:09:51.300
you can translate all the theorems that we
have proved you can just use all the theorems
00:09:51.300 --> 00:09:57.680
that we have proved so far to get theorems
for the case when the domain includes the
00:09:57.680 --> 00:10:01.279
includes the point at infinity, okay. So let
me write this down.
00:10:01.279 --> 00:10:16.330
So the next thing is let me use a different
colour normal convergence
00:10:16.330 --> 00:10:40.590
at infinity. So let f n of z be a sequence
of continuous functions I do not even need
00:10:40.590 --> 00:10:58.149
continuous let me just say sequence of functions
on a domain D of the extended complex plane
00:10:58.149 --> 00:11:14.010
with infinity in the domain, okay and taking
values in the extended complex plane, okay.
00:11:14.010 --> 00:11:23.200
So basically you can think of f n as a sequence
defined on and you know an open on a domain
00:11:23.200 --> 00:11:28.120
on the Riemann Sphere and taking values on
the Riemann Sphere, if you think of the extended
00:11:28.120 --> 00:11:33.220
complex plane is Riemann Sphere you can think
that your store is a domain it is an open
00:11:33.220 --> 00:11:38.870
connected set on the Riemann Sphere, okay
your D is on the Riemann Sphere it is an open
00:11:38.870 --> 00:11:42.070
connected set that contains a north pole which
is supposed to correspond to the point at
00:11:42.070 --> 00:11:47.000
infinity and you have these functions each
of these functions are defined on that D and
00:11:47.000 --> 00:11:50.959
they are talking values again in the Riemann
Sphere which means they can take the value
00:11:50.959 --> 00:11:52.920
infinity, okay.
00:11:52.920 --> 00:11:59.149
So you have to take a sequence of functions
so when do we say that this sequence converges
00:11:59.149 --> 00:12:15.089
normally on D? So let me write that down we
say that f n converges normally on D on D
00:12:15.089 --> 00:12:29.389
if number 1 f n so the sequence f n converges
normally on D minus infinity which is D minus
00:12:29.389 --> 00:12:38.230
infinity is a domain in the complex plane,
okay and you know defining convergence on
00:12:38.230 --> 00:12:41.420
a domain in the complex plane is something
that we have already done it is just uniform
00:12:41.420 --> 00:12:45.340
convergence on compact subsets, okay.
00:12:45.340 --> 00:13:02.560
And and this and is very very important number
2, okay f n of 1 by w okay converges normally
00:13:02.560 --> 00:13:13.850
in a neighbourhood of w equal to 0. So here
is the so the second statement is what actually
00:13:13.850 --> 00:13:19.560
takes care of normal convergence at infinity
and you know it is very beautiful because
00:13:19.560 --> 00:13:26.000
normal convergence at infinity means you should
ensure uniform convergence on compact subsets
00:13:26.000 --> 00:13:30.280
of infinity, okay but the problem is there
is no compact subset of infinity that you
00:13:30.280 --> 00:13:34.810
can think off in the usual complex plane you
can think of it only under Riemann Sphere.
00:13:34.810 --> 00:13:39.090
So if you want it to translate it back to
the usual complex plane you have to translate
00:13:39.090 --> 00:13:44.839
from neighbourhoods of infinity to neighbourhoods
of 0 by making this inversion z going to 1
00:13:44.839 --> 00:13:53.700
z going to 1 by z which is w, okay so you
replace z by 1 by w, alright. So this is the
00:13:53.700 --> 00:13:58.530
definition, alright. Now comes now you see
this definition is peace wise what you have
00:13:58.530 --> 00:14:06.050
done is you have got normal convergence outside
infinity that is the first statement because
00:14:06.050 --> 00:14:10.670
outside infinity in the domain is again the
domain in the usual complex plane you have
00:14:10.670 --> 00:14:12.140
no problems with that.
