WEBVTT
Kind: captions
Language: en
00:00:51.039 --> 00:00:57.469
Alright, so what we are going to do now is
we are going to show that if you have a normal
00:00:57.469 --> 00:01:03.219
converging sequence of meromorphic functions,
then the limit function is either a meromorphic
00:01:03.219 --> 00:01:08.590
function or it is identically the function
of infinity, okay.
00:01:08.590 --> 00:01:14.020
And the importance of this theorem is that
in the limit you can only get a meromorphic
00:01:14.020 --> 00:01:16.090
function and nothing worse, okay.
00:01:16.090 --> 00:01:22.140
Because you see meromorphic function is something
that has special kind of singularities, there
00:01:22.140 --> 00:01:24.180
are is only poles as singularities.
00:01:24.180 --> 00:01:28.880
But when you take a limit of meromorphic functions,
anything could have happened, we could have
00:01:28.880 --> 00:01:36.030
got a function with, for example essentially
singularity, such a thing could have happened,
00:01:36.030 --> 00:01:37.030
okay.
00:01:37.030 --> 00:01:44.599
Or you could have even got a function non-isolated
similarities but all these things do not happen.
00:01:44.599 --> 00:01:48.920
The nice thing happens, namely if you take
a normal limit of meromorphic functions, you
00:01:48.920 --> 00:01:50.429
can get a meromorphic function.
00:01:50.429 --> 00:01:54.759
And the worst thing that can happen is that
it is a function which is identically infinity,
00:01:54.759 --> 00:01:56.920
therefore, okay.
00:01:56.920 --> 00:02:07.759
So let us go ahead into the idea is similar
to the idea that you know if you take
00:02:07.759 --> 00:02:14.180
a sequence of holomorphic functions, and then
you take a normal limit of holomorphic functions.
00:02:14.180 --> 00:02:21.720
Then the limit function is either absolutely
holomorphic, okay, it is holomorphic, perfectly
00:02:21.720 --> 00:02:27.860
holomorphic or it is identically infinity,
okay.
00:02:27.860 --> 00:02:30.050
So let me start like this.
00:02:30.050 --> 00:02:45.480
Let fn be a sequence of maps from D to C union
infinity, D inside the complex plane is a
00:02:45.480 --> 00:02:52.470
domain, okay, of course nonempty set, nonempty
open connected set.
00:02:52.470 --> 00:02:57.200
And I am taking maps with values in C union
infinity which is an external complex plane
00:02:57.200 --> 00:03:01.510
and the reason for that is, of course you
know we allow the value infinity now.
00:03:01.510 --> 00:03:17.819
And so suppose, so you know suppose,
so let me write it here, suppose fn, suppose
00:03:17.819 --> 00:03:28.810
fn tends to f, suppose fn tends to f point
wise, point wise on D, right.
00:03:28.810 --> 00:03:41.220
Suppose it is a point wise convergence, so
that means that f is also here, so your f
00:03:41.220 --> 00:03:47.569
is also here, so let me write it somewhere
here.
00:03:47.569 --> 00:03:55.430
f is also a map from D to C union infinity
and fn tends to f point wise on D, you have
00:03:55.430 --> 00:04:00.309
to be, when you see point wise, it means for
every point.
00:04:00.309 --> 00:04:08.919
And when you say every point, it means that
fn of z tends to f of z for all z in D, okay.
00:04:08.919 --> 00:04:12.599
But then what does fn of z tends to f of z
for all z in D mean?
00:04:12.599 --> 00:04:19.120
It means that the distance between fn of z
and f of z tends to 0, okay.
00:04:19.120 --> 00:04:25.230
As n tends to infinity for each z in D . And
what distance are you going to use, you just
00:04:25.230 --> 00:04:29.920
cannot use the usual Euclidean distance, okay,
because you are allowing the value infinity,
00:04:29.920 --> 00:04:31.860
you have to use spherical metric.
00:04:31.860 --> 00:04:35.260
The distance under the spherical distance
has to be used.
00:04:35.260 --> 00:04:47.800
So let me write that down, i.e. fn z tends
to fz as n tends to infinity for all z in
00:04:47.800 --> 00:04:49.340
D .
00:04:49.340 --> 00:04:59.620
And that is supposed to mean that the spherical
distance of fn of z, spherical distance between
00:04:59.620 --> 00:05:07.720
fn z and f of z, that tends to 0 as n tends
to infinity for every z in D . This is what
00:05:07.720 --> 00:05:10.530
point wise convergence D means.
00:05:10.530 --> 00:05:16.480
And of course you have to use spherical distance
because the function values might be one of
00:05:16.480 --> 00:05:22.510
the points to which we are measuring from
or to which you are measuring the distance
00:05:22.510 --> 00:05:27.190
could be the point at infinity, so then you
will have to use the spherical metric, okay.
00:05:27.190 --> 00:05:30.280
That we have denoted by D sub S, all right.
00:05:30.280 --> 00:05:36.560
So you have a sequence of functions defined
on a domain D and at this moment they are
00:05:36.560 --> 00:05:38.100
only maps, okay.
00:05:38.100 --> 00:05:41.790
I am not even saying that they are continuous
or something like that but then we have been
00:05:41.790 --> 00:05:48.660
looking at so many subsets of this collection
of maps, so the 1st important subsets is those
00:05:48.660 --> 00:05:56.590
which are continuous maps from D to C union
infinity, these are the continuous ones.
00:05:56.590 --> 00:05:59.960
So I will write CTS for continuous maps.
00:05:59.960 --> 00:06:04.890
And continuous maps from D to C union infinity
makes sense because D is anyways a domain,
00:06:04.890 --> 00:06:09.590
it is a topological space, it is a subspace
of the complex plane.
00:06:09.590 --> 00:06:14.840
And C union infinity is a very nice topological
space, in fact even complete compact matrix
00:06:14.840 --> 00:06:16.170
space, okay.
00:06:16.170 --> 00:06:21.770
Because it is, basically it is isomorphic
to the Riemann sphere by the stereo graphic
00:06:21.770 --> 00:06:23.250
projection.
