WEBVTT
Kind: captions
Language: en
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Alright so let us continue with whatever we
were doing, see our aim as a told you is to
00:01:38.729 --> 00:01:44.870
prove the Picard theorem for which we need
to look at families of Meromorphic functions,
00:01:44.870 --> 00:01:54.150
so we need to do topology on a space of Meromorphic
functions okay and so I told you in the previous
00:01:54.150 --> 00:01:59.571
lectures that you know if you want to do topology
on a space of functions and you go and try
00:01:59.571 --> 00:02:07.030
to draw inspiration from usual topology than
for example if you look at real valued or
00:02:07.030 --> 00:02:12.280
complex valued bounded functions continuous
functions on a topological space then the
00:02:12.280 --> 00:02:18.050
topology is done by looking at uniform convergence
okay.
00:02:18.050 --> 00:02:30.860
So we define convergence in the space by uniform
convergence but now if you try to it inspiration
00:02:30.860 --> 00:02:35.940
from this and come to complex analysis and
then you are working of course with holomorphic
00:02:35.940 --> 00:02:41.960
functions that is analytic functions and then
otherwise we want to be a little bit more
00:02:41.960 --> 00:02:45.530
general and we want to work with Meromorphic
functions and Meromorphic functions you know
00:02:45.530 --> 00:02:50.900
are analytic or holomorphic functions accept
for an isolated set of points where they have
00:02:50.900 --> 00:02:55.390
only poles singularities okay, so you want
to work with such functions at the point is
00:02:55.390 --> 00:03:02.170
that even if you take analytic functions okay
and you look at sequence of analytic functions
00:03:02.170 --> 00:03:03.680
converging okay.
00:03:03.680 --> 00:03:10.400
Normally what happens is that you do not get
uniform convergence okay but you get only
00:03:10.400 --> 00:03:17.650
normal convergence, so when you come from
a topology to complex analysis you must remember
00:03:17.650 --> 00:03:22.320
that you should not work with uniform convergence
but you should work with uniform convergence
00:03:22.320 --> 00:03:27.090
restricted to compact sets and that is called
normal convergence okay that is the 1st point
00:03:27.090 --> 00:03:33.230
then the 2nd point is this pathology that
you know you can have a domain in which you
00:03:33.230 --> 00:03:41.079
can have a decent sequence of analytic functions
which goes to infinity everywhere okay. Now
00:03:41.079 --> 00:03:48.959
so for example I was discussing about this
domain which is the exterior of the unit disk
00:03:48.959 --> 00:03:56.840
and I said you take the you know the sequence
of functions given by Z power N, so the 1st
00:03:56.840 --> 00:04:01.299
function is Z the 2nd function is Z square
and so on and then this sequence you know
00:04:01.299 --> 00:04:06.410
it converges point wise to the function which
is infinity okay.
00:04:06.410 --> 00:04:13.430
So and the point is that is convergence and
normally if you are if you are only thinking
00:04:13.430 --> 00:04:19.290
of a 1st course in complex analysis we will
say that this sequence diverges okay because
00:04:19.290 --> 00:04:24.240
if you take any complex number Z with modulus
greater than 1 and you look at the sequence
00:04:24.240 --> 00:04:28.240
Z power n that is not going to go to limit
it is going to go to infinity okay you will
00:04:28.240 --> 00:04:33.960
simply say this is the sequence diverges but
the fact is that we want to think of this
00:04:33.960 --> 00:04:39.729
as the convergence sequence and there are
2 reasons for this the 1st thing is that it
00:04:39.729 --> 00:04:46.919
converges to the constant function which is
infinity at every point outside the unit disk
00:04:46.919 --> 00:04:51.390
and we have to allow the value infinity because
we want to think of Meromorphic options because
00:04:51.390 --> 00:04:58.389
say if you are thinking of Meromorphic functions
then you can define the value at a pole to
00:04:58.389 --> 00:05:02.770
be infinity because that is the limit that
you get as you approach the poles okay.
00:05:02.770 --> 00:05:06.801
So we have to allow the value infinity and
once you allow the value infinity then you
00:05:06.801 --> 00:05:12.800
must also allow the constant function which
is infinity and if you go by that reasoning
00:05:12.800 --> 00:05:20.389
then the sequence Z power n actually tends
to the constant function infinity and that
00:05:20.389 --> 00:05:29.759
is the reason that we say that Z power n tends
to the constant function infinity it tends
00:05:29.759 --> 00:05:38.449
to the constant function infinity outside
the unit disk okay and in the exterior of
00:05:38.449 --> 00:05:44.910
the unit disk alright and of course unit circle
is not include we are looking at the exterior
00:05:44.910 --> 00:05:54.330
of the unit circle and then the point is that
if you want to talk about this convergence
00:05:54.330 --> 00:06:01.030
in terms of a metric okay then you will have
to worry about giving a metric on the extended
00:06:01.030 --> 00:06:05.520
complex plane and I told you that that metric
that you can use is the spherical metric there
00:06:05.520 --> 00:06:07.970
is a so-called spherical metric and what is
the spherical metric?
00:06:07.970 --> 00:06:12.590
You take 2 points in the extended complex
plane take their images in on the Riemann
00:06:12.590 --> 00:06:17.080
sphere under the stereographic projection
and then use spherical metric on the Riemann
00:06:17.080 --> 00:06:22.509
sphere, the spherical metric on the Riemann
sphere is just it is the length of the minor
00:06:22.509 --> 00:06:30.090
arc of the big circle passing through those
2 points on the Riemann sphere okay, so in
00:06:30.090 --> 00:06:34.560
this way you get spherical metric on the extended
complex plane and then with respect to this
00:06:34.560 --> 00:06:39.699
spherical metric I can actually defined the
distance between any 2 points in the extended
00:06:39.699 --> 00:06:43.629
plane, so for example I can have one point
which is a point in the complex plane I have
00:06:43.629 --> 00:06:46.999
I can have the other point to be a point at
infinity and I can still have this notion
00:06:46.999 --> 00:06:53.570
of distance okay and it is with respect to
this metric the spherical metric at the convergence
00:06:53.570 --> 00:06:56.710
is actually normal okay.