00:14:12.140 --> 00:14:16.100
And the second part of the definition says
you have convergence normal convergence at
00:14:16.100 --> 00:14:21.000
infinity, okay that is what the second one
says because you know after all studying f
00:14:21.000 --> 00:14:26.800
n of 1 by w at w equal to 0 neighbourhood
of w equal to 0 is a same of studying f of
00:14:26.800 --> 00:14:32.089
z in the neighbourhood of infinity, alright.
But because it is a neighbourhood of 0 I know
00:14:32.089 --> 00:14:40.120
what converges normally means okay fine.
00:14:40.120 --> 00:14:47.370
So this is the right definition and this definition
works so I will put this as Def for definition
00:14:47.370 --> 00:14:52.980
so this definition works. But then there are
certain remarks that need to be made so that
00:14:52.980 --> 00:14:59.600
you know you realize that you always make
a definition if a definition has to suit a
00:14:59.600 --> 00:15:02.790
particular condition you make a twist in the
definition when you have to check whether
00:15:02.790 --> 00:15:04.279
the definition is consistent.
00:15:04.279 --> 00:15:08.610
So one of the thing that you can ask is the
following. Since I have defined the normal
00:15:08.610 --> 00:15:14.510
convergence a sequence in two pieces can it
happen that on each piece I get a different
00:15:14.510 --> 00:15:21.149
limit function? You can ask that and the answer
is no, you cannot if you are working with
00:15:21.149 --> 00:15:25.980
continuous functions it cannot happen because
of continuity, okay so this definition will
00:15:25.980 --> 00:15:30.279
work properly with continuous functions, okay
so let me write this down.
00:15:30.279 --> 00:15:43.910
Suppose so here is a remark so the remark
is suppose each f n of z is continuous on
00:15:43.910 --> 00:15:58.350
D, okay
then
00:15:58.350 --> 00:16:11.490
f n converges to a continuous function f on
D, okay so you get only one function, okay
00:16:11.490 --> 00:16:22.199
and what is the similar statements hold, similar
statements hold okay let me say that because
00:16:22.199 --> 00:16:30.740
I need to expand on that, okay. So what is
the proof? The proof is that well you see
00:16:30.740 --> 00:16:42.930
on D minus infinity, okay f n of z will converge
to well let us say f of z alright and f z
00:16:42.930 --> 00:16:55.100
is continuous this is because of normal due
to normal convergence you know a normal limit
00:16:55.100 --> 00:16:58.410
of continuous function is continuous and mind
you should consider the limit function f also
00:16:58.410 --> 00:17:06.320
to be a function with values in the extended
complex plane, okay.
00:17:06.320 --> 00:17:10.089
And so this is because of part 1 of the definition,
now part 2 of the definition will tell you
00:17:10.089 --> 00:17:17.630
that in the neighbourhood of the origin f
n of 1 by w will also converge to something
00:17:17.630 --> 00:17:31.640
and I will call that as g of w, okay so also
on a neighbourhood of w equal to 0 f n of
00:17:31.640 --> 00:17:41.880
1 by w converges to g n of w sorry g of w
so I in principle I should expect to get function
00:17:41.880 --> 00:17:55.600
g and again g is continuous at 0 and in fact
g is continuous everywhere again because again
00:17:55.600 --> 00:18:01.230
I am using just the fact that normal limit
of continuous function is continuous, okay.
00:18:01.230 --> 00:18:12.720
So there is only one thing that for w not
equal to 0, okay z equal to 1 by w is not
00:18:12.720 --> 00:18:23.010
infinity, okay so z lies in D minus infinity,
okay and therefore what will happen is by
00:18:23.010 --> 00:18:29.600
the uniqueness of limits of a sequence point
wise you will get that f of z will be equal
00:18:29.600 --> 00:18:43.100
to g of what is this g of 1 by z okay so this
implies for w not equal to 0, okay f of 1
00:18:43.100 --> 00:18:50.763
by w is same a g of so maybe I should call
this as you know here I have to I will have
00:18:50.763 --> 00:18:57.650
to say the following thing I should call this
as g n of w, okay I should call g n of w as
00:18:57.650 --> 00:19:05.770
f n of w and g n of w tends to g of , okay
and the point is that this g of w is f of
00:19:05.770 --> 00:19:14.730
1 by f of 1 by w is g of w that is what I
am getting, okay if I put z equal to 1 by
00:19:14.730 --> 00:19:15.730
w, alright.