00:06:23.250 --> 00:06:28.210
So you have this and then there were smaller
subsets, more important subsets, there was
00:06:28.210 --> 00:06:34.850
also the subset of meromorphic functions on
D . So this is the subset of meromorphic functions
00:06:34.850 --> 00:06:40.800
on D and so this is meromorphic.
00:06:40.800 --> 00:06:49.280
So meromorphic means that you know these are
functions which are analytic, that is holomorphic
00:06:49.280 --> 00:06:53.240
except for subset which is a subset of necessarily
isolated points.
00:06:53.240 --> 00:06:59.200
And at each of those points the function on
the phone, okay.
00:06:59.200 --> 00:07:02.950
So analytic except for pole, that is what
meromorphic means.
00:07:02.950 --> 00:07:09.330
And we have seen that this set of meromorphic
functions is actually a field, it is a field
00:07:09.330 --> 00:07:13.530
extension of the complex numbers, algebraically
it is a field.
00:07:13.530 --> 00:07:17.870
And then there is further smaller subsets
of holomorphic functions on D or analytic
00:07:17.870 --> 00:07:20.680
functions on D . So let me write here analytic.
00:07:20.680 --> 00:07:26.100
That is a small subset and of course these
are functions which are analytic everywhere
00:07:26.100 --> 00:07:29.780
on D, there is they do not have any singularities,
right.
00:07:29.780 --> 00:07:39.380
And what we have, what we have seen is that
all the fns are in H of D, then f is either
00:07:39.380 --> 00:07:43.880
in H of D or f is identically infinity, that
is what we have seen, okay.
00:07:43.880 --> 00:07:46.960
So let me, so let me write that here.
00:07:46.960 --> 00:08:00.270
We saw is all the fns are in H of D, then
f is in H of D or f is identically infinity.
00:08:00.270 --> 00:08:08.560
This is what we are seen in the last lectures
and it is a nice thing because it says that
00:08:08.560 --> 00:08:13.800
sequence of holomorphic functions, analytic
functions can either go to infinity or it
00:08:13.800 --> 00:08:16.180
will go to analytic functions, that is all.
00:08:16.180 --> 00:08:19.680
You do not get, you do not get something weird
in between.
00:08:19.680 --> 00:08:23.620
for example you do not even get a meromorphic
function which is not holomorphic.
00:08:23.620 --> 00:08:25.389
Okay.
00:08:25.389 --> 00:08:30.360
You remember we used, basically we used 2
important things, we used the invariances
00:08:30.360 --> 00:08:36.169
the spherical metric that between version
and the other thing we used was the Hurwitz
00:08:36.169 --> 00:08:38.520
theorem, okay, the theorem of Hurwitz.
00:08:38.520 --> 00:08:47.820
Alright, now what I want to say is that similar
theorem is true if if you take fns to
00:08:47.820 --> 00:08:49.360
be meromorphic.
00:08:49.360 --> 00:08:56.880
So if you take, what we will show today is
that if all the fns are in M of D, then either
00:08:56.880 --> 00:09:00.170
f even M of D or f is identically infinity.
00:09:00.170 --> 00:09:04.070
So you are just extending this from the holomorphic
case to the meromorphic case, that is what
00:09:04.070 --> 00:09:05.589
we want to do, okay.
00:09:05.589 --> 00:09:07.130
And that is also very nice thing.
00:09:07.130 --> 00:09:13.320
See the point is that, you know why all these
theorems are important is that, you see on
00:09:13.320 --> 00:09:19.579
the one hand they are very believable, I mean
you expect them to happen because under normal
00:09:19.579 --> 00:09:22.730
convergence good properties are preserved,
okay.
00:09:22.730 --> 00:09:26.810
You know from basic analysis, if you have
normal convergence, which is locally uniform
00:09:26.810 --> 00:09:29.230
convergence, so essentially it is uniform
convergence locally.
00:09:29.230 --> 00:09:34.629
And under uniform convergence all nice things
happen.
00:09:34.629 --> 00:09:38.500
If you take uniform, if you take uniform limit
of continuous functions, you get a continuous
00:09:38.500 --> 00:09:43.019
function, if you take uniform limit of analytic
functions, you get analytic functions.
00:09:43.019 --> 00:09:50.920
If you take uniform limit of integrals, okay,
then that is the same as integrating the uniform
00:09:50.920 --> 00:09:55.730
limit, you can interchange the limit an integral,
you can interchange, you can do term wise
00:09:55.730 --> 00:10:00.019
differentiation if you are working with the
series which is uniformly converging.
00:10:00.019 --> 00:10:03.220
So uniform convergence is a very nice thing
to suffer under uniform convergence you expect
00:10:03.220 --> 00:10:04.259
everything to go smoothly.
00:10:04.259 --> 00:10:09.230
So it is correct to expect that if you take
a sequence of holomorphic functions, if it
00:10:09.230 --> 00:10:16.750
converges normally, that is locally uniformly,
then the limit function is also holomorphic,
00:10:16.750 --> 00:10:19.079
that is correct to expect and that is what
happens.
00:10:19.079 --> 00:10:22.579
Similarly if you have sequence of meromorphic
functions, if it converges normally to limit
00:10:22.579 --> 00:10:25.170
function, the limit function is also meromorphic.
00:10:25.170 --> 00:10:29.060
The only thing that you have to worry about
is the other extreme possibility which is
00:10:29.060 --> 00:10:34.480
that the function becomes infinity, that is
a possibility you cannot ignore, okay.
00:10:34.480 --> 00:10:38.779
But that is the worst thing that will happen
and nothing in between these 2 happens.
00:10:38.779 --> 00:10:45.950
So it is important that, this is very very
important because you know it should not happen
00:10:45.950 --> 00:10:50.630
that I start with a series of meromorphic
functions of holomorphic functions.
00:10:50.630 --> 00:10:54.309
If there are holomorphic functions, they do
not have any singularities, if they are meromorphic
00:10:54.309 --> 00:10:56.310
functions they have poles, okay.
00:10:56.310 --> 00:11:00.410
But in the limit suddenly I should not get
a function which is crazy enough to have say
00:11:00.410 --> 00:11:01.910
essential singularities.
00:11:01.910 --> 00:11:07.869
Or I should not get a function which has a
singularity which is nonisolated, I mean such
00:11:07.869 --> 00:11:09.970
horrible thing should not happen.
