WEBVTT
Kind: captions
Language: en
00:00:51.100 --> 00:00:59.140
So you see we are looking at compactness of
families of functions, okay we want to look
00:00:59.140 --> 00:01:06.740
at Meromorphic functions alright and so you
know what we did last time was Montelâ€™s
00:01:06.740 --> 00:01:15.320
Theorem, okay which was for analytic functions,
alright. So let me recall this so that I will
00:01:15.320 --> 00:01:24.490
give you the background for the formulation
of the version of Montel's Theorem for Meromorphic
00:01:24.490 --> 00:01:32.750
functions which goes by the name of Marty's
Theorem, okay and once you have that then
00:01:32.750 --> 00:01:41.020
we can go ahead and try to we get closer to
the proof of the Picard Theorems, alright.
00:01:41.020 --> 00:01:55.080
So you see so you can recall so maybe I will
put the tittle as Marty's Theorem so let me
00:01:55.080 --> 00:02:20.900
put here version of Montel's Theorem for Meromorphic
functions, okay. So let us recall see so the
00:02:20.900 --> 00:02:29.450
idea is the following so you have D inside
the complex plane its domain it is an open
00:02:29.450 --> 00:02:41.019
connected set non-empty ofcourse, okay and
you have script F is family of analytic functions
00:02:41.019 --> 00:02:54.400
on D so this is this is subset of holomorphic
functions on D, okay and mind you these are
00:02:54.400 --> 00:03:00.820
ofcourse certainly continuous functions on
D with complex values, alright.
00:03:00.820 --> 00:03:08.959
So you have this family of holomorphic functions
on D and then the whole idea is you want to
00:03:08.959 --> 00:03:14.450
worry about the compactness of these familiar
functions, okay and compactness in what sense
00:03:14.450 --> 00:03:24.229
compactness can be you know in general compactness
is the same as sequential compactness, okay
00:03:24.229 --> 00:03:33.880
as a general philosophy and so and you should
expect therefore you want conditions for which
00:03:33.880 --> 00:03:41.299
given a sequence of functions in F in the
family script F you want to pick out a convergent
00:03:41.299 --> 00:03:43.450
subsequence, okay.
00:03:43.450 --> 00:03:57.479
Now the point is that you know ofcourse since
you are working with holomorphic functions,
00:03:57.479 --> 00:04:05.610
okay just convergence is not a very useful
thing you would like to have a kind of convergence
00:04:05.610 --> 00:04:12.430
that will reserve properties of original functions
which are converging to the limit function
00:04:12.430 --> 00:04:18.100
and therefore the best kind of convergence
you can expect is uniform convergence but
00:04:18.100 --> 00:04:21.400
ofcourse uniform convergence is too much to
expect what you will normally get is only
00:04:21.400 --> 00:04:25.250
normal convergence, okay that is you will
get uniform convergence restricted only to
00:04:25.250 --> 00:04:27.070
compact subsets, alright.
00:04:27.070 --> 00:04:36.780
So that is the background and so what is it
that you have? See on the one hand you want
00:04:36.780 --> 00:04:52.030
F is normally sequentially compact and what
do I mean by this? I mean that given any sequence
00:04:52.030 --> 00:05:01.400
in F, I can find subsequence which converges
normally that means which converges uniformly
00:05:01.400 --> 00:05:18.100
on compact subsets, okay so let me write that
any sequence in F admits a subsequence that
00:05:18.100 --> 00:05:31.960
converges
uniformly on compact subsets on compact subsets
00:05:31.960 --> 00:05:46.850
of D, okay so this is this is compactness
for us, okay that and that given any sequence
00:05:46.850 --> 00:05:50.720
you can extract a convergent subsequence but
the convergence is only with respect to the
00:05:50.720 --> 00:05:55.130
normal convergence, okay so this is what you
want.