00:19:15.730 --> 00:19:28.910
So but then you see and both are continuous
at 0, okay both are continuous at 0 mind you
00:19:28.910 --> 00:19:37.990
because of continuity therefore limit w tends
to 0, okay. So what this will tell you is
00:19:37.990 --> 00:19:50.280
that you know the so this will tell you that
f of infinity will be g of 0, okay so this
00:19:50.280 --> 00:20:05.370
implies that the limit function the limit
function is f of z even at infinity so you
00:20:05.370 --> 00:20:09.690
get a unique limit, okay that is the whole
point.
00:20:09.690 --> 00:20:14.770
So this normal convergence defining it peace
wise that is one outside infinity and the
00:20:14.770 --> 00:20:18.671
other one in the neighbourhood of infinity
by translating to a neighbourhood of 0, okay
00:20:18.671 --> 00:20:23.280
though it is a two piece definition normal
convergence will still give rise to only one
00:20:23.280 --> 00:20:27.940
function but mind you all functions are being
taken with values in the extended complex
00:20:27.940 --> 00:20:35.750
plane and that in the target the metric you
are using is always a spherical metric, okay.
00:20:35.750 --> 00:20:43.159
So you know unless you give a argument like
this, okay things could go round see after
00:20:43.159 --> 00:20:51.570
all I have defined normal normality separately
in two pieces normality I mean normal convergence
00:20:51.570 --> 00:20:59.299
in two pieces, okay and then it could happen
that on each piece I could get different limits
00:20:59.299 --> 00:21:03.460
atleast what this tells you that it will not
happen if your functions are continuous, okay
00:21:03.460 --> 00:21:07.020
and that is going to be the case because we
are going to deal only with analytic functions
00:21:07.020 --> 00:21:12.100
or meromorphic functions and you know analytic
functions are ofcourse continuous and even
00:21:12.100 --> 00:21:17.830
if you recall even if you take an analytic
function at infinity by definition it is a
00:21:17.830 --> 00:21:22.880
function which is continuous at infinity you
know and by version of the Riemann’s removable
00:21:22.880 --> 00:21:26.440
similarities theorem saying that function
of analytic at infinity means that it should
00:21:26.440 --> 00:21:28.260
be bounded at infinity, okay.
00:21:28.260 --> 00:21:33.390
So it means that the function value at infinity
is a finite complex number, okay so analytic
00:21:33.390 --> 00:21:38.020
function at infinity make sense. So analytic
functions on a domain in the extended complex
00:21:38.020 --> 00:21:42.409
plane containing the point at infinity make
sense for us, okay. And similarly meromorphic
00:21:42.409 --> 00:21:49.250
functions also make sense because what is
the meromorphic function on a domain which
00:21:49.250 --> 00:21:56.690
contains a point at infinity it is you see
it is supposed to be a meromorphic function
00:21:56.690 --> 00:22:02.510
on the punctured domain with the punctured
infinity and you invert the variable and that
00:22:02.510 --> 00:22:07.500
should give me a meromorphic function at the
origin, okay always you go to saying that
00:22:07.500 --> 00:22:14.299
a function is meromorphic at infinity is a
saying f of z is meromorphic at infinity,
00:22:14.299 --> 00:22:19.080
okay in the neighbourhood of infinity is same
as f of 1 by z is meromorphic at z equal to
00:22:19.080 --> 00:22:20.080
0.