00:11:09.970 --> 00:11:13.670
And you can expect such horrible things should
not happen because of normal convergence,
00:11:13.670 --> 00:11:17.529
which is local uniform convergence and this
is exactly what we are trying to show in these
00:11:17.529 --> 00:11:18.529
lectures.
00:11:18.529 --> 00:11:21.959
I mean this is, so you know this is always
part of mathematics, you guess that something
00:11:21.959 --> 00:11:25.209
nice will happen but then to prove it you
will have to do some work, he will have to
00:11:25.209 --> 00:11:26.930
use theorems, okay.
00:11:26.930 --> 00:11:33.399
So, well, so let us see.
00:11:33.399 --> 00:11:39.089
So 1st of all I want to say that see if you
take the 1st thing I want you to understand
00:11:39.089 --> 00:11:45.490
is that if all these fns were not
just maps but suppose they were continuous,
00:11:45.490 --> 00:11:50.059
okay, the f also becomes continuous.
00:11:50.059 --> 00:11:54.920
Because uniform limit of continuous function
is continuous and a normal limit is a local
00:11:54.920 --> 00:11:56.870
uniform limit, all right.
00:11:56.870 --> 00:12:05.110
Therefore, so let me write that down, if,
so let me use a different colour.
00:12:05.110 --> 00:12:20.069
See if all the, if all fn are continuous functions
from D to C union infinity and the fns converge
00:12:20.069 --> 00:12:33.509
to f normally, normally on D
and this always means with respect to the
00:12:33.509 --> 00:12:35.240
spherical metric.
00:12:35.240 --> 00:12:41.300
So you know here I must tell you that in the
previous slide here when I say
00:12:41.300 --> 00:12:47.681
if fn, if fn are all holomorphic then f is
holomorphic or f is identically infinity here.
00:12:47.681 --> 00:12:53.410
Of course I am assuming that the convergence
is not just point wise but it is in fact normal.
00:12:53.410 --> 00:13:01.259
fn converges to f normally, this is very important,
this is very very important.
00:13:01.259 --> 00:13:04.990
Of course normal convergence I did not mention
it then but it is important, okay.
00:13:04.990 --> 00:13:12.760
So point wise convergence is useless, point
wise limits cannot be as good as you want
00:13:12.760 --> 00:13:13.760
them to be, okay.
00:13:13.760 --> 00:13:18.029
But normal convergence is very very important
here, right.
00:13:18.029 --> 00:13:22.689
So suppose you take all the fns are continuous
and assume that the convergence is normal
00:13:22.689 --> 00:13:32.679
on D, then of course f is also continuous
and this is just the, the very well-known
00:13:32.679 --> 00:13:37.619
fact that the uniform limit of continuous
function is continuous, all right, you know
00:13:37.619 --> 00:13:38.619
that.
00:13:38.619 --> 00:13:43.550
And the point is that normal limit is something
that is locally a uniform limit, okay, so
00:13:43.550 --> 00:13:49.199
it means that given any point, okay, because
you are working on an open set, you can always
00:13:49.199 --> 00:13:54.959
find the closed disc surrounding centred with
that point which is inside your open said,
00:13:54.959 --> 00:13:55.959
okay.
00:13:55.959 --> 00:14:00.660
And such a closed disc is of course compact
and normal convergence promises you that the
00:14:00.660 --> 00:14:02.850
convergence will be uniform on that closed
disc.
00:14:02.850 --> 00:14:10.759
In particular it will be uniform on the open
disc which is a subset of the closed disc.
00:14:10.759 --> 00:14:15.360
So whenever you have uniform convergence on
a set, you will always have, you will automatically
00:14:15.360 --> 00:14:18.920
have uniform converges on any subset, okay.
00:14:18.920 --> 00:14:24.699
So you get locally uniform converges and therefore
the locally the limit function f will become
00:14:24.699 --> 00:14:26.470
continuous.
00:14:26.470 --> 00:14:31.290
And if you and continuity is a local property,
so if a function is locally continuous, discontinuous,
00:14:31.290 --> 00:14:33.420
if you have global functions.
00:14:33.420 --> 00:14:39.759
So that is why f is continuous, now what is
very very important is the following thing.
00:14:39.759 --> 00:14:47.629
If you add this condition that, now notice
that since you are working with values in
00:14:47.629 --> 00:14:54.079
the extended plane, this is very very important,
not only fn makes sense, 1 by fn also makes
00:14:54.079 --> 00:14:55.689
sense.
00:14:55.689 --> 00:15:00.000
And not only f makes sense, 1 by f also makes
ends, okay.
00:15:00.000 --> 00:15:22.630
So you see so let me say this,
node then that 1 by fn also makes sense
00:15:22.630 --> 00:15:26.660
and 1 by f also makes sense.
00:15:26.660 --> 00:15:32.910
So let me say, makes sense, so let me put
something more general.
00:15:32.910 --> 00:15:42.740
1 by fn certainly makes sense as map from
D to C union infinity, okay.
00:15:42.740 --> 00:15:53.819
1 by fn of z is very well-defined, if fn of
z is infinity, then 1 by fn of z is 0, if
00:15:53.819 --> 00:15:59.170
fn of z is 0, then 1 by fn of z is infinity
and if fn of z is a nonzero complex number
00:15:59.170 --> 00:16:02.149
than 1 by fn of z is the inverse of that complex
numbers.
00:16:02.149 --> 00:16:03.149
So it is very well-defined.
00:16:03.149 --> 00:16:07.639
So this one by fn is also, these 1 by fns
also makes sense, okay.
00:16:07.639 --> 00:16:14.889
And not only that similarly 1 by f also
makes sense, okay.
00:16:14.889 --> 00:16:25.209
Any function on the domain which has values
in the extended plane has an inverse, okay.
00:16:25.209 --> 00:16:29.110
for example, if you take the worst-case which
is the function which is identically 0, you
00:16:29.110 --> 00:16:33.750
do not have to worry, its inverse is the function
which is identically infinity.
00:16:33.750 --> 00:16:37.369
But it is not an inverse in algebraic sense
that you know the function multiplied by the
00:16:37.369 --> 00:16:39.429
inverse will not give you 1.