00:06:56.710 --> 00:07:00.550
So here this is the point that we have to
understand if you take the sequence of function
00:07:00.550 --> 00:07:07.470
at power and in the domain mod Z greater than
1 then the sequence converges normally to
00:07:07.470 --> 00:07:13.789
the constant function infinity okay and this
is the viewpoint it is very important okay
00:07:13.789 --> 00:07:18.900
and what I am trying to tell you today is
that you know this is the only pathology that
00:07:18.900 --> 00:07:25.629
can occur. If you take a domain in the complex
plane and suppose you have sequence of analytic
00:07:25.629 --> 00:07:33.469
functions which converge normally to a function
on the domain okay and assume that you allow
00:07:33.469 --> 00:07:38.080
this function the limit function to take the
value infinity suppose you assume you allow
00:07:38.080 --> 00:07:46.639
that then is very beautiful that either the
limit function is completely analytic or it
00:07:46.639 --> 00:07:51.050
is completely infinity you do not get anything
in between.
00:07:51.050 --> 00:07:56.900
So you know you do not get this kind of horrible
pathology that you have sequence of analytic
00:07:56.900 --> 00:08:03.330
function and they tend to for example a Meromorphic
function so that will not happen, you cannot
00:08:03.330 --> 00:08:08.009
have a sequence of analytic function on a
domain they converge normally okay that is
00:08:08.009 --> 00:08:12.069
they converge uniformly on compact subsets
of the domain and in the limit you get low
00:08:12.069 --> 00:08:18.460
behold you suddenly get a Meromorphic function
that will not happen, so so this is very very
00:08:18.460 --> 00:08:27.380
important thing it tells you that things are
going on well as per usual intuition something
00:08:27.380 --> 00:08:32.690
does not you must always see whenever you
work with some extraordinary cases you should
00:08:32.690 --> 00:08:38.060
always be careful, so for example when I am
allowing a function to take the value infinity
00:08:38.060 --> 00:08:42.300
the function can very well be a Meromorphic
function because Meromorphic function takes
00:08:42.300 --> 00:08:44.490
the value infinity at the poles.
00:08:44.490 --> 00:08:51.450
So it can happen that I have a sequence of
analytic functions on a domain it is converging
00:08:51.450 --> 00:08:57.390
normally what limit function, now if I allow
the limit function take the value infinity
00:08:57.390 --> 00:09:04.410
okay and the limit function could very well
be even Meromorphic okay but the fact is that
00:09:04.410 --> 00:09:10.170
does not happen, what happens is either the
limit function is analytic that is the good
00:09:10.170 --> 00:09:16.680
thing and in the bad case the worst thing
happens the limit function is always infinity
00:09:16.680 --> 00:09:23.860
okay and the reason or this is 2 important
facts, one fact is the theorem of Hurwitz
00:09:23.860 --> 00:09:31.810
for analytic functions, the other thing is
the so-called symmetry of the spherical metric
00:09:31.810 --> 00:09:37.180
with respect to inversion okay, so these are
the 2 facts and this is what I wanted to concentrate
00:09:37.180 --> 00:09:39.050
today upon.
00:09:39.050 --> 00:09:55.589
So I will start with the following thing,
so let maps D, C union infinity so look at
00:09:55.589 --> 00:10:07.840
this set let this be the set of maps or just
functions from D which is it is a domain
00:10:07.840 --> 00:10:16.250
D is a domain in the complex plane to the
extended complex plane, so these are the important
00:10:16.250 --> 00:10:26.840
things I am allowing the value infinity alright
and so what I am going to do is there are
00:10:26.840 --> 00:10:33.190
many subset of this which I am interested
in, so let me write them down among this is
00:10:33.190 --> 00:10:41.430
inside this is I will put this C of D C union
infinity and what is this C of C? This is
00:10:41.430 --> 00:10:54.080
the set of continuous maps, this is the set
of continuous maps okay and you know when
00:10:54.080 --> 00:10:58.300
I say this is a set of continuous maps am
looking at all those maps which are continuous
00:10:58.300 --> 00:11:04.940
this domain D to C union infinity and mind
you C union infinity as a topology.
00:11:04.940 --> 00:11:11.470
So C union infinity has 1 point compactification
topology and with this topology it is holomorphic
00:11:11.470 --> 00:11:16.010
to the Riemann sphere under the stereographic
projection and in fact it is in fact even
00:11:16.010 --> 00:11:22.470
a metric space you can take either the spherical
metric or the cordial metric on the Riemann
00:11:22.470 --> 00:11:27.220
sphere and transport it to C union infinity
via the stereographic projection and so C
00:11:27.220 --> 00:11:32.310
union infinity becomes even a metric space,
it becomes a complete metric space it is compact
00:11:32.310 --> 00:11:37.140
it is very nice okay. So it does make sense
to look at continuous maps into a topological
00:11:37.140 --> 00:11:39.890
space or a metric space so it discontinues
in that sense okay.
00:11:39.890 --> 00:11:45.380
So the point I want you to understand is that
now if you look at the set of continuous maps
00:11:45.380 --> 00:11:51.030
into C union infinity then the constant map
which is equal to infinity okay that is the
00:11:51.030 --> 00:11:57.330
map that takes every point to infinity that
is also continuous because mind you always
00:11:57.330 --> 00:12:02.130
constant maps are always continuous okay,
so the function which is infinity uniformly
00:12:02.130 --> 00:12:06.470
that is a continuous function mind you, so
this is the point that you have to carefully
00:12:06.470 --> 00:12:14.810
notice this is what we introduce and in particular
what happens is that this contains the set
00:12:14.810 --> 00:12:19.690
m of D the set of Meromorphic functions on
D.