00:22:20.080 --> 00:22:23.980
So you always translate back to at infinity
you always translate back to at neighbourhood
00:22:23.980 --> 00:22:29.510
at 0 so it make sense, okay. So meromorphic
functions on an extend on a domain in the
00:22:29.510 --> 00:22:34.030
extended complex plane containing the point
at infinity make sense and all these functions
00:22:34.030 --> 00:22:38.600
with values in the extended complex plane
also make sense. So we are in a perfect situation
00:22:38.600 --> 00:22:42.130
and all these functions are all continuous,
mind you meromorphic functions are continuous
00:22:42.130 --> 00:22:45.900
because you allow at a pole you define the
function value to be infinity and you allow
00:22:45.900 --> 00:22:50.320
the infinity to be in the co-domain of the
function, okay so we are in the right set
00:22:50.320 --> 00:22:51.320
up.
00:22:51.320 --> 00:22:54.940
Now what we need to know is we need to check
each of those theorems that we proved we need
00:22:54.940 --> 00:23:03.140
to deduce at those theorems also work for
such a domain, okay so here is the first point.
00:23:03.140 --> 00:23:24.130
So theorem let D be a domain in C union infinity
and f n a sequence so a sequence of holomorphic
00:23:24.130 --> 00:23:42.440
functions or analytic functions functions
on D taking values in C considered as subsets
00:23:42.440 --> 00:23:46.090
of C union infinity, okay.
00:23:46.090 --> 00:24:06.990
If f n converges to f normally on D then either
f is analytic on D or f is identically infinity
00:24:06.990 --> 00:24:14.720
on D, okay. so you see the point is that we
have already proved this theorem when D does
00:24:14.720 --> 00:24:19.450
not contain the point infinity when D is a
subset of D usual complex plane we have already
00:24:19.450 --> 00:24:27.370
proved this theorem that is you take a sequence
of analytic functions and you assume that
00:24:27.370 --> 00:24:31.830
it converges normally then the limit function
has to be either analytic or it will be identically
00:24:31.830 --> 00:24:34.070
infinity, okay and you can get anything in
between.
00:24:34.070 --> 00:24:40.049
For example you cannot get a strictly meromorphic
function in between that, okay and the reason
00:24:40.049 --> 00:24:44.840
is a pole cannot pop up at infinity I mean
in the limit and if you remember this was
00:24:44.840 --> 00:24:52.120
because if you invert the variable 0 cannot
pop up at the limit because of so is working
00:24:52.120 --> 00:24:58.630
in the background, okay so we will have to
only worry about the case when D contains
00:24:58.630 --> 00:25:02.590
infinity that is the extension we are particular
interested in so let me write that down.
00:25:02.590 --> 00:25:17.539
We have already proved this for infinity not
belonging to D so we are the essential thing
00:25:17.539 --> 00:25:25.679
that so let the essential thing is we allow
infinity to belong to D, okay. Now go back
00:25:25.679 --> 00:25:32.169
to the definition of normal convergence for
a domain containing the point infinity what
00:25:32.169 --> 00:25:36.460
is the definition? First thing is you throw
out infinity and on the remaining thing which
00:25:36.460 --> 00:25:39.590
is a domain in the complex plane there is
normal convergence, okay.
00:25:39.590 --> 00:25:46.429
And the other thing is that you take a neighbourhood
of 0 and look at the functions of the variable
00:25:46.429 --> 00:25:52.870
invertible in the neighbourhood of 0, okay.
So now so let me write that down now for D
00:25:52.870 --> 00:26:08.100
minus infinity f n will converge to f normally
so either f is identically infinity on D minus
00:26:08.100 --> 00:26:20.260
infinity or f is analytic on D minus infinity,
okay this is because of the fact that D minus
00:26:20.260 --> 00:26:25.350
infinity is the usual domain the usual complex
plane and for the usual complex plane we have
00:26:25.350 --> 00:26:29.620
proved such a theorem, okay whenever you have
normal convergence of analytic functions the
00:26:29.620 --> 00:26:36.820
limit is either identically infinity or it
is identically or it is uniformly an analytic
00:26:36.820 --> 00:26:37.820
function, okay.
00:26:37.820 --> 00:26:45.039
But you know if f is identically infinity
on D minus infinity it will also be infinity
00:26:45.039 --> 00:26:53.640
at infinity because f is continuous, okay.