00:16:39.429 --> 00:16:44.739
You should not go ahead and write the function
0 multiplied by the function infinity is equal
00:16:44.739 --> 00:16:46.399
to1, that is not correct, okay.
00:16:46.399 --> 00:16:49.680
But this is a convention, to the convention,
okay.
00:16:49.680 --> 00:16:55.970
So what we say is if you take a function which
is identically 0, then its inverse function
00:16:55.970 --> 00:16:58.610
is defined to be the function which is identically
infinity.
00:16:58.610 --> 00:17:02.819
But it is not an inverse in the algebraic
sense, you have to be careful about that,
00:17:02.819 --> 00:17:03.819
okay.
00:17:03.819 --> 00:17:06.650
fine but what is the advantage of this?
00:17:06.650 --> 00:17:13.250
Advantage of this is that you know if
you are looking at meromorphic functions,
00:17:13.250 --> 00:17:17.870
then the inverses of meromorphic functions
are also meromorphic functions, that is the
00:17:17.870 --> 00:17:18.870
advantage.
00:17:18.870 --> 00:17:25.770
So now what I want to tell you is that you
see if, suppose I assume that fn converges
00:17:25.770 --> 00:17:30.870
to f normally, okay, then come on D, with
respect to the spherical metric, then it also
00:17:30.870 --> 00:17:39.470
happens that one by fn converges to 1 by f
normally on D with respect to the spherical
00:17:39.470 --> 00:17:40.470
metric.
00:17:40.470 --> 00:17:45.020
That will also happen, that is just because
of the fact that the spherical metric is invariant
00:17:45.020 --> 00:17:47.440
with respect to inversion, okay.
00:17:47.440 --> 00:18:01.670
So let me write that, note also that 1 by
fn converges to 1 by f normally on D with
00:18:01.670 --> 00:18:13.120
respect to spherical metric, because the distance,
spherical distance between fn of z and f of
00:18:13.120 --> 00:18:20.120
z is the same as the spherical distance between
1 by fn of z and 1 by f of z.
00:18:20.120 --> 00:18:24.040
This is just the invariance of the spherical
metric under inversion, okay.
00:18:24.040 --> 00:18:30.380
I told you that the inversion on the complex
plane, if you transport it via the stereo
00:18:30.380 --> 00:18:34.940
graphic projection to the Riemann sphere,
what what it will give you is it will give
00:18:34.940 --> 00:18:37.070
you rotation of the Riemann sphere.
00:18:37.070 --> 00:18:42.960
It will give you a rotation of the Riemann
sphere by 180 degrees, okay.
00:18:42.960 --> 00:18:52.520
So and under under rotation, under rotations
of a sphere, the distance, the spherical distance
00:18:52.520 --> 00:18:56.330
between 2 points on the sphere will not change
obviously.
00:18:56.330 --> 00:19:00.040
So this is something that we have seen before.
00:19:00.040 --> 00:19:05.080
So the point is that it is beautiful, if fn
converges to f normally, then one batsman
00:19:05.080 --> 00:19:09.470
converges to also 1 by f normally and this
is, these are equivalent mind you.
00:19:09.470 --> 00:19:17.940
fn converges to f, if and only if one by fn
converges to 1 by f, okay, it is a simultaneous
00:19:17.940 --> 00:19:18.940
statement.
00:19:18.940 --> 00:19:22.020
these 2 simultaneous statements
and both of them are equivalent, okay.
00:19:22.020 --> 00:19:25.950
So the point is that, therefore what I am
trying to tell you is the philosophy is as
00:19:25.950 --> 00:19:26.950
follows.
00:19:26.950 --> 00:19:31.710
Whenever you are looking at functions with
values in the extended plane, always think
00:19:31.710 --> 00:19:37.920
automatically of the reciprocal functions
also, the inverse of those functions.
00:19:37.920 --> 00:19:43.660
They also make sense, okay, that is the advantage.
00:19:43.660 --> 00:19:51.050
now what I want to tell you is that you see,
suppose, now let me do the following thing.
00:19:51.050 --> 00:20:01.960
Suppose all the, suppose all the functions
fn were all meromorphic, okay, what I want
00:20:01.960 --> 00:20:09.250
to actually say is that f will be meromorphic
and that is important result that we want
00:20:09.250 --> 00:20:10.250
to see.
00:20:10.250 --> 00:20:11.850
So now let me, let me assume that.
00:20:11.850 --> 00:20:24.480
Suppose fn are all meromorphic functions
only, okay, then the theorem is that f is
00:20:24.480 --> 00:20:26.410
also meromorphic only.
00:20:26.410 --> 00:20:36.370
So here is a theorem, then f is meromorphic
or the other extreme case is that f is identically
00:20:36.370 --> 00:20:38.420
infinity, okay.
00:20:38.420 --> 00:20:42.280
So this is the theorem, okay.
00:20:42.280 --> 00:20:51.960
If fn is meromorphic, each fn is meromorphic,
than the normal limit of fn namely f, that
00:20:51.960 --> 00:20:55.200
is meromorphic or it is identically infinity.
00:20:55.200 --> 00:21:01.440
What it means is that the limit function will
have, if it all it has singularity is, they
00:21:01.440 --> 00:21:05.550
will only be isolated, they will be poles
and you will not get anything worse than that,
00:21:05.550 --> 00:21:06.550
okay.
00:21:06.550 --> 00:21:14.310
now what I wanted to understand is that, you
see 1st and foremost fn are meromorphic, okay,
00:21:14.310 --> 00:21:22.670
and that will imply that 1 by fn are also
meromorphic, okay.
00:21:22.670 --> 00:21:28.670
See because, see reciprocal of a meromorphic
function is also a meromorphic function, right.
00:21:28.670 --> 00:21:39.510
And the, you see, so this, so this makes sense
for.
00:21:39.510 --> 00:21:47.440
All this is okay if, with a small modification,
all right.
00:21:47.440 --> 00:21:53.790
This is provided fn is not identically infinity,
okay, because a function which is identically
00:21:53.790 --> 00:22:05.231
infinity, we do not, I mean we do not, you
know it is an extra functions at we add, all
00:22:05.231 --> 00:22:06.231
right.