00:12:19.690 --> 00:12:24.291
You take any Meromorphic functions on D, what
is a Meromorphic function? A Meromorphic function
00:12:24.291 --> 00:12:34.560
as a holomorphic function on D minus an isolated
set of points for which the given function
00:12:34.560 --> 00:12:38.921
have poles at which the given function has
poles. So you take a Meromorphic function
00:12:38.921 --> 00:12:47.750
on D okay it is a holomorphic function outside
you know isolated set of points but at each
00:12:47.750 --> 00:12:53.550
of those isolated set of points in D the function
has a pole but I can define the value of the
00:12:53.550 --> 00:12:57.700
function to be infinity at a pole because
that is the limit I get as I approach a pole
00:12:57.700 --> 00:13:04.400
okay that is the definition of pole okay and
therefore this a Meromorphic function becomes
00:13:04.400 --> 00:13:08.690
a continuous function in to C union infinity
that is the point that is another critical
00:13:08.690 --> 00:13:14.730
point you have to understand okay by including
the value infinity you are making the Meromorphic
00:13:14.730 --> 00:13:20.260
function continuous on the whole domain you
are making it continuous even at the poles
00:13:20.260 --> 00:13:28.010
okay and do not confuse that continuity with
the usual continuity because the usual continuity
00:13:28.010 --> 00:13:30.170
is with respect to complex numbers.
00:13:30.170 --> 00:13:34.540
The target is complex numbers and you do not
allow the value infinity okay but this continuity
00:13:34.540 --> 00:13:38.260
is different this continuity is with respect
to the extended complex plane okay this is
00:13:38.260 --> 00:13:46.001
something that you have to clearly distinguish
okay because you know a Meromorphic function
00:13:46.001 --> 00:13:52.930
in the usual sense cannot be continuous at
a pole because at the pole it becomes infinity
00:13:52.930 --> 00:13:59.640
okay but and that is because you do not allow
it to take the value infinity, you do not
00:13:59.640 --> 00:14:03.890
think of infinity as a value if you are only
thinking of complex values but now since I
00:14:03.890 --> 00:14:07.850
have added the extra point at infinity the
Meromorphic function becomes also continuous
00:14:07.850 --> 00:14:13.750
at infinity that is the point you must notice
and then of course the further subset of H
00:14:13.750 --> 00:14:19.570
of D this is the subset of all holomorphic
functions okay or analytic functions and of
00:14:19.570 --> 00:14:23.310
course you know holomorphic or analytic functions
are also by default they are also included
00:14:23.310 --> 00:14:24.310
as Meromorphic functions.
00:14:24.310 --> 00:14:29.481
So Meromorphic function is a function which
can either be analytic or if it has singularities,
00:14:29.481 --> 00:14:36.090
the singularities must only be isolated and
they must be only poles that is the definition
00:14:36.090 --> 00:14:41.750
okay. So the definition of Meromorphic includes
the definition of holomorphic okay, so let
00:14:41.750 --> 00:14:58.670
me write that so this is m of D is Meromorphic
functions and H of D is holomorphic functions
00:14:58.670 --> 00:15:05.310
and so you know basically what we want to
do is we want to do topology on this set,
00:15:05.310 --> 00:15:10.590
the set of Meromorphic functions alright on
the domain D and I told you this topology
00:15:10.590 --> 00:15:19.230
has to be done with respect to I mean
we say that we do topology while trying to
00:15:19.230 --> 00:15:24.290
study convergence okay, so the convergence
that we should think of is normal convergence.
00:15:24.290 --> 00:15:30.480
So the topology corresponds to working with
normal convergence okay and what we saw last
00:15:30.480 --> 00:15:36.630
time is that if you take the domain to be
the exterior of the unit circle okay if you
00:15:36.630 --> 00:15:40.899
take the domain to be the exterior of the
unit circle and you take the sequence Z n,
00:15:40.899 --> 00:15:49.290
Z power n okay namely Z square, Z cube then
that is a sequence in H of D that is the sequence
00:15:49.290 --> 00:15:55.420
of holomorphic functions and it converges
normally to what? It converges normally to
00:15:55.420 --> 00:16:05.260
the constant function infinity which is in
this set about this script C D C union infinity
00:16:05.260 --> 00:16:15.920
it is here okay and so I want to say the following
thing, you must have studied this in the 1st
00:16:15.920 --> 00:16:20.920
course in topology but the idea is essentially
the same namely that whenever you have a uniform
00:16:20.920 --> 00:16:24.650
limit okay the limit function is also continuous
okay.
00:16:24.650 --> 00:16:30.850
So that is the 1st thing that I want to that
is the 1st thing I want to prove or recall
00:16:30.850 --> 00:16:39.980
at least and mind you here we are not working
with uniform convergence on the whole domain.
00:16:39.980 --> 00:16:46.060
Whatever convergence we are working with is
only normal convergence but that is good enough
00:16:46.060 --> 00:16:50.070
because you know normal convergence will also
give locally uniform convergence, so locally
00:16:50.070 --> 00:16:55.800
you can still do the same thing that you would
do if you had uniform convergence everywhere
00:16:55.800 --> 00:17:02.420
okay and that is good enough or local properties
like continuity, analyticity, et cetera that
00:17:02.420 --> 00:17:04.910
is good enough okay.
00:17:04.910 --> 00:17:20.000
So here is a lemma; if f n is a sequence of
continuous maps from the domain D into C union
00:17:20.000 --> 00:17:33.370
infinity and f n tends to f which is thought
of as a map D to C union infinity so, so that
00:17:33.370 --> 00:17:43.270
means when I say f n tends to f point wise,
so this is point wise on D okay that means
00:17:43.270 --> 00:17:54.190
f n of Z tends to the value f of Z as n tends
to infinity for each Z in D okay and this
00:17:54.190 --> 00:18:11.820
convergence is normal on D then f is also
a continuous map D to C union infinity, so
00:18:11.820 --> 00:18:21.039
I am just saying that are normal limit of
continuous map is continuous and why should
00:18:21.039 --> 00:18:27.150
this be true? Because you know uniform limit
of continuous map is continuous and if you
00:18:27.150 --> 00:18:33.310
have a normal limit is actually a uniform
limit locally and therefore the limit function
00:18:33.310 --> 00:18:37.950
is continuous locally but continuity is a
local property therefore if something is continuous
00:18:37.950 --> 00:18:39.820
locally then it is continues.