So we have to only take care of the situation
00:26:53.640 --> 00:26:58.679
when f is not identically infinity and proof
that f is analytic on even at infinity so
00:26:58.679 --> 00:27:03.880
infinity is the only problem, okay. So let
me write that down if f is identically infinity
00:27:03.880 --> 00:27:17.799
on D minus infinity then by continuity of
f on D, f is identically infinity on all of
00:27:17.799 --> 00:27:23.390
D because f of infinity will become infinity,
okay.
00:27:23.390 --> 00:27:37.120
So if f is not identically infinity, infinity
becomes an isolated singular point
00:27:37.120 --> 00:27:42.890
of f okay because mind you f is analytic on
D minus infinity D minus infinity is the neighbourhood
00:27:42.890 --> 00:27:48.490
of infinity it is a deleted neighbourhood
of infinity D minus infinity is deleted neighbourhood
00:27:48.490 --> 00:27:53.830
of infinity, okay and f is analytic on that
that means f is analytic in neighbourhood
00:27:53.830 --> 00:27:59.490
of infinity that means infinity is an isolated
singular point for f and to check that f is
00:27:59.490 --> 00:28:03.480
analytic at infinity I have to only check
f is bounded at infinity but why is that true
00:28:03.480 --> 00:28:09.730
that is because I have normal convergence
at infinity, okay which is normal convergence
00:28:09.730 --> 00:28:13.760
if you change the variable to 1 by w and look
at w equal to 0, okay.
00:28:13.760 --> 00:28:17.020
Now we will use the second part of the definition
of normal convergence which is compact convergence
00:28:17.020 --> 00:28:21.730
I mean all convergence at infinity we use
that and then you are done, okay.
00:28:21.730 --> 00:28:42.559
Now we also have we have also f n of 1 by
w converges to f of 1 by w in normally
00:28:42.559 --> 00:28:53.710
normally in a neighbourhood of 0 of w equal
to 0 this is the second part of the definition,
00:28:53.710 --> 00:29:02.760
okay and mind you that is a that is you could
have taken a you could have taken a small
00:29:02.760 --> 00:29:07.980
enough open disk at the origin which will
be a domain it will be connected, okay in
00:29:07.980 --> 00:29:11.950
fact it could be simply connected, okay.
00:29:11.950 --> 00:29:23.909
And now on that on that domain okay you look
at the f n’s okay the point is that each
00:29:23.909 --> 00:29:33.220
of the f n's is also analytic at infinity
you see what look at what was given to me
00:29:33.220 --> 00:29:43.590
what was given to us, we have started with
sequence of holomorphic functions on D and
00:29:43.590 --> 00:29:48.809
we are looking at infinity and we have assumed
infinity belongs to D it means that each f
00:29:48.809 --> 00:29:55.440
n is already analytic at infinity each f n
is already analytic at infinity, okay that
00:29:55.440 --> 00:30:00.710
means f n of z is analytic at z equal to infinity
that means f n of z is bounded at z equal
00:30:00.710 --> 00:30:05.190
to infinity that means f n of 1 by w is bounded
at w equal to 0.
00:30:05.190 --> 00:30:11.520
And because the f is the normal limit of the
f n's f of 1 by w will also be bounded at
00:30:11.520 --> 00:30:16.380
w equal to 0 and that is the same as saying
that f is analytic at infinity and you are
00:30:16.380 --> 00:30:34.520
done, okay so that is it. So let me write
this down. Now since since each f n is analytic
00:30:34.520 --> 00:31:00.549
at infinity, each f n of w is analytic at
0, so f of 1 by w is bounded at 0, which means
00:31:00.549 --> 00:31:14.260
f is analytic at infinity. Thus is f is not
identically infinity then f is also holomorphic
00:31:14.260 --> 00:31:23.690
on D
and that is the proof, okay.