00:05:55.130 --> 00:06:03.260
Now how do you what what are the necessary
and sufficient conditions for this to happen
00:06:03.260 --> 00:06:07.250
what are the necessary and sufficient conditions
for this to happen? So you know if you want
00:06:07.250 --> 00:06:17.830
to go from here to here if you want to go
from here to here alright so you will get
00:06:17.830 --> 00:06:38.380
immediately that F is normally uniformly bounded,
okay so that means that you given this family
00:06:38.380 --> 00:06:44.910
of functions in scripts F and if you restrict
it to any compact set then that family will
00:06:44.910 --> 00:06:46.300
have uniform bound.
00:06:46.300 --> 00:06:51.590
So there is some positive number such that
the modulus of the functions is bounded by
00:06:51.590 --> 00:07:01.340
that number and this will work for all functions
and for all points on that compact set, okay.
00:07:01.340 --> 00:07:04.520
So this is something that you will get and
the other thing you will get is you will also
00:07:04.520 --> 00:07:14.930
get that F is the family is equicontinuous,
okay so plus so let me write this F is so
00:07:14.930 --> 00:07:29.370
for some reason let me write it at a distance
so I will write here F is equicontinuous.
00:07:29.370 --> 00:07:40.240
So how does this come? These two implications
come because of the they basically come because
00:07:40.240 --> 00:07:45.680
of the Arzela-Ascoli Theorem, okay so this
is so let me write this let me put AA here
00:07:45.680 --> 00:07:57.050
where you know I will put AA here also where
let me write here as a legend AA stands for
00:07:57.050 --> 00:08:03.370
Arzela-Ascoli Theorem you know this is just
the Arzela-Ascoli Theorem. See what Arzela-Ascoli
00:08:03.370 --> 00:08:07.560
Theorem says is that you know if you are looking
at continuous complex or real valued functions
00:08:07.560 --> 00:08:11.510
on a compact metric space, okay.
00:08:11.510 --> 00:08:21.130
Then the condition that such a family of functions
is compact, okay is equivalent to that family
00:08:21.130 --> 00:08:30.150
of functions being uniformly bounded and equicontinuous
that is Arzela-Ascoli Theorem. So if you now
00:08:30.150 --> 00:08:34.950
look at it if you now look at this thing that
I have assumed on the this property on the
00:08:34.950 --> 00:08:45.180
left side which says that this family is sequentially
compact if I restrict to any compact subset,
00:08:45.180 --> 00:08:52.430
okay. So if I take capital K a compact subset
of D and I restrict this family to that, okay
00:08:52.430 --> 00:08:58.320
then I am looking at a family of continuous
functions on a compact metric space, okay
00:08:58.320 --> 00:09:03.329
any compact subset of D is also a metric space
it is a compact metric space metric is just
00:09:03.329 --> 00:09:07.240
metric on D restricted to that subset, okay.
00:09:07.240 --> 00:09:14.550
And then by the Arzela-Ascoli Theorem that
the family script F will become actually compact
00:09:14.550 --> 00:09:25.839
you know with respect to the topology given
by the supremum norm, okay the supremum norm
00:09:25.839 --> 00:09:31.839
is defined and you have the metric induced
by that norm and the topology induced by the
00:09:31.839 --> 00:09:37.190
metric so with respect to that this family
script F actually becomes a compact family,
00:09:37.190 --> 00:09:43.550
okay it becomes a compact subset of points,
okay and you are now considering this family
00:09:43.550 --> 00:09:53.850
inside C K, C namely the set of all continuous
functions on the compact set K with values
00:09:53.850 --> 00:09:59.209
in C, okay because analytic functions are
ofcourse continuous, okay.
00:09:59.209 --> 00:10:05.870
So then by the Arzela-Ascoli Theorem what
will happen is that on that compact set K
00:10:05.870 --> 00:10:13.040
what will happen is that this family will
be uniformly bounded, okay that is bounded
00:10:13.040 --> 00:10:18.670
with respect to the sup norm which is uniform
boundedness and you will get equicontinuity.
00:10:18.670 --> 00:10:24.800
So and since this and equicontinuity is something
that is needs to be checked at every point
00:10:24.800 --> 00:10:31.100
so you will get equicontinuity for the family
the whole family throughout all of D, okay
00:10:31.100 --> 00:10:36.350
to check equicontinuity at all of D at every
point of D I just have to check equicontinuity
00:10:36.350 --> 00:10:41.139
at each point of D and to check at each point
of D it is enough to check on each compact
00:10:41.139 --> 00:10:45.470
subset of D even a point is a compact subset
if you want, okay.