00:22:06.231 --> 00:22:08.961
We do not consider it to be meromorphic, we
consider to be an extreme case, the function
00:22:08.961 --> 00:22:11.130
which is identically infinity, okay.
00:22:11.130 --> 00:22:16.120
But its inverse will be the function which
is identically 0 and that is a nice constant
00:22:16.120 --> 00:22:18.520
function, it is the holomorphic function also,
okay.
00:22:18.520 --> 00:22:25.710
So when I write this, I am assuming that the
fn that I am looking at is not identically
00:22:25.710 --> 00:22:31.260
infinity and when you may ask what if fn is
identically infinity, in that case 1 by fn
00:22:31.260 --> 00:22:35.250
is defined to be the function that is identically
0 and that is by definition it is a constant
00:22:35.250 --> 00:22:36.500
function, the holomorphic okay.
00:22:36.500 --> 00:22:41.700
So that case is always, you know you keep
it separate when a function is identically
00:22:41.700 --> 00:22:43.490
infinity, okay.
00:22:43.490 --> 00:22:49.480
And also this case that if fn is identically
0, then 1 by fn is defined to be identically
00:22:49.480 --> 00:22:51.820
infinity, all right.
00:22:51.820 --> 00:22:58.440
And so you know you should also, I also
throughout the case where fn is not identically
00:22:58.440 --> 00:23:08.060
0, when fn is identically 0, okay, I throw
out that case, because then 1 by fn becomes
00:23:08.060 --> 00:23:11.980
the function which is identically infinity
and the function which is that typically infinity
00:23:11.980 --> 00:23:14.980
is not considered as the meromorphic function,
okay.
00:23:14.980 --> 00:23:21.860
Because it has, meromorphic function should
have only poles only at, you know it has only
00:23:21.860 --> 00:23:24.150
poles and they should be an isolated set of
points.
00:23:24.150 --> 00:23:31.160
If suppose it goes to infinity but it cannot
go to infinity everywhere, right.
00:23:31.160 --> 00:23:40.310
now what I want you to understand is that,
see I want you to see that therefore, since
00:23:40.310 --> 00:23:51.190
1 by fns are all meromorphic, 1 by fns are,
including the case of 1 by fn being infinity,
00:23:51.190 --> 00:23:55.270
okay, all these 1 by fns are of course continuous.
00:23:55.270 --> 00:23:59.930
Mind you because the set of meromorphic functions
is actually contained in the set of all continuous
00:23:59.930 --> 00:24:01.070
maps.
00:24:01.070 --> 00:24:07.590
It is continuous because mind you at the poles
you are allowing the value infinity for a
00:24:07.590 --> 00:24:08.590
function.
00:24:08.590 --> 00:24:14.200
So what you must understand is that all these
1 by fns are in fact continuous functions
00:24:14.200 --> 00:24:21.220
and therefore if you take, if fn tends normally
to f, then because of the invariance of the
00:24:21.220 --> 00:24:27.550
spherical metric under inversion, 1 by fn
will tend normally to 1 by f and 1 by f will
00:24:27.550 --> 00:24:32.540
certainly be continuous, that because each
one by fn is certainly continuous.
00:24:32.540 --> 00:24:38.040
Even is 1 by fn is a function which is identically
infinity, discontinuous, mind you, even the
00:24:38.040 --> 00:24:39.060
extreme case.
00:24:39.060 --> 00:24:51.310
So what I wanted to, fn tends to f
normally implies 1 by fn tends to 1 by f normally,
00:24:51.310 --> 00:24:52.780
this happens.
00:24:52.780 --> 00:25:00.880
And this implies that this 1 by f is certainly
a continuous function from D to C union infinity
00:25:00.880 --> 00:25:08.950
and this includes the case when 1 by fn is,
I mean when, when 1 by f is even identically
00:25:08.950 --> 00:25:11.720
the function which is infinity.
00:25:11.720 --> 00:25:17.030
Mind you the function which is identically
infinity is here but it is not here, did not
00:25:17.030 --> 00:25:20.570
in the meromorphic functions by definition,
right.
00:25:20.570 --> 00:25:25.140
fine, so what we want to show is that, we
want to show this 1 by f is meromorphic, that
00:25:25.140 --> 00:25:26.960
is what we want to show.
00:25:26.960 --> 00:25:36.690
now I want you to, now let me say another
thing that I wanted to say but I did not.
00:25:36.690 --> 00:25:42.500
And that is, that how do you see 1 by fn is
meromorphic, it is very very simple, you look
00:25:42.500 --> 00:25:51.340
at the places where the function fn has poles,
this is an isolated set of points and 1 by
00:25:51.340 --> 00:25:55.340
fn will have 0 that those points, okay.
00:25:55.340 --> 00:26:01.000
Is a function has a pole at the point, then
it is reciprocal will have 0 at that point
00:26:01.000 --> 00:26:04.650
and order of the 0 will be exactly equal to
the order of the pole, okay.
00:26:04.650 --> 00:26:09.100
And if a function has a 0 at a point, then
its reciprocal will have a pole.
00:26:09.100 --> 00:26:17.150
So you see if you take fn to be a meromorphic
function, then there is, it is analytic outside
00:26:17.150 --> 00:26:20.560
isolated set of points, which are poles, okay.
00:26:20.560 --> 00:26:26.960
And outside and isolated set of points you
will get a open subset of D and inside that
00:26:26.960 --> 00:26:30.310
again the set of zeros will again be an isolated
set of points.
00:26:30.310 --> 00:26:33.960
Because you see the set of zeros of an analytic
function is always isolated.
00:26:33.960 --> 00:26:38.610
And this isolated set of points which are
the zeros of fn, they will be the isolated
00:26:38.610 --> 00:26:41.950
set of points which are the poles of 1 by
fn.
00:26:41.950 --> 00:26:47.150
So 1 by fn has only poles as singularities.
00:26:47.150 --> 00:26:50.500
So 1 by fn is meromorphic, you have to understand
that, okay.
00:26:50.500 --> 00:27:00.290
So now you see what I want to say is that,
so we had defined this set D infinity of f.
00:27:00.290 --> 00:27:07.440
This D infinity of f is the set of points
in D where f takes the value infinity.