00:18:39.820 --> 00:18:45.490
If you have a global map which is locally
continuous, continuous on every on sufficiently
00:18:45.490 --> 00:18:53.889
small open sets okay which cover the domain
then it is continuous okay so this is a very
00:18:53.889 --> 00:18:59.590
the proof of this lemma is just trivial I
mean if you assume the fact that the uniform
00:18:59.590 --> 00:19:03.179
limit of continuous functions is continuous
okay, so this is something that you can easily
00:19:03.179 --> 00:19:11.710
deduce, so let me pinpoint the important fact
here if you take a point in D okay then you
00:19:11.710 --> 00:19:18.000
can choose a sufficiently small disk surrounding
that point to lie in D that is because D is
00:19:18.000 --> 00:19:23.669
a domain mind you D is an open connected set,
so D is an open set so if you give me a point
00:19:23.669 --> 00:19:28.580
of D there is a whole disk surrounding that
point which is lying inside D and if I make
00:19:28.580 --> 00:19:33.769
the radius of disk small enough I can ensure
that even the boundary of that disk lies inside
00:19:33.769 --> 00:19:35.580
D okay.
00:19:35.580 --> 00:19:41.590
Now if I include the boundary to the disk
then it becomes compact set because it is
00:19:41.590 --> 00:19:46.639
closed and bounded set and on a compact set
I know that this convergence is actually uniform
00:19:46.639 --> 00:19:50.360
because I have been given normal convergence.
Normal convergence means at whenever you look
00:19:50.360 --> 00:19:54.620
at the convergence on compact subset it is
uniform okay therefore if you give me any
00:19:54.620 --> 00:20:04.070
point I can find a sufficiently small disk
on whose closure the convergence is uniform,
00:20:04.070 --> 00:20:10.009
so in particular it is also uniform on sufficiently
small disk okay that means that the limit
00:20:10.009 --> 00:20:16.629
function on sufficiently small disk is continuous
but then I can cover the whole space by such
00:20:16.629 --> 00:20:21.379
small disk, so the limit function becomes
continues everywhere okay that is the proof
00:20:21.379 --> 00:20:29.120
alright. So I want to write down the proof
I have told you in words, now comes the very
00:20:29.120 --> 00:20:39.690
important theorems, so here is the theorem,
the theorem is that so let me tell you in
00:20:39.690 --> 00:20:45.240
this lemma we have been only worrying about
continuous maps alright, now you can ask this
00:20:45.240 --> 00:20:50.350
question because you know in the previous
slide.
00:20:50.350 --> 00:20:55.460
If you look at the previous slide I have giving
you these 2 subsets is one subset which is
00:20:55.460 --> 00:20:59.840
m of D which is the Meromorphic functions
and then there is this other subset which
00:20:59.840 --> 00:21:05.169
is H of D which is a holomorphic function
and then you can ask what will happen to the
00:21:05.169 --> 00:21:13.799
limit function if the given sequence of functions
is in m of D or H of D, so you can ask so
00:21:13.799 --> 00:21:19.440
what we have just now proved in this lemma
is that a normal limit of continuous function
00:21:19.440 --> 00:21:22.370
is continuous alright that is what the lemma
says.
00:21:22.370 --> 00:21:26.539
Now we can ask is a normal limit of Meromorphic
functions Meromorphic that is one question
00:21:26.539 --> 00:21:32.840
can ask then the other question you can ask
is, is the normal limit of holomorphic that
00:21:32.840 --> 00:21:36.879
is analytic functions analytic you can ask
that but you know you already have seen an
00:21:36.879 --> 00:21:43.669
exception, we have seen an example of a normal
limit of holomorphic analytic functions which
00:21:43.669 --> 00:21:48.539
is going to the constant function infinity
okay. Now this is the only exception that
00:21:48.539 --> 00:21:53.580
is the beautiful thing, so in the case of
an analytic function and that is a theorem,
00:21:53.580 --> 00:22:00.899
so if you are looking at sequence of holomorphic
or analytic functions suppose it converges
00:22:00.899 --> 00:22:06.039
normally to a limit function then the limit
function is either completely holomorphic
00:22:06.039 --> 00:22:10.860
that is completely analytic or it is completely
infinity it is a constant function infinity
00:22:10.860 --> 00:22:12.869
and there is nothing in between okay.
00:22:12.869 --> 00:22:19.759
So this is very good behaviour, so what it
means is that you know if you are the
00:22:19.759 --> 00:22:26.970
exception whenever you are working with
respect to the spherical metric, that is you
00:22:26.970 --> 00:22:31.590
are working with respect to extended complex
plane namely you are allowing the value infinity
00:22:31.590 --> 00:22:38.669
okay then you must allow the exceptional case
that is sequence of holomorphic function tends
00:22:38.669 --> 00:22:46.269
to the constant function infinity uniformly
on compact sets okay that is the only exception
00:22:46.269 --> 00:22:51.860
that is what the theorem says, so for example
you do not have this further pathologies like
00:22:51.860 --> 00:22:58.260
you know you have a sequence of analytic functions
and in the limit you get a Meromorphic function.
00:22:58.260 --> 00:23:02.490
Suddenly a pole pops up in the limit you know
such kind of horrible behaviour does not happen
00:23:02.490 --> 00:23:03.490
okay.
00:23:03.490 --> 00:23:08.950
Now that is very very important that tells
you that you know the behaviour is very good
00:23:08.950 --> 00:23:13.970
right so otherwise you would have been worried
if you can find a sequence of analytic functions
00:23:13.970 --> 00:23:17.950
okay which is converging normally to a limit
function and the limit function suddenly starts
00:23:17.950 --> 00:23:24.340
having poles you know of course intuitively
you do not expect that to happen but how do
00:23:24.340 --> 00:23:28.820
you prove that such a thing does not happen
and that is what the theorem says okay. So
00:23:28.820 --> 00:23:40.999
let me write that down theorem: if f n is
a sequence of holomorphic functions on H of
00:23:40.999 --> 00:23:59.769
D and f n converges to f normally on D then
either f is also holomorphic or f is identically
00:23:59.769 --> 00:24:08.179
infinity that is all, you do not get the intermediate
is of f b in Meromorphic okay.