00:10:45.470 --> 00:10:51.980
So equicontinuity will fall out and restricted
to that compact subset I also have uniform
00:10:51.980 --> 00:10:56.879
boundedness, okay therefore restricted to
compact subsets I have uniform boundedness
00:10:56.879 --> 00:11:01.760
and that is normal uniform boundedness so
that is how I get these two implications,
00:11:01.760 --> 00:11:09.389
okay but the serious thing is to go so that
the serious thing is to go the other way round,
00:11:09.389 --> 00:11:19.972
okay so starting with starting with you know
the Arzela-Ascoli Theorem in another direction
00:11:19.972 --> 00:11:25.949
tells you that is you are on a compact metric
space and you are looking at continuous functions
00:11:25.949 --> 00:11:31.819
complex valued or real valued, if you know
these collection of functions is uniformly
00:11:31.819 --> 00:11:38.050
bounded and it is equicontinuous then your
family is compact, okay.
00:11:38.050 --> 00:11:46.749
Now so if I start go from this direction suppose
I assume F is normally uniformly bounded script
00:11:46.749 --> 00:11:52.740
F is normally uniformly bounded namely this
condition so I will purposely change colour
00:11:52.740 --> 00:12:03.179
because I want to emphasize something else.
So you know I take this I take this condition
00:12:03.179 --> 00:12:09.459
script F is normally uniformly bounded alright
then the beautiful thing is we do not have
00:12:09.459 --> 00:12:13.819
to add equicontinuity that is the big deal,
the big deal is you can go from here to here
00:12:13.819 --> 00:12:23.449
directly as a theorem and this is the this
is Montel's Theorem so this is Montel's Theorem
00:12:23.449 --> 00:12:32.269
that is you start with a normal uniformly
bounded family of analytic functions, okay
00:12:32.269 --> 00:12:40.920
then it is normally sequentially compact that
means you can given given a sequence you can
00:12:40.920 --> 00:12:44.480
extract a subsequence which converges normally,
alright.
00:12:44.480 --> 00:12:51.809
And mind you so I have I have to tell you
few things here this is this Montel Theorem
00:12:51.809 --> 00:13:00.480
is stronger than Arzela-Ascoli, okay in the
following sense. See what Arzela-Ascoli Theorem
00:13:00.480 --> 00:13:06.970
will say is that you know if you restrict
yourself to a compact subset of D if you take
00:13:06.970 --> 00:13:12.749
a particular compact subset of D, okay and
suppose I have this condition that this family
00:13:12.749 --> 00:13:17.480
script F is normally uniformly bounded then
family script F will become uniformly bounded
00:13:17.480 --> 00:13:22.670
on that compact subset and again I will get
equicontinuity and I will apply the Arzela-Ascoli
00:13:22.670 --> 00:13:28.209
Theorem and I will be able to pick a subsequence
which converges uniformly on that compact
00:13:28.209 --> 00:13:30.139
subset, okay.
00:13:30.139 --> 00:13:41.329
And the ofcourse equicontinuity comes because
of the derivatives being bounded which comes
00:13:41.329 --> 00:13:47.499
as a result of the Cauchy integral formulas
and some estimations, okay and the Cauchy
00:13:47.499 --> 00:13:52.040
estimates of the first derivative, okay but
let us forget that for the moment basically
00:13:52.040 --> 00:13:58.660
what happens is you are able to for every
compact subset if I start with the sequence
00:13:58.660 --> 00:14:03.850
in the family F for every compact subset I
am able to get a convergent subsequence I
00:14:03.850 --> 00:14:10.139
am able to pick a subsequence such that on
that compact subset the convergence is uniform.