00:31:23.690 --> 00:31:31.591
So you see you are able to extend the theorem
that we already proved that a normal limit
00:31:31.591 --> 00:31:37.940
of analytic functions can either be analytic
or it will be identically infinity even if
00:31:37.940 --> 00:31:42.540
your domain contains infinity you are able
to do that, okay and the technical point was
00:31:42.540 --> 00:31:47.400
to deal with normal convergence at infinity
and that is cleverly done by translating a
00:31:47.400 --> 00:31:51.590
neighbourhood of infinity into a neighbourhood
of 0 by inverting the variable which is the
00:31:51.590 --> 00:31:56.519
usual philosophy that we have always been
using to study the point at infinity, okay.
00:31:56.519 --> 00:32:02.420
Fine so the next see the same kind of argument
will give the corresponding theorem for you
00:32:02.420 --> 00:32:07.169
know it will give you the corresponding theorem
for meromorphic functions, okay. So let me
00:32:07.169 --> 00:32:27.010
write that down theorem. Let f n be a sequence
of meromorphic functions
00:32:27.010 --> 00:32:51.679
on the domain D in the extended complex plane.
Suppose f n converges normally on D then either
00:32:51.679 --> 00:33:14.980
the limit f equal to limit f n is identically
infinity on D or it is meromorphic on D, okay.
00:33:14.980 --> 00:33:26.870
So
so this is just extending the previous theorem
00:33:26.870 --> 00:33:32.059
to the meromorphic but now the point is that
essentially you want to do deal with the domain
00:33:32.059 --> 00:33:36.470
which contains the point at infinity and what
is the proof? Proof is exactly the same, okay
00:33:36.470 --> 00:33:47.620
the proof is exactly the same let me write
it out. If D does not contain infinity, this
00:33:47.620 --> 00:34:13.109
has already been proved, so assume D contains
infinity, okay. We have that f is meromorphic
00:34:13.109 --> 00:34:17.230
on D minus infinity, okay.
00:34:17.230 --> 00:34:23.100
So again let me stop and say couple of things
please remember that when you say normal convergence
00:34:23.100 --> 00:34:30.560
now, okay it is uniform convergence on compact
subsets, alright but either you must look
00:34:30.560 --> 00:34:38.250
at compact subsets of D minus infinity or
you should look at compact subsets of a neighbourhood
00:34:38.250 --> 00:34:48.730
of the origin with the variable inverted that
is the point, okay and in both cases for the
00:34:48.730 --> 00:34:54.810
variable you are using only the eucledian
metric but for the you values you are using
00:34:54.810 --> 00:34:59.520
the spherical metric, okay you have to remember
that that is a big difference.
00:34:59.520 --> 00:35:10.550
And the other important thing is that you
know
00:35:10.550 --> 00:35:14.770
you are trying to make use of the theorems
that you have already proved for a domain
00:35:14.770 --> 00:35:20.020
in the usual complex plane and trying to reduce
the corresponding theorems when the domain
00:35:20.020 --> 00:35:25.010
contains the point at infinity, okay so these
are things that you should highlight in your
00:35:25.010 --> 00:35:32.770
mind, okay. So you see so let me write repeat
what I said if infinity is not in D then D
00:35:32.770 --> 00:35:39.220
is the usual domain and you know for a limit
of meromorphic functions the limit can either
00:35:39.220 --> 00:35:42.300
be identically infinity or it can be meromorphic,
okay.
00:35:42.300 --> 00:35:46.990
And mind you this is a very important thing,
you know it tells you the normal limits are
00:35:46.990 --> 00:35:51.450
good because after all what is meromorphic
function? A meromorphic function is a function
00:35:51.450 --> 00:35:56.681
is analytic except for poles, okay but when
a function goes to a limit the limit function
00:35:56.681 --> 00:36:02.099
could be horrible, see the limit function
could have been an analytic function with
00:36:02.099 --> 00:36:06.089
non-pole singularities it could have been
an analytic function with essential singularities
00:36:06.089 --> 00:36:12.030
or even worse the limit could have been an
analytic function with non-isolated singularities
00:36:12.030 --> 00:36:15.790
such horrible things could happen but the
fact is normal convergence prevents that.