00:24:08.179 --> 00:24:19.090
So this is the theorem and this is intuitively
this looks fine but the big deal with serious
00:24:19.090 --> 00:24:25.419
mathematics is to some statements which are
intuitive and then the bigger deal is to really
00:24:25.419 --> 00:24:31.840
prove them you giving prove is very important
part of mathematics. So let us go to this
00:24:31.840 --> 00:24:43.619
so the proof requires a couple of things
1st of all the proof requires the so-called
00:24:43.619 --> 00:24:49.470
invariance of the spherical metric with respect
to , so let me explain that so that is one
00:24:49.470 --> 00:24:53.409
ingredient of the proof, the other ingredient
of the proof is so-called Hurwitz’s theorem
00:24:53.409 --> 00:24:57.169
which probably you have seen it in the 1st
course in complex analysis but I do not expect
00:24:57.169 --> 00:25:00.919
many people to have seen it in the 1st course
in complex analysis, so I will tell you what
00:25:00.919 --> 00:25:04.730
the theorem is, it is a pretty simple theorem
it has got to do with uniform convergence
00:25:04.730 --> 00:25:10.179
and it has got to do with the argument principal
which you might have seen okay, so so let
00:25:10.179 --> 00:25:19.230
me start with this start with the invariance
of the spherical metric with respect to inversion.
00:25:19.230 --> 00:25:30.269
So let us again recall, so let me put this
here may be I will use a different color,
00:25:30.269 --> 00:25:52.679
so invariance of the spherical metric
with respect to inversion, this is a very
00:25:52.679 --> 00:26:01.190
important fact, so here is my so here is what
I am going to say so you see you have this
00:26:01.190 --> 00:26:09.039
so let me again draw this stereographic
projection, so here is the this is x y plane
00:26:09.039 --> 00:26:15.950
this is the complex plane which is identified
with the x y plane and then you have this
00:26:15.950 --> 00:26:32.280
well you
have the Riemann sphere again okay and of
00:26:32.280 --> 00:26:48.120
course there is this 3rd axis which…so here
is the 3rd one this is u because of course
00:26:48.120 --> 00:26:56.159
you know I am reserving I reserving Z to be
x plus i y, so I called in the 3rd axis as
00:26:56.159 --> 00:27:07.399
u and now you see you take 2 points Z 1 and
say Z 2 okay on the complex plane and in fact
00:27:07.399 --> 00:27:12.580
you know then you get the stereographic projection
on the Riemann sphere.
00:27:12.580 --> 00:27:20.049
So you get these points so you get this point
P 1 and you get this point P 2 okay and of
00:27:20.049 --> 00:27:25.629
course you know you get P 1 by so this is
the North pole and you join North pole to
00:27:25.629 --> 00:27:31.330
P 1 then it should go and hit Z 1 you join
North pole to P 2 and it has to go and hit
00:27:31.330 --> 00:27:40.309
Z 2 okay, so this is the definition of stereographic
projection and what is the spherical distance
00:27:40.309 --> 00:27:55.539
between Z 1 and Z 2 this is actually the spherical
distance well let me call this as a S
00:27:55.539 --> 00:28:06.700
2 this is the real 2 sphere this is a standard
topological notation, so let me put this put
00:28:06.700 --> 00:28:15.779
S 2 here between P 1 and P 2 and what is a
thing I have written this on the right.
00:28:15.779 --> 00:28:28.190
This DS superscript S 2 is actually the geodesic
distance, it is the distance along that great
00:28:28.190 --> 00:28:32.240
circle passing through P 1 and P 2 you know
if you have a sphere and if you take 2 points
00:28:32.240 --> 00:28:38.580
of the sphere if there are 2 distinct points
as only one big circle, circle of largest
00:28:38.580 --> 00:28:42.549
radius that passes through those 2 points
and lies on the sphere that is called the
00:28:42.549 --> 00:28:49.010
great circle passing through those 2 points
okay and then what you do is that you
00:28:49.010 --> 00:28:54.549
have these 2 points on a circle and they divide
the circle into a smaller portion, a larger
00:28:54.549 --> 00:29:00.350
portion in general and you take the length
of the smaller arc okay, so that is what this
00:29:00.350 --> 00:29:05.529
distance here is and that is defined to be
equal to the distance of the spherical distance
00:29:05.529 --> 00:29:11.919
between the point Z 1 and Z 2 okay and well
this is spherical distance.
00:29:11.919 --> 00:29:21.200
Now the point is that you know I need also
to write something here, so so let me do the
00:29:21.200 --> 00:29:39.580
following thing let me take another color,
so you have also so if I write it out in a
00:29:39.580 --> 00:29:49.799
so let me write it somewhere here, so
you have S 2 and you have this stereographic
00:29:49.799 --> 00:29:55.899
projection with C union infinity, so C union
infinity is extended complex plain there is
00:29:55.899 --> 00:30:02.360
a point at infinity added to that set of complex
numbers and this infinity goes to the North
00:30:02.360 --> 00:30:05.799
pole mind you under the stereographic injection,
so what I have written here is the stereographic
00:30:05.799 --> 00:30:19.700
projection and so here P 1 goes to Z 1 and
P 2 goes to Z 2 okay and this is the situation
00:30:19.700 --> 00:30:24.919
that we have but on C union infinity there
is automorphism okay there is a self-isomorphism,
00:30:24.919 --> 00:30:31.179
there is an isomorphism at least as a set
and what is that isomorphism? That is inverse
00:30:31.179 --> 00:30:36.369
okay so on C union infinity you have the map
Z going to 1 by Z.
00:30:36.369 --> 00:30:42.159
Z going to 1 by Z makes sense on C union infinity
where you declare 0 to go to infinity and
00:30:42.159 --> 00:30:47.559
you declare infinity to go to 0, so it interchanges
0 and infinity but for points which are different
00:30:47.559 --> 00:30:53.019
from 0 and infinity you know for a complex
number which is different from 0 and infinity
00:30:53.019 --> 00:30:57.600
of course when we say complex number infinity
is not allowed, so if it is nonzero complex
00:30:57.600 --> 00:31:02.950
number than it is reciprocal is also a nonzero
complex number, so the point is that on this
00:31:02.950 --> 00:31:08.460
right side here which is the C union infinity
there is this map Z going to 1 by Z and this
00:31:08.460 --> 00:31:14.370
map Z going to 1 by Z is in fact a homeomorphism
because you know it is continuous and it remains
00:31:14.370 --> 00:31:20.389
continuous even if you give C union infinity
of course you know Z going to 1 by Z is continuous
00:31:20.389 --> 00:31:24.330
in the punctured complex plane that is if
you take the complex plane and remove the
00:31:24.330 --> 00:31:29.799
origin Z going to 1 by Z is actually you know
holomorphic isomorphism because it is an injectable
00:31:29.799 --> 00:31:36.279
holomorphic map and you do not have to go
too deep because it is inverses itself for
00:31:36.279 --> 00:31:38.830
Z going to 1 by Z the inverse map is itself
okay.