00:14:10.139 --> 00:14:16.399
But the point is if I change the compact subset
the subsequence can change, okay if I apply
00:14:16.399 --> 00:14:21.809
only the Arzela-Ascoli Theorem if I change
the compact subset then the subsequence can
00:14:21.809 --> 00:14:26.369
change but the Montel Theorem is very strong
what it says is that I can uniformly find
00:14:26.369 --> 00:14:31.779
a single subsequence of the original sequence
which will converge uniformly on every compact
00:14:31.779 --> 00:14:37.879
subset it will work for every compact subset,
okay that is the power in the Montel Theorem
00:14:37.879 --> 00:14:46.220
and if you remember this we got this by diagonalization
argument, okay we covered the domain D by
00:14:46.220 --> 00:14:55.329
an increasing sequence of compact sets, okay
which fill out the domain and on each member
00:14:55.329 --> 00:15:00.449
of this sequence of compact sets we picked
out a convergent subsequence using Arzela-Ascoli
00:15:00.449 --> 00:15:09.809
Theorem, okay and then we wrote down this
metric of convergent subsequences by you know
00:15:09.809 --> 00:15:16.749
for the first compact set in the sequence
we picked out a subsequence in the sequence
00:15:16.749 --> 00:15:23.509
of sets covering the space D we picked out
one sequence from the original sequence we
00:15:23.509 --> 00:15:25.720
from Arzela-Ascoli, okay.
00:15:25.720 --> 00:15:31.370
Then from this sequence we picked out another
subsequence which will work on the next bigger
00:15:31.370 --> 00:15:38.279
compact subsets and then we went on like this
and all these compact subsets eventually filled
00:15:38.279 --> 00:15:44.569
their union filled the whole of D, okay and
the diagonal sequence gave us a sequence of
00:15:44.569 --> 00:15:50.470
the original sequence which will converge
uniformly on every compact subset of D because
00:15:50.470 --> 00:15:56.569
every compact subset of D is contained in
one of the members of this sequence of compact
00:15:56.569 --> 00:16:01.449
sets that we increasing sequence of compact
sets that we constructed to cover D, okay.
00:16:01.449 --> 00:16:07.470
So you see we have got this very strong statement
from Montel's Theorem, okay so that is one
00:16:07.470 --> 00:16:11.809
point you have to remember, okay and the other
important point about Montel's Theorem is
00:16:11.809 --> 00:16:17.740
that you do not worry about you really do
not worry about equicontinuity, okay and this
00:16:17.740 --> 00:16:21.369
is basically because you are working with
analytic functions. So what is happening is
00:16:21.369 --> 00:16:31.850
that this condition that the family is normally
uniformly bounded tells you that if you take
00:16:31.850 --> 00:16:38.739
the family of derivatives of these functions
then that family of derivatives is also uniform
00:16:38.739 --> 00:16:41.529
normally uniformly bounded, okay.
00:16:41.529 --> 00:16:54.649
And so let me write this so I will write this
as F prime is normally uniformly bounded
00:16:54.649 --> 00:17:05.329
so there is this thing here in between, okay
so when I write ofcourse when I write F prime
00:17:05.329 --> 00:17:11.579
I mean the set of all I mean all those derivatives
of functions f which are in small f which
00:17:11.579 --> 00:17:21.240
are in script F, okay so so script F prime
is just the derivatives of the functions in
00:17:21.240 --> 00:17:27.179
script F, okay and mind you the functions
in script F are all analytic therefore the
00:17:27.179 --> 00:17:32.309
derivatives you know if you take a function
which is analytic then all orders of derivatives
00:17:32.309 --> 00:17:35.240
of that function exist and they are also analytic,
okay.
00:17:35.240 --> 00:17:41.620
So script F prime is also a bonafied family
of holomorphic functions on the same domain,
00:17:41.620 --> 00:17:49.679
okay and the point is that that is normally
uniformly bounded, okay and that is because
00:17:49.679 --> 00:17:55.100
that is simply because of the fact that the
derivative of a function can be expressed
00:17:55.100 --> 00:18:00.019
in terms of the function using the Cauchy
integral formula, okay and therefore if the
00:18:00.019 --> 00:18:05.169
original functions are uniformly bounded then
the derivatives are also uniformly bounded
00:18:05.169 --> 00:18:09.210
on closed on sufficiently small closed disks,
okay.