00:27:07.440 --> 00:27:14.900
So D infinity of S is, so let me write that,
it is a set of all z belonging to D such that
00:27:14.900 --> 00:27:24.500
f of z equal to infinity and this is just
f inverse of infinity and it is a closed set,
00:27:24.500 --> 00:27:30.190
it is a closed subset of D, okay.
00:27:30.190 --> 00:27:42.760
And well and it is closed because
the point infinity is the closed point in
00:27:42.760 --> 00:27:47.620
the extended plane, because it corresponds
to the north pole on the Riemann sphere, okay,
00:27:47.620 --> 00:27:48.960
under stereo graphic projection.
00:27:48.960 --> 00:27:52.000
And the inverse image of a closed set under
continuous function is closed, f is continuous,
00:27:52.000 --> 00:27:55.260
and so D infinity of f is a closed set.
00:27:55.260 --> 00:28:02.030
And the point is that if, if you assume that
f is not identically infinity, then you are
00:28:02.030 --> 00:28:06.950
just saying that D, this is the same as saying
that D infinity of f is a proper subset of
00:28:06.950 --> 00:28:19.690
T. D infinity of f is a proper subset, okay,
because if D infinity of f is all of D, then
00:28:19.690 --> 00:28:23.130
it is the same as saying f is identically
infinity.
00:28:23.130 --> 00:28:28.310
So D infinity of f is a proper subset of D
. What we want to show is that we want to
00:28:28.310 --> 00:28:31.150
show f is analytic except for poles.
00:28:31.150 --> 00:28:37.340
So you want to show that this D infinity of
f which is a closed set, D s are certainly
00:28:37.340 --> 00:28:40.690
points where f takes the value infinity, okay.
00:28:40.690 --> 00:28:45.260
So you want to show that they are all poles
and in particular you want to show that this
00:28:45.260 --> 00:28:49.850
D infinity of f, mind you, it is isolated.
00:28:49.850 --> 00:28:57.860
That is the, see that is the important point,
it could be connected, it could be a curve,
00:28:57.860 --> 00:29:01.690
after all it is a subset of a domain in the
complex plane.
00:29:01.690 --> 00:29:05.780
A subset of, a domain is an open set and an
open set can contain curves.
00:29:05.780 --> 00:29:13.050
It can even contain a sequence of points which
has a limit, it can contain so many things.
00:29:13.050 --> 00:29:20.060
So it is not even clear that D infinity is
isolated, is an isolated set of points, okay.
00:29:20.060 --> 00:29:25.320
So that we have to prove, he that is the important
part, okay, that you to observe.
00:29:25.320 --> 00:29:33.830
So of course you know the, the way you handle
it is by looking at 1 by f, okay.
00:29:33.830 --> 00:29:42.090
Because 1 by f is now there for you to, to,
to play with because 1 by f is already continuous,
00:29:42.090 --> 00:29:43.840
you see that is the point.
00:29:43.840 --> 00:29:47.860
So you have it here, 1 by f is here, it is
already continuous.
00:29:47.860 --> 00:29:51.690
All right, new and used it, all right.
00:29:51.690 --> 00:29:57.450
And 1 by fn converges normally to 1 by f,
so you can use that.
00:29:57.450 --> 00:29:58.980
So let me say the following thing.
00:29:58.980 --> 00:30:04.850
Let us take, let us take this D minus D infinity.
00:30:04.850 --> 00:30:11.640
If you take this D minus D infinity, this
is would take f of D minus D infinity, this
00:30:11.640 --> 00:30:16.980
will go into the complex plane, okay.
00:30:16.980 --> 00:30:21.180
Because you have thrown out D infinity which
is a set of points where f takes the value
00:30:21.180 --> 00:30:25.430
infinity, so at the other point which is in
the point of D minus D infinity, f will take
00:30:25.430 --> 00:30:26.430
value other than infinity.
00:30:26.430 --> 00:30:30.810
So the only values other than infinity in
the external complex plane complex values.
00:30:30.810 --> 00:30:40.241
So that means, and mind your D minus D infinity
is open, D minus D infinity is open because
00:30:40.241 --> 00:30:43.100
D infinity is closed, all right.
00:30:43.100 --> 00:30:49.220
And now you have the function f, you have
the function f which is defined on this open
00:30:49.220 --> 00:30:55.760
set D minus D infinity and it is taking complex
values and what is this f, this f is again
00:30:55.760 --> 00:31:03.010
a normal limit of fn. f is a normal limit
of fn or all of D, so it is also a normal
00:31:03.010 --> 00:31:08.770
limit of fn on a subset of D, okay.
00:31:08.770 --> 00:31:15.820
If fn tends to f normally on D itself, so
fn will tend to f normally or any subset of
00:31:15.820 --> 00:31:21.570
D. So since D minus D infinity is a subset
of D, fn will tend to f normally on D minus
00:31:21.570 --> 00:31:22.570
D infinity.
00:31:22.570 --> 00:31:28.240
And what is this f on D minus D infinity,
it is a complex valued function, all right.
00:31:28.240 --> 00:31:30.690
Just let me think for a moment.
00:31:30.690 --> 00:31:37.990
I am trying to look at, I am trying to locate
the point where f is a common has values,
00:31:37.990 --> 00:31:38.990
takes the value infinity.
00:31:38.990 --> 00:31:46.070
So I am trying to look at a point z0 in D
infinity, so I want to say it is a pole.
00:31:46.070 --> 00:31:51.900
So I want to say is z0 belongs to D infinity,
I want to say f has a pole z0.
00:31:51.900 --> 00:31:57.580
But the way to verify that f has a pole at
z0 is to, is equivalent to verifying that
00:31:57.580 --> 00:32:00.090
1 by f has a 0 at z0.
00:32:00.090 --> 00:32:05.270
So I will have to show that 1 by f has a 0,
it will get 0 at z0, okay.
00:32:05.270 --> 00:32:10.980
But then what I will have to show is that
1 by f is analytic in the neighbourhood of
00:32:10.980 --> 00:32:16.200
z0, in the Delta neighbourhood of z0, I will
have to show that 1st.
00:32:16.200 --> 00:32:24.440
And I must make sure, and then I will get
that z0 is an isolated 0 of 1 by f.