00:31:38.830 --> 00:31:44.779
We apply the map twice you get identity okay,
so it is a holomorphic isomorphism it is an
00:31:44.779 --> 00:31:49.419
analytic isomorphism C minus 0 to C minus
0 but the point is if you include 0 then you
00:31:49.419 --> 00:31:53.649
have to also include infinity and you have
to send 0 to infinity and you have to send
00:31:53.649 --> 00:31:59.590
infinity to 0 and therefore you get Bijective
map of the extended complex plane and that
00:31:59.590 --> 00:32:05.899
map is actually a homeomorphism okay with
respect to the one-point compactification
00:32:05.899 --> 00:32:14.169
topology on C union infinity we can check
that, so but you see this Z going to 1 by
00:32:14.169 --> 00:32:19.730
Z which is inversion is a homeomorphism on
the extended plane.
00:32:19.730 --> 00:32:28.450
Now if you transport that is via the stereographic
projection you will get you will get homeomorphism
00:32:28.450 --> 00:32:35.340
of the real sphere S 2 which is a Riemann
sphere okay because you know if 2 spaces are
00:32:35.340 --> 00:32:42.649
isomorphic then if 2 topological space are
isomorphic that is holomorphic if one
00:32:42.649 --> 00:32:48.669
one space you have a automorphism a homeomorphism,
self-homeomorphism then this isomorphism will
00:32:48.669 --> 00:32:55.539
transport it and give rise to a self-homeomorphism
to the other space okay, so this inversion
00:32:55.539 --> 00:33:01.899
which is a homeomorphism of the self-homeomorphism
of the extended plane will give you a self-homeomorphism
00:33:01.899 --> 00:33:08.830
Riemann sphere okay and guess what it is?
You know what address it is, you can check
00:33:08.830 --> 00:33:16.759
it, it is actually nothing but rotation of
the Riemann sphere okay it is the rotation
00:33:16.759 --> 00:33:28.799
of the Riemann sphere about the x axis okay
and you can see that that is because you see
00:33:28.799 --> 00:33:35.600
you take any point on the x-axis okay you
take any point on the x-axis say for example
00:33:35.600 --> 00:33:42.830
it the point 1 on the x-axis that corresponds
the complex number 1 okay.
00:33:42.830 --> 00:33:51.679
Where does it go it goes back to 1 alright
and the point minus 1 goes back to minus 1
00:33:51.679 --> 00:33:58.529
and you know the stereographic projection
is such that for every point on the unit circle,
00:33:58.529 --> 00:34:02.279
the point on the Riemann sphere is the same
as the point on the unit circle for the stereographic
00:34:02.279 --> 00:34:06.539
projection, this geographic projection induces
a bisection on the unit circle because the
00:34:06.539 --> 00:34:12.429
unit circle lies also on the complex plane,
it lies also on the Riemann sphere on S 2
00:34:12.429 --> 00:34:21.860
and this stereographic projection fixes point
wise it fixes the unit circle, so this inversion
00:34:21.860 --> 00:34:29.590
is going to induce some homeomorphism of the
Riemann sphere that is going to fix plus 1
00:34:29.590 --> 00:34:38.840
and minus 1 okay and look at what happens
to a point at infinity and infinity goes to
00:34:38.840 --> 00:34:41.450
0 and 0 goes to infinity okay.
00:34:41.450 --> 00:34:46.500
Now what does this translate to the Riemann
sphere you see on the Riemann sphere infinity
00:34:46.500 --> 00:34:54.169
corresponds to the North pole, 0 corresponds
to the South pole okay the point 0 corresponds
00:34:54.169 --> 00:35:00.790
to the South pole because you see the 0 is
here and you know what is the stereographic
00:35:00.790 --> 00:35:08.720
projection of 0 it is you have to take the
lines joining the North pole to 0 and then
00:35:08.720 --> 00:35:14.160
look at the unique point on the Riemann sphere
where this line hits and that will be the
00:35:14.160 --> 00:35:19.670
South pole okay, so 0 on the complex plane
corresponds to South pole on the Riemann sphere.
00:35:19.670 --> 00:35:25.880
So you know this inversion which sends 0 to
infinity and infinity to 0 on the extended
00:35:25.880 --> 00:35:29.430
plane when you translate into the Riemann
sphere it will send North pole to the South
00:35:29.430 --> 00:35:31.310
pole and South pole to the North pole.
00:35:31.310 --> 00:35:36.160
Now you can imagine this map what it is doing?
it is switching the North and South poles
00:35:36.160 --> 00:35:40.720
and it is fixing plus or minus 1, so it is
a rotation about the x-axis that is what is
00:35:40.720 --> 00:35:47.950
happening okay. You can use your analytic
geometry and you can actually write out equations
00:35:47.950 --> 00:35:53.310
and check that this is actually a rotation
by 180 degrees of this sphere with respect
00:35:53.310 --> 00:36:01.799
to the x-axis okay and now what does this
tell you, this tells you that if I take the
00:36:01.799 --> 00:36:09.039
point Z 1 and Z 2 okay and I take the spherical
distance between them and if I take the inverse
00:36:09.039 --> 00:36:13.549
points 1 by Z 1 and 1 by Z 2 and take the
spherical distance between the inverse points
00:36:13.549 --> 00:36:18.599
that will be the same because this spherical
distances are being measured by looking at
00:36:18.599 --> 00:36:24.470
the corresponding points on the sphere and
the inversion corresponds to rotation of the
00:36:24.470 --> 00:36:30.819
sphere but if you take any 2 points on the
sphere, the spherical distance between those
00:36:30.819 --> 00:36:34.970
2 points that is not going to change if I
rotate the sphere that is invariant under
00:36:34.970 --> 00:36:36.859
rotation of the sphere.