00:18:09.210 --> 00:18:15.529
So the moral of the story is that the derivatives
are normally uniformly bounded and because
00:18:15.529 --> 00:18:20.809
the derivatives are normally uniformly bounded
you know this is the philosophy that I told
00:18:20.809 --> 00:18:27.110
you last time whenever the derivatives are
bounded uniformly then the original family
00:18:27.110 --> 00:18:32.910
is equicontinuous, okay because you just have
to integrate, okay so this so there is another
00:18:32.910 --> 00:18:38.220
implication that is going like this whenever
the derivatives whenever the family of derivatives
00:18:38.220 --> 00:18:42.200
is uniformly bounded then the original family
is equicontinuous.
00:18:42.200 --> 00:18:48.720
So what happens is that because I assumed
that the family is normally uniformly bounded
00:18:48.720 --> 00:18:53.580
I am also getting equicontinuity the way I
am getting equicontinuity is because I am
00:18:53.580 --> 00:18:59.179
getting actually normal uniform boundedness
of the family of derivatives that is the whole
00:18:59.179 --> 00:19:03.210
point and the reason I am able to get this
is because of the Cauchy integral formula
00:19:03.210 --> 00:19:08.809
because of the Cauchy estimates, okay. So
this is how this is how everything works.
00:19:08.809 --> 00:19:20.840
Now what is that so what is that we want to
do with meromorphic functions okay so if you
00:19:20.840 --> 00:19:28.520
now if you are see so far we are working here
in the set of all holomorphic functions I
00:19:28.520 --> 00:19:35.360
mean analytic functions, alright but you know
you want to work with meromorphic functions
00:19:35.360 --> 00:19:40.340
the problem with that is that if you are working
with meromorphic functions then you are going
00:19:40.340 --> 00:19:43.309
to allow the value infinity, okay.
00:19:43.309 --> 00:19:51.990
So you are going to take values not if you
are going to take a meromorphic function you
00:19:51.990 --> 00:19:58.899
cannot just considered it as a function into
complex numbers because then at a pole you
00:19:58.899 --> 00:20:05.850
cannot define it. Whereas if you considered
it as a function into the extended complex
00:20:05.850 --> 00:20:10.309
plane, okay then at a pole you can define
the function value to be infinity and still
00:20:10.309 --> 00:20:13.289
keep the function continuous even at a pole,
okay.
00:20:13.289 --> 00:20:19.700
So if you are working with meromorphic functions
you want to do a same you want to have the
00:20:19.700 --> 00:20:28.539
same kind of theorem, okay then you know it
is little it is little troublesome somehow
00:20:28.539 --> 00:20:35.080
you can see that in this whole game you have
to pass through this red box that I have put
00:20:35.080 --> 00:20:41.460
here which is that the derivatives are normally
uniformly bounded, okay and that see will
00:20:41.460 --> 00:20:49.320
work for analytic functions as it is not work
for meromorphic functions because the problem
00:20:49.320 --> 00:20:55.750
will be at the poles at a pole I cannot apply
any kind of Cauchy integral formula I cannot
00:20:55.750 --> 00:21:00.309
express in fact even derivative at a pole
is not defined it is a singular point, okay
00:21:00.309 --> 00:21:02.250
so I am in trouble.
00:21:02.250 --> 00:21:08.490
And you know in order to overcome this we
had introduced this concept of spherical derivatives,
00:21:08.490 --> 00:21:17.960
okay so that is what we are going to use.
So in fact we will get this theorem that now
00:21:17.960 --> 00:21:24.730
you again take a domain in the complex plane,
you take family of (holo) not holomorphic
00:21:24.730 --> 00:21:30.750
but meromorphic functions on the domain but
mind you you are now considering this as functions
00:21:30.750 --> 00:21:35.370
not into the complex plane but functions into
the extended complex plane, okay.