00:36:15.790 --> 00:36:21.800
See normal convergence you know is always
a it is locally uniform convergence, okay
00:36:21.800 --> 00:36:26.990
because every point has a neighbourhood which
is compact, okay so at every point you can
00:36:26.990 --> 00:36:30.970
find a neighbourhood compact neighbourhood
where you will have uniform convergence and
00:36:30.970 --> 00:36:34.270
therefore on that neighbourhood also you will
have uniform convergence should it is locally
00:36:34.270 --> 00:36:40.290
uniform convergence normal convergence and
therefore because of the local uniformness
00:36:40.290 --> 00:36:43.890
everything nice happens you know the moment
you have uniform convergence limits of continuous
00:36:43.890 --> 00:36:48.119
functions are continuous, limits of analytic
functions are analytic and so on.
00:36:48.119 --> 00:36:55.190
But so this is happening globally, alright
and there is one more thing that I have to
00:36:55.190 --> 00:37:01.180
tell you that here the moment I assume f n
is sequence of meromorphic functions on the
00:37:01.180 --> 00:37:06.720
domain which contains a point at infinity
and I assume that it converges but mind you
00:37:06.720 --> 00:37:11.869
unique limit function is already defined that
is because you know the f n's are meromorphic
00:37:11.869 --> 00:37:16.609
and therefore are continuous and I told you
that whenever you take a continuous limit
00:37:16.609 --> 00:37:20.780
of continuous functions if you take a normal
limit then the limit function is continuous
00:37:20.780 --> 00:37:28.290
even though your definition of normal convergence
has been split into two pieces, one for the
00:37:28.290 --> 00:37:32.930
domain infinity punctured and the other for
the neighbourhood of infinity part of the
00:37:32.930 --> 00:37:34.440
neighbourhood of the origin, okay.
00:37:34.440 --> 00:37:41.310
So even this existence of f a uniform function
f a single function f is because of the continuity
00:37:41.310 --> 00:37:46.380
of all these functions because meromorphic
functions are continuous functions considered
00:37:46.380 --> 00:37:49.829
as functions into the extended complex plane
that is something that you should not forget,
00:37:49.829 --> 00:37:57.079
okay so alright so if you take the domain
which contains the point at infinity then
00:37:57.079 --> 00:38:01.490
D minus infinity is a usual domain in the
usual complex plane and you have already proved
00:38:01.490 --> 00:38:10.760
the theorem for that so the function f is
meromorphic on D union infinity or f is identically
00:38:10.760 --> 00:38:16.941
infinity on D minus infinity, okay so this
is something that we have already proved we
00:38:16.941 --> 00:38:22.560
have that f is meromorphic on D minus infinity
or f is identically infinity on D minus infinity,
00:38:22.560 --> 00:38:23.690
alright.
00:38:23.690 --> 00:38:29.270
And again the same old argument if f is identically
infinity on D minus infinity then it has to
00:38:29.270 --> 00:38:33.640
be infinity at infinity because of continuity
of f, okay so we have to only deal with the
00:38:33.640 --> 00:38:42.320
condition when f is not identically infinity,
right. So let me write that down. So if ya
00:38:42.320 --> 00:38:55.240
so if f is identically infinity on D minus
infinity then f is identically infinity on
00:38:55.240 --> 00:39:07.300
D as f is continuous at continuous on D in
fact continuous at infinity, okay.
00:39:07.300 --> 00:39:23.450
So if f is not identically infinity we have
f is meromorphic on D minus infinity, okay
00:39:23.450 --> 00:39:33.730
and therefore infinity becomes singular point,
okay infinity is certainly a singular point
00:39:33.730 --> 00:39:42.560
because the problem now is slightly more complicated
as it tends it looks you can say infinity
00:39:42.560 --> 00:39:47.589
is a singular point but you cannot immediately
say that infinity is an isolated singular
00:39:47.589 --> 00:39:51.260
point that is the point that is the issue.