00:36:36.859 --> 00:36:41.210
So the moral of the story is that this spherical
distance between Z 1 and Z 2 is the same as
00:36:41.210 --> 00:36:46.440
spherical distance between 1 by Z 1 and Z
2. In other words the spherical metric is
00:36:46.440 --> 00:36:50.299
invariant for the inversion that is a very
important fact which you are going to use
00:36:50.299 --> 00:37:02.859
the proof okay. So let me write this down
so let me say let me write somewhere here,
00:37:02.859 --> 00:37:07.160
so diagrammatically what you are going to
have is that so this is Z 1 and let us say
00:37:07.160 --> 00:37:17.369
that this is Z 2 sorry this is so if this
is Z 1 then this is…so 1 by Z 1 is going
00:37:17.369 --> 00:37:26.780
to lie somewhere here and if this is Z 2 then
1 by Z 2 is going to lie somewhere here and
00:37:26.780 --> 00:37:39.829
this 1 by Z 1 and 1 by Z 2 are going to correspond
to points P 1 prime and P 2 prime on the Riemann
00:37:39.829 --> 00:37:48.900
sphere and the fact is that the spherical
distance between…the spherical distance
00:37:48.900 --> 00:37:59.210
on S 2 between P 1 prime and P 2 prime is
the same as spherical distance on S 2 between
00:37:59.210 --> 00:38:05.680
P 1 and P 2 because P 1 prime and P 2 prime
are just gotten from P 1 and P 2 by rotation
00:38:05.680 --> 00:38:14.460
by 180 degrees and so in other words I am
saying that if I draw this arc here and if
00:38:14.460 --> 00:38:18.609
I draw this arc here they are of the same
length.
00:38:18.609 --> 00:38:22.230
Mind you this is a prospective drawing, so
they do not really look to be of the same
00:38:22.230 --> 00:38:31.059
length when I draw it on this but they are
of the same length essentially and well
00:38:31.059 --> 00:38:42.380
what you see this guy here on top of it is
the spherical distance between 1 by Z 1 and
00:38:42.380 --> 00:38:51.030
1 by Z 2 and this guy here the bottom is the
spherical distance between Z 1 and Z 2 is
00:38:51.030 --> 00:38:56.730
okay and this spherical distance is invariant
under inversion and the way I have drawn it
00:38:56.730 --> 00:39:00.420
I have taken Z 1 and Z 2 in the complex plane
but you can make one of them or even both
00:39:00.420 --> 00:39:06.869
of them infinity, it still works okay. So
the moral of the story is therefore so let
00:39:06.869 --> 00:39:14.609
me also mention this P 1 goes to Z 1, P 2
goes to Z 2 and of course, so let me write
00:39:14.609 --> 00:39:24.109
it here P 1 prime goes to Z 1 prime, P 2 prime
corresponds sorry so Z 1 prime is actually
00:39:24.109 --> 00:39:32.660
1 by Z 1 and P 2 prime goes to Z 2 prime which
is 1 by Z 2 okay.
00:39:32.660 --> 00:39:54.289
So let me write the statement here so the
holomorphic that is analytic isomorphism
00:39:54.289 --> 00:40:14.630
Z going to 1 over Z from C minus 0 to C minus
00:40:14.630 --> 00:40:31.140
extends to homeomorphism
C union infinity to C union infinity and this
00:40:31.140 --> 00:40:48.780
is inversion where you send infinity goes
to 0 and 0 goes to infinity
00:40:48.780 --> 00:41:03.089
and the homeomorphism that it induces via
the stereographic projection on S 2. S 2 to
00:41:03.089 --> 00:41:22.220
S 2 that it induces via the stereographic
projection
00:41:22.220 --> 00:41:44.670
is just rotation of S 2 about the x axis by
180 degrees okay, so this is the fact that
00:41:44.670 --> 00:41:53.289
you need to check. You can write it out it
is a very simple exercise in analytic geometry
00:41:53.289 --> 00:42:01.030
if you want but the point is that distances
on the sphere will not change if you rotate
00:42:01.030 --> 00:42:13.880
the sphere, so the upshot is the thus or all
points Z 1, Z 2 in C union infinity the spherical
00:42:13.880 --> 00:42:19.760
distance between Z 1 and Z 2 is the same as
the spherical distance between 1 by Z 1 and
00:42:19.760 --> 00:42:23.569
1 by Z 2.
00:42:23.569 --> 00:42:32.309
So this is the fact that we want, so it uses
the fact that if you…distances on the sphere
00:42:32.309 --> 00:42:40.460
will not change if you rotate the sphere and
the rotation of the sphere by 180 degree is
00:42:40.460 --> 00:42:48.230
about the x-axis corresponds to actually it
translates to inversion for the extended plane
00:42:48.230 --> 00:42:57.490
okay, so the spherical metric is inversion
variant, so let me write that in words i.e.
00:42:57.490 --> 00:43:21.270
the spherical metric d S is inversion in variant
okay, so this is one fact that we need to
00:43:21.270 --> 00:43:29.500
use okay I need to use this fact this fact
will be used in trying to prove our theorem
00:43:29.500 --> 00:43:35.950
that you know if you have a sequence of analytic
functions on a domain. Either it converges
00:43:35.950 --> 00:43:44.470
to an analytic function or it converges to
the constant function infinity provided you
00:43:44.470 --> 00:43:50.410
assume the convergence is normal many form
on compact subset okay. So this is one fact,
00:43:50.410 --> 00:43:56.700
the other fact is the very important Hurwitz’s
theorem okay which I will try to explain next.
00:43:56.700 --> 00:44:03.589
So the next thing that I needed is Hurwitz’s
theorem that is another thing that I need
00:44:03.589 --> 00:44:18.970
Hurwitz’s theorem okay, so what is this
Hurwitz’s theorem? So basically the theorem
00:44:18.970 --> 00:44:24.330
is very very simple, what the theorem says
is that it tells you something that is very
00:44:24.330 --> 00:44:29.270
very believable. What it says is that you
know if you take a normal limit of analytic
00:44:29.270 --> 00:44:35.569
functions, holomorphic functions then of course
you know the limit function suppose a limit
00:44:35.569 --> 00:44:43.880
function is complex value not it doesn’t
take the value infinity, so we are in the
00:44:43.880 --> 00:44:48.490
setup of an undergraduate course of complex
analysis where all complex function takes
00:44:48.490 --> 00:44:53.369
only values in C okay and you are not allowing
the value infinity okay.