00:21:35.370 --> 00:21:39.259
And when you consider it as functions into
the extended complex plane mind you the target
00:21:39.259 --> 00:21:43.950
plane is not complex plane it is the extended
complex plane and the extended complex plane
00:21:43.950 --> 00:21:49.050
has been made into a metric space by putting
the spherical metric and with respect to the
00:21:49.050 --> 00:21:53.200
spherical metric it is a compact metric space,
it is a beautiful metric space, it is just
00:21:53.200 --> 00:21:58.769
the Riemann sphere with the spherical metric
on that, okay alright.
00:21:58.769 --> 00:22:05.130
And now what you do is you get this version
of the Montel the correct version of the Montel's
00:22:05.130 --> 00:22:09.799
Theorem for meromorphic functions will tell
you that now you again take a family of meromorphic
00:22:09.799 --> 00:22:16.630
functions the condition that it is normally
sequentially compact is equivalent to saying
00:22:16.630 --> 00:22:21.960
that the spherical derivatives are bounded
that is it, okay. So what I want you to understand
00:22:21.960 --> 00:22:31.549
is that it is rather funny when you move from
the holomorphic version of the Montel Theorem
00:22:31.549 --> 00:22:36.929
to the meromorphic version of the Montelâ€™s
Theorem which is called Marty's Theorem, okay
00:22:36.929 --> 00:22:47.899
your condition changes from the normal boundedness
of the of the family of functions to the normal
00:22:47.899 --> 00:22:53.639
boundedness of the derivatives but what derivatives
spherical derivatives that is the big change,
00:22:53.639 --> 00:22:59.270
okay and with that everything works, okay
so that is what I am going to state next.
00:22:59.270 --> 00:23:18.919
So here is Marty's Theorem. Let D in C be
a domain ofcourse non-empty as usual and script
00:23:18.919 --> 00:23:29.230
F is a family of meromorphic functions on
D, okay mind you this is a subspace of the
00:23:29.230 --> 00:23:36.890
continuous functions on D with values in the
extended complex plane, okay you have to remember
00:23:36.890 --> 00:23:43.299
this this is very very important we are considering
meromorphic when you say meromorphic function
00:23:43.299 --> 00:23:48.059
you are allowing the value infinity, okay
otherwise you will not continuity at the pole
00:23:48.059 --> 00:23:53.559
at poles, okay that is very very important,
okay.
00:23:53.559 --> 00:24:16.130
Then F is sequentially is normally sequentially
compact compact i.e, every sequence in script
00:24:16.130 --> 00:24:39.940
F in script F admits a subsequence that converges
uniformly normally that is uniformly on compact
00:24:39.940 --> 00:24:53.870
subsets
of D if and only if
00:24:53.870 --> 00:25:01.970
so I will write F hash, what is F hash? This
is the collection of spherical derivatives
00:25:01.970 --> 00:25:06.990
of the functions in F so we use prime for
derivative when it is an analytic function
00:25:06.990 --> 00:25:13.309
when it is not an analytic function but it
is a meromorphic function we use hash, okay
00:25:13.309 --> 00:25:18.399
which is which is the notation we introduced
earlier so this is set of f hash such that
00:25:18.399 --> 00:25:42.629
f belongs to script F is normally uniformly
bounded that is uniformly bounded on on compact
00:25:42.629 --> 00:25:58.760
subsets, okay so this is Marty's Theorem so
this is meromorphic version of Montel's Theorem
00:25:58.760 --> 00:25:59.830
meromorphic version of Montel's Theorem.
00:25:59.830 --> 00:26:17.580
So let me write that here
00:26:17.580 --> 00:26:29.049
and the big deal in this statement is essentially
to say that instead of requiring that the
00:26:29.049 --> 00:26:34.750
original family of functions is normally uniformly
bounded which is what the original Montel
00:26:34.750 --> 00:26:43.809
Theorem want you know needed you shift to
the spherical derivatives of these functions
00:26:43.809 --> 00:26:46.159
that is the difference, okay.