00:32:24.440 --> 00:32:32.610
1 by f being analytic in the neighbourhood
of z0 will tell me that f is analytic in the
00:32:32.610 --> 00:32:37.670
Delta neighbourhood of z0 and z0 is a pole.
00:32:37.670 --> 00:32:44.510
And if, and now, this will tell me that f
is meromorphic, this is what I will have to
00:32:44.510 --> 00:32:45.510
do.
00:32:45.510 --> 00:32:52.760
So how do I show that, see the ideas of the
proof are similar to the ideas that we used
00:32:52.760 --> 00:32:54.720
in the earlier lectures, okay.
00:32:54.720 --> 00:32:58.120
So let me do the following thing.
00:32:58.120 --> 00:33:07.180
So D infinity of f is actually D0 of 1 by
f, okay.
00:33:07.180 --> 00:33:17.970
So D0 of 1 by f is the set of all z belonging
to D, such that 1 by f of z is 0, okay.
00:33:17.970 --> 00:33:23.020
Mind you 1 by f makes sense, it is a continuous
map, okay.
00:33:23.020 --> 00:33:27.960
And therefore 1 by f of z equal to 0 will
be the locus of points where 1 by f takes
00:33:27.960 --> 00:33:31.880
the value, it is inverse image of 0 under
1 by f, which is continuous.
00:33:31.880 --> 00:33:34.480
So it is a, mind you it is a closed set, okay.
00:33:34.480 --> 00:33:39.680
So the singlepoint 0 is a closed set in C
union infinity, any singlepoint is a closed
00:33:39.680 --> 00:33:40.680
set.
00:33:40.680 --> 00:33:47.821
So this is, so if you want let me write this,
this is just 1 by f inverse of 0, okay, so
00:33:47.821 --> 00:33:51.270
take a point z0 in D infinity of f.
00:33:51.270 --> 00:34:01.500
Take a point z0 in D infinity of f, then what
happens, this is the point where f takes the
00:34:01.500 --> 00:34:06.110
value infinity, so 1 by f takes the value
0, okay.
00:34:06.110 --> 00:34:12.129
Since 1 by f take the value 0 and 1 by f discontinuous,
okay, there is a small disc surrounding z0,
00:34:12.129 --> 00:34:20.260
in fact even a closed disc surrounding z0,
where 1 by f is bounded, okay, because
00:34:20.260 --> 00:34:21.260
it is just continuity.
00:34:21.260 --> 00:34:30.000
C1 by f is a map from, it is a continuous
map from D to C union infinity, okay.
00:34:30.000 --> 00:34:37.629
And if you use continuous, so in particular
it means a discontinuous at z0 also, okay.
00:34:37.629 --> 00:34:43.599
That means that and 1 by f of z0 is 0, okay.
00:34:43.599 --> 00:34:46.520
So what it means, what is continuous, what
will continuity tell you, it will tell you
00:34:46.520 --> 00:34:52.909
that given an Epsilon greater than 0, there
exists a Delta greater than 0 such that if
00:34:52.909 --> 00:35:00.240
you make the distance between, so
I can use the usual distance because now I
00:35:00.240 --> 00:35:07.410
am going to, because I am on D, if you make
the distance between z and z0 less than Delta,
00:35:07.410 --> 00:35:08.600
okay.
00:35:08.600 --> 00:35:15.240
Then the distance, the spherical metric between
f of 1 by f of z and 1 by f of z0, by the
00:35:15.240 --> 00:35:20.290
way which is 0, this quantity is 0, that can
be made less than Epsilon.
00:35:20.290 --> 00:35:24.050
This is just continuous, definition of continuity
of 1 by f at z0.
00:35:24.050 --> 00:35:33.080
Given an Epsilon, okay, I can make 1 by fz
to be as close as I want to 0 to within a
00:35:33.080 --> 00:35:34.550
maximum distance of 0.
00:35:34.550 --> 00:35:41.110
If I choose z sufficiently closed z0 and how
sufficiently close, that is what is decided
00:35:41.110 --> 00:35:43.860
by the Delta, that is done, okay.
00:35:43.860 --> 00:35:49.610
And you know I can, I can, I can even put
less than or equal to, that is because you
00:35:49.610 --> 00:35:55.850
know I can choose a smaller disc, such that,
smaller disc centred at z0, radius is Delta,
00:35:55.850 --> 00:36:01.790
such that the closed disc including the circle,
the boundary circle, that also like inside
00:36:01.790 --> 00:36:05.220
D, because D is after all an open set.
00:36:05.220 --> 00:36:13.330
I can always choose this disc sufficiently
small in D, okay, because D is an open set
00:36:13.330 --> 00:36:20.190
and then I make sure that even the boundary
of the disc is inside D, I do that, right.
00:36:20.190 --> 00:36:28.610
And you see, what does, what does this tell
you, this tells you that you see, the distance
00:36:28.610 --> 00:36:40.580
between 1 by f and 0, 1 by f is very close
to 0 and you also have this fact that 1 by
00:36:40.580 --> 00:36:48.800
fns, they converge to 1 by f, mind you, normally.
00:36:48.800 --> 00:36:56.761
So the 1st thing I want you to understand
is that, since 1 by fn converges to 1 by f
00:36:56.761 --> 00:37:04.030
normally, 1 by fn will converge to 1 by f
uniformly on this closed disc, that is because
00:37:04.030 --> 00:37:05.490
this closed disc is compact.
00:37:05.490 --> 00:37:13.520
And because of that all the 1 by fns beyond
a certain stage, they will also take values
00:37:13.520 --> 00:37:16.830
in the neighbourhood of 0, okay.
00:37:16.830 --> 00:37:24.850
And that means that in a neighbourhood of
0 all the 1 by, in the neighbourhood of z0,
00:37:24.850 --> 00:37:32.530
okay, all the 1 by fns beyond a certain stage,
they are all going to be bounded.
00:37:32.530 --> 00:37:36.700
And that means that they are all going to
be analytic, because mind you about the more
00:37:36.700 --> 00:37:41.080
in the lab and fns were all meromorphic.
00:37:41.080 --> 00:37:46.750
So 1 by fns are also meromorphic but you know
a meromorphic function, if it is bounded at
00:37:46.750 --> 00:37:50.660
a point, then that has to be a good point.