00:44:53.369 --> 00:44:58.069
So suppose the sequence of analytic functions
converges normally to a function which takes
00:44:58.069 --> 00:45:02.539
values only in C then you know already that
this limit function is already analytic there
00:45:02.539 --> 00:45:06.490
is something that you know I have already
explain to you that a normal limit of analytic
00:45:06.490 --> 00:45:11.460
functions is analytic provided you make sure
that the limit function takes values only
00:45:11.460 --> 00:45:18.680
in C okay this is again something that you
have seen in the 1st course in complex analysis
00:45:18.680 --> 00:45:21.970
basically using Morera's theorem and Clausius
theorem okay.
00:45:21.970 --> 00:45:27.569
So the limit function is analytic, now what
Hurwitz’s theorem says is that if you take
00:45:27.569 --> 00:45:33.750
a 0 of the limit function okay and mind you
the limit function is analytic and you know
00:45:33.750 --> 00:45:39.540
an analytic function has isolated zeros unless
it is identically 0 and a non-constant analytic
00:45:39.540 --> 00:45:45.760
function if you take the zeros they are isolated
there is a very important fact and for example
00:45:45.760 --> 00:45:52.440
it is as powerful as identity theorem which
says that if 2 analytic functions are having
00:45:52.440 --> 00:46:01.270
the same values on a convergence set of points
even at the limit including the limit which
00:46:01.270 --> 00:46:06.150
is also a point of analyticity then the functions
have to be identically the same that is identity
00:46:06.150 --> 00:46:12.450
theorem and that is equivalent to this theorem
that the zeros of an analytic functions or
00:46:12.450 --> 00:46:16.549
isolated, a non-constant analytic function
zeros are isolated okay.
00:46:16.549 --> 00:46:25.490
So you take normal sequence of analytic functions,
you take the limit function it is analytic
00:46:25.490 --> 00:46:29.720
and you take a 0 of the limit function then
what Hurwitz’s theorem says is that this
00:46:29.720 --> 00:46:38.020
0 of the limit function is from zeros of the
functions of the original sequence. In fact
00:46:38.020 --> 00:46:42.059
if you take a 0 of the limit function it has
to have certain order you know the zeros always
00:46:42.059 --> 00:46:47.210
have a certain order this way you will have
a 0 of order 1 or order 2 and so on, so if
00:46:47.210 --> 00:46:55.380
the order of 0 is capital n Hurwitz’s theorem
says that because this limit function has
00:46:55.380 --> 00:47:02.910
come as a normal limit of a sequence, what
it will say is that beyond a certain stage
00:47:02.910 --> 00:47:08.529
each of the functions in your sequence will
also have n zeros.
00:47:08.529 --> 00:47:13.779
The same number of zeros with multiplicity
as the 0 of the limit function that you are
00:47:13.779 --> 00:47:20.680
looking at and these zeros and this will happen
in a neighbourhood of that 0 of the limit
00:47:20.680 --> 00:47:26.440
function and what will happen is slowly as
n tense to infinity these zeros will cluster
00:47:26.440 --> 00:47:30.920
and cluster and cluster and come closer and
closer and in the limit they will all the
00:47:30.920 --> 00:47:35.619
0 of the limit function. So basically what
Hurwitz’s theorem says is that the 0 of
00:47:35.619 --> 00:47:40.770
the limit function is actually coming as limit
of zeros of the original functions beyond
00:47:40.770 --> 00:47:46.619
a certain stage and the way this is done is
very precisely the sense that even multiplicities
00:47:46.619 --> 00:47:50.760
are taken care of that is essentially what
Hurwitz’s theorem is okay. This is how you
00:47:50.760 --> 00:47:58.710
say it in words but of course when you technically
write it down it looks pretty more difficult
00:47:58.710 --> 00:47:59.750
okay.
00:47:59.750 --> 00:48:14.299
So now let me write this statement of the
theorem one suppose f n converges to f uniformly
00:48:14.299 --> 00:48:41.369
normally on D, D domain and f n and f
are all holomorphic functions on D and Z naught
00:48:41.369 --> 00:49:03.720
belonging to D is zero of f of order n greater
than 0 then there exist a row such that
00:49:03.720 --> 00:49:18.170
f n of Z has n 0 in mod Z minus Z naught less
than go for n sufficiently large, so when
00:49:18.170 --> 00:49:38.060
I say n zeros counted with multiplicity. In
this disk for n sufficiently large and of
00:49:38.060 --> 00:49:48.559
course mod Z minus Z naught less than row
is in D, so you have to choose row small enough
00:49:48.559 --> 00:50:07.269
that is such a row and further and these zeros
sets converge to Z naught as an tends to infinity,
00:50:07.269 --> 00:50:15.000
so this is Hurwitz’s theorem in notations
okay.
00:50:15.000 --> 00:50:22.269
So what it says is that 0 Z naught of the
limit function, the limit analytic function
00:50:22.269 --> 00:50:34.589
f is the limit point of zeros of the functions
in the sequence and the fact is that even
00:50:34.589 --> 00:50:38.940
the multiplicities are taken care of. Mind
you multiplicity is a very important because
00:50:38.940 --> 00:50:43.480
you know multiplicity is you have to count
zeros with multiplicity, you cannot just count
00:50:43.480 --> 00:50:50.799
zeros just as points it may be a 0. A function
may have a 0 at a point but it may have a
00:50:50.799 --> 00:50:55.260
certain multiplicity that is the order of
0 you have to count the order also okay. So
00:50:55.260 --> 00:50:59.710
that is why the multiplicities are important
these n zeros they need not be n distinct
00:50:59.710 --> 00:51:04.799
points there could be lesser than n points
with some points having zeros of higher order
00:51:04.799 --> 00:51:09.680
okay, so multiplicity is very important, so
this is Hurwitz’s theorem okay and the prove
00:51:09.680 --> 00:51:15.691
of this theorem essentially uses the argument
principle okay. It uses argument principle
00:51:15.691 --> 00:51:23.730
and it uses uniform convergence okay. So I
will try to give a proof of that in the…I
00:51:23.730 --> 00:51:25.429
will give a sketch of that proof in the next
talk.