00:39:51.260 --> 00:39:56.810
See infinity is a singular point because in
if you take the if you take D minus infinity
00:39:56.810 --> 00:40:04.010
there are only poles on D minus infinity f
is you know it is meromorphic. So on D minus
00:40:04.010 --> 00:40:09.810
infinity it is there are poles and poles are
ofcourse isolated. So as far as D minus infinity
00:40:09.810 --> 00:40:14.750
is concerned all the singular points are isolated
but infinity itself may be a non-isolated
00:40:14.750 --> 00:40:22.280
singular point it could happen poles could
accumulate at infinity. If you have a sequence
00:40:22.280 --> 00:40:27.000
of singular points going to a point then that
point is not an cannot be an isolated singular
00:40:27.000 --> 00:40:29.390
point so you have this problem.
00:40:29.390 --> 00:40:35.370
But then that will not happen because of the
normal convergence at infinity because what
00:40:35.370 --> 00:40:39.660
is happening at infinity is being controlled
by what is happening when the variable is
00:40:39.660 --> 00:40:44.470
inverted in a neighbourhood of 0, okay.
00:40:44.470 --> 00:40:59.060
So let me say this now in neighbourhood of
w equal to 0, f n of 1 by w converges to f
00:40:59.060 --> 00:41:09.849
of 1 by w normally and what I want you to
understand is that you have changed the variable
00:41:09.849 --> 00:41:18.090
from z to 1 by w alright and if you are looking
at z not equal to infinity or looking at w
00:41:18.090 --> 00:41:28.710
not equal to 0, okay and the point I want
to make is that even at w equal to 0 I want
00:41:28.710 --> 00:41:36.050
to say that the f n's okay they are continue
to be meromorphic that is because that is
00:41:36.050 --> 00:41:37.050
already given to you.
00:41:37.050 --> 00:41:45.290
See what is given to you is that each of these
functions is meromorphic on D, so infinity
00:41:45.290 --> 00:41:51.630
the point infinity which is in D can either
be a pole by itself for each of the f n's
00:41:51.630 --> 00:41:58.640
or it can be a point of analyticity, infinity
cannot be any verse, okay. So that means each
00:41:58.640 --> 00:42:05.170
of this f n's of 1 by w are meromorphic in
a neighbourhood of w equal to 0, okay but
00:42:05.170 --> 00:42:09.393
this neighbourhood of w equal to 0 on that
neighbourhood that is the usual neighbourhood
00:42:09.393 --> 00:42:12.410
in the complex plane and you have a normal
convergence of the sequence of meromorphic
00:42:12.410 --> 00:42:16.520
functions therefore the limit functions can
be either identically infinity or it can be
00:42:16.520 --> 00:42:21.030
meromorphic but the limit function is not
identically infinity because f is not identically
00:42:21.030 --> 00:42:23.119
infinity, okay.
00:42:23.119 --> 00:42:28.500
So f of 1 by w has to be you know a meromorphic
function at w equal to 0 that means you are
00:42:28.500 --> 00:42:34.600
saying f of z is meromorphic at infinity and
you are done, okay. So you escape and you
00:42:34.600 --> 00:42:39.839
get the proof of the statement that you want.
So let me write that down so let me write
00:42:39.839 --> 00:43:08.930
this and since each f n is each f n z is meromorphic
on D, f n of 1 by w is meromorphic on neighbourhood
00:43:08.930 --> 00:43:22.730
of w equal to 0, so f of 1 by w is meromorphic
at 0 at w equal to 0 as f is not identically
00:43:22.730 --> 00:43:41.310
infinity, so f is so f of z is also meromorphic
at infinity and we are done, okay and so you
00:43:41.310 --> 00:43:47.000
this horrible thing of infinity being a non-isolated
singular point for f does not happen. So you
00:43:47.000 --> 00:43:52.640
see we are still in very nice in a very nice
situation, okay.
00:43:52.640 --> 00:43:58.920
So what I need to do is what we need to do
next is try to generalize a Montel Theorem
00:43:58.920 --> 00:44:02.619
and Marty’s Theorem for the case of meromorphic
functions and we will do that in the next
00:44:02.619 --> 00:44:02.730
lecture.