00:26:46.159 --> 00:26:56.750
Now you see what does this say in retrospect
I mean what it says in retrospect is that
00:26:56.750 --> 00:27:03.120
in principle it says that if you take a family
of even if you take a family of analytic functions,
00:27:03.120 --> 00:27:11.299
okay even if you take a family of analytic
functions the condition that the the usual
00:27:11.299 --> 00:27:22.230
derivatives are normally uniformly bounded
is also equivalent to the normal sequential
00:27:22.230 --> 00:27:27.080
compactness of the family that is the big
deal, the big deal is you know if you go back
00:27:27.080 --> 00:27:36.860
to this diagram, okay we had this we had this
red box here which said that the derivatives
00:27:36.860 --> 00:27:42.470
the usual derivatives which in this case are
they make sense because functions are analytic
00:27:42.470 --> 00:27:45.890
the usual derivatives are normally uniformly
bounded they are uniformly bounded on compact
00:27:45.890 --> 00:27:47.120
subsets, okay.
00:27:47.120 --> 00:27:58.710
Now this itself this itself is good enough
to give you normal sequential compactness,
00:27:58.710 --> 00:28:07.340
okay but there is only there is only one small
issue since the compactness is I mean since
00:28:07.340 --> 00:28:15.450
the convergence is with respect to functions
which can take the value infinity the convergence
00:28:15.450 --> 00:28:19.470
point wise convergence is with respect to
the spherical metric that is the difference,
00:28:19.470 --> 00:28:26.049
okay. So what it means is it means the following
suppose I have a family of analytic functions
00:28:26.049 --> 00:28:33.110
on a domain how do I decide that this family
is compact, okay that is it is normally sequentially
00:28:33.110 --> 00:28:40.470
compact one direct way is use the usual Montel
Theorem for which I need all the functions
00:28:40.470 --> 00:28:44.549
in the family to be normally uniformly bounded
to be I must be able to find uniform bound
00:28:44.549 --> 00:28:47.350
for this family on every compact subset.
00:28:47.350 --> 00:28:57.760
There is another way the other way is verify
that the derivatives the family of derivatives
00:28:57.760 --> 00:29:04.740
of these functions that is normally uniformly
bounded if you verify that okay then what
00:29:04.740 --> 00:29:14.961
happens because of this meromorphic version
of Montel's Theorem that family of see if
00:29:14.961 --> 00:29:24.139
the usual derivatives are uniformly bounded
on a set then the spherical derivatives are
00:29:24.139 --> 00:29:30.249
also be uniformly bounded on the set, okay
that is because of the way in which the spherical
00:29:30.249 --> 00:29:31.269
derivatives are defined.
00:29:31.269 --> 00:29:40.880
See how is the spherical derivative defined
see it is defined like this for for f meromorphic
00:29:40.880 --> 00:29:50.960
function, recall that the spherical derivative
of f at z you know it is defined as 2 times
00:29:50.960 --> 00:30:00.870
mod f dash of z divided by 1 plus mod f z
the whole square this is the definition of
00:30:00.870 --> 00:30:04.919
spherical derivative this is how spherical
derivative is defined, okay this is how we
00:30:04.919 --> 00:30:06.350
define spherical derivative.
00:30:06.350 --> 00:30:13.970
And ofcourse you know there is an issue I
have used a mod f dash in the numerator that
00:30:13.970 --> 00:30:19.100
makes sense only if f dash exist and therefore
I can write this only at point z which are
00:30:19.100 --> 00:30:25.860
not poles but at poles what happens at poles
you know we did this in an earlier lecture
00:30:25.860 --> 00:30:30.929
at poles you extend the spherical derivative
by continuity and what happens is that the
00:30:30.929 --> 00:30:36.929
spherical derivative will become 0 at a pole
of higher order, okay and at a pole of order
00:30:36.929 --> 00:30:42.499
1 namely at a simple pole the spherical derivative
is 2 divided by the modulus of the residue
00:30:42.499 --> 00:30:45.249
of the function at that pole, okay.