00:37:50.660 --> 00:38:00.470
If a meromorphic function is bounded in the
neighbourhood of a point, okay, then you know,
00:38:00.470 --> 00:38:08.710
it cannot, it can assume, that point has to
be a good point by the Riemann’s removable
00:38:08.710 --> 00:38:10.390
singularity’s theorem.
00:38:10.390 --> 00:38:16.510
What can happen in the neighbourhood of that
point, I have some poles because it is a meromorphic
00:38:16.510 --> 00:38:19.450
function but it cannot have a pole because
at the pole it will take the value infinity.
00:38:19.450 --> 00:38:25.170
I am putting the restriction that it is taking
value close to 0, so it means that in the
00:38:25.170 --> 00:38:30.840
neighbourhood of z0, all these fns, 1 by fns,
they are all going to be analytic.
00:38:30.840 --> 00:38:36.510
And since all the 1 by fns are analytic and
they converge uniformly to 1 by f in this
00:38:36.510 --> 00:38:40.990
disc centred at z0, 1 by f becomes analytic
at z 0.
00:38:40.990 --> 00:38:47.090
And once 1 by f becomes analytic at z0, z0
becomes a 0 of 1 by f which is now in analytic
00:38:47.090 --> 00:38:48.710
function, therefore it is isolated.
00:38:48.710 --> 00:38:58.530
So z0 becomes an isolated 0 of 1 by f and
that is, that will tell you that z0 will be
00:38:58.530 --> 00:39:01.150
an isolated pole for f.
00:39:01.150 --> 00:39:07.350
So what this argument tells you is that every
z0 where f takes the value infinity is actually
00:39:07.350 --> 00:39:10.550
a pole of f and therefore f is meromorphic.
00:39:10.550 --> 00:39:13.220
And that, that ends the proof, okay.
00:39:13.220 --> 00:39:15.050
So let me write that down.
00:39:15.050 --> 00:39:17.560
So let me write these things in words.
00:39:17.560 --> 00:39:45.380
note that 1 by f is bounded near
0 in more z minus z0 n equal to Delta, also
00:39:45.380 --> 00:39:56.260
1 by fn converges to 1 by f uniformly, so
I am abbreviating it as ufly on mod of z minus
00:39:56.260 --> 00:40:01.410
z0 less than equal to Delta because of normal
conversation because mod of z minus z0 less
00:40:01.410 --> 00:40:03.890
than equal to Delta is compact.
00:40:03.890 --> 00:40:21.110
So for n sufficiently large, the 1 by fns
are bounded near 0 in mod of z minus z0 lead
00:40:21.110 --> 00:40:27.590
than or equal to Delta, okay.
00:40:27.590 --> 00:40:55.000
And, and hence, if you want Riemann’s removable
similarities theorem, z0 is 0 of 1 by f and
00:40:55.000 --> 00:41:01.380
of course there I am saying that 1 by f is,
1 by f is analytic because 1 by fns are all
00:41:01.380 --> 00:41:07.130
analytic, beyond a certain stage all the 1
by fns are analytic, 1 by fns tend to 1 by
00:41:07.130 --> 00:41:11.270
f and it is uniform convergence, therefore
1 by f is analytic.
00:41:11.270 --> 00:41:14.630
So 1 by f is analytic and it has a 0 at z0.
00:41:14.630 --> 00:41:17.570
So it is an isolated zero, okay.
00:41:17.570 --> 00:41:36.960
note that 1 by fn, n sufficiently large, are
analytic, so 1 by f is analytic being
00:41:36.960 --> 00:41:49.230
a normal limit, being a uniform limit, so
uf is abbreviation for uniform.
00:41:49.230 --> 00:41:54.970
So all this happens in mod z minus z0 less
than equal to Delta, you may need to make
00:41:54.970 --> 00:41:58.240
Delta smaller if you want but that is not
a problem.
00:41:58.240 --> 00:42:16.380
The point is that, z0 is thus a zero of the
analytic function 1 by f in model of z minus
00:42:16.380 --> 00:42:27.670
z0 in mod of z minus z0 less than equal to
Delta, so it is isolated.
00:42:27.670 --> 00:42:46.720
Thus f of z has a pole at z0, so f is actually
meromorphic function and that is the end of
00:42:46.720 --> 00:42:47.720
the proof.
00:42:47.720 --> 00:42:53.330
So it is a very nice fact that if you take
a normal limit of meromorphic functions, barring
00:42:53.330 --> 00:43:01.170
the extreme case that the normal limit is
identically infinity, the limit is again meromorphic
00:43:01.170 --> 00:43:04.130
function, okay.
00:43:04.130 --> 00:43:07.250
So that ends the proof of this fact, okay.
00:43:07.250 --> 00:43:12.040
And what we saw last class was that if you
put the additional condition that fns are
00:43:12.040 --> 00:43:17.690
all holomorphic and you assume that the limit
function is not identically infinity, then
00:43:17.690 --> 00:43:19.570
the limit function is also holomorphic.
00:43:19.570 --> 00:43:24.540
And why is that too, now you can see, that
if you take the fns additionally to be holomorphic,
00:43:24.540 --> 00:43:28.220
you know and if it is not, and the limit is
not identically infinity, you know, by whatever
00:43:28.220 --> 00:43:31.940
you proof, now you know that the limit is
meromorphic, okay.
00:43:31.940 --> 00:43:36.150
But if it is really meromorphic at a certain
point, namely if it has a full letter certain
00:43:36.150 --> 00:43:41.060
point, then what will happen is that it is
reciprocal will have a 0.
00:43:41.060 --> 00:43:46.560
And Hurwitz theorem will say that if 1 by
f has a 0, then 1 by fns will start having
00:43:46.560 --> 00:43:49.290
0 as n becomes large.
00:43:49.290 --> 00:43:54.490
But the moment 1 by fns start having zeros,
fns will start having poles and that is not
00:43:54.490 --> 00:43:56.570
possible if you are assuming fns to be holomorphic.
00:43:56.570 --> 00:44:02.140
Therefore you see that all the, if all the
fns are holomorphic, then f also has to be
00:44:02.140 --> 00:44:05.590
holomorphic, or it has to be identically infinity,
okay.
00:44:05.590 --> 00:44:06.119
So I will stop here.