00:30:45.249 --> 00:30:55.399
So therefore this so in particular you know
if you look at it in a very logical kind of
00:30:55.399 --> 00:31:01.340
way even if you take the function which is
identically infinity, okay mind you that function
00:31:01.340 --> 00:31:07.470
is also there in because we are considering
functions with you know values possibly being
00:31:07.470 --> 00:31:12.880
infinity also. If you take the function which
is always infinity that function is also by
00:31:12.880 --> 00:31:18.529
definition one for which you have to define
the spherical derivative and the spherical
00:31:18.529 --> 00:31:25.100
derivative will be 0, okay so this is something
that we make as a default definition, okay.
00:31:25.100 --> 00:31:30.149
And this so I am just trying to say if you
take the function which is uniformly infinity
00:31:30.149 --> 00:31:35.440
whose infinity at every point on your domain
that function is also included and that function
00:31:35.440 --> 00:31:41.250
also spherical derivative is also included
spherical derivative is 0, the way you think
00:31:41.250 --> 00:31:48.840
about it is a that you know usually the derivative
should be 0 if the function is constant after
00:31:48.840 --> 00:31:52.889
all the function which is infinity at every
point is just the constant function infinity
00:31:52.889 --> 00:31:56.509
so you should expect the derivative to be
0 that is one way of looking at it.
00:31:56.509 --> 00:32:01.539
The other way of looking at it is what is
the spherical derivative? If you take a meromorphic
00:32:01.539 --> 00:32:10.419
function you treat it as a map into the Riemann
sphere, okay and what it does is it is the
00:32:10.419 --> 00:32:19.100
magnification factor of the length of the
image I mean it is a magnification factor
00:32:19.100 --> 00:32:24.450
that you will have to put in to calculate
the length of the image of an arch under this
00:32:24.450 --> 00:32:29.570
mapping. So suppose you have an arch on the
suppose you have a suppose you have a an arch
00:32:29.570 --> 00:32:34.909
or contour on the complex plain in your domain
where your meromorphic function is defined,
00:32:34.909 --> 00:32:42.090
okay and you take its image under this meromorphic
function it will land on the Riemann sphere
00:32:42.090 --> 00:32:45.529
where I am thinking of the extended complex
plane as the Riemann sphere.
00:32:45.529 --> 00:32:49.980
So I am going to get an arch on the Riemann
sphere, ofcourse this arch can pass through
00:32:49.980 --> 00:32:55.029
infinity, it will pass through infinity if
the original arch in the plane pass through
00:32:55.029 --> 00:33:00.140
some poles of your meromorphic function wherever
original arch in your complex plane hit a
00:33:00.140 --> 00:33:03.961
pole the image will hit on the north pole
on the Riemann sphere which corresponds to
00:33:03.961 --> 00:33:05.720
the point at infinity, okay.
00:33:05.720 --> 00:33:10.059
And if you take the image arch how do you
get the length of the image arch, what you
00:33:10.059 --> 00:33:16.740
will do is you will integrate the meromorphic
function not the meromorphic function in fact
00:33:16.740 --> 00:33:22.350
you will integrate spherical derivative along
the original arch and you will get the length
00:33:22.350 --> 00:33:26.899
of the image arch. So the length of the spherical
derivative is a magnification factor, okay
00:33:26.899 --> 00:33:32.009
and if this if your original function is just
the function which is constant function infinity
00:33:32.009 --> 00:33:36.550
then it is going to map your whole domain
onto a point, okay if you take the function
00:33:36.550 --> 00:33:41.080
which is constant function infinity then your
whole domain is going to be collapsed to the
00:33:41.080 --> 00:33:42.919
point which corresponds to the north pole.
00:33:42.919 --> 00:33:47.610
So any arch is going to be collapsed to a
single point, okay so what is the magnification
00:33:47.610 --> 00:33:51.990
factor 0 and that should be the spherical
derivative. So this is another way of saying
00:33:51.990 --> 00:33:56.230
that you know if you take if you take the
constant function infinity you must think
00:33:56.230 --> 00:34:00.050
of the spherical derivative of that to be
0, okay that is another point that you will
00:34:00.050 --> 00:34:06.200
have to remember in mind. So but in any case
the spherical derivative as it is is always
00:34:06.200 --> 00:34:10.600
a continuous function and that is the reason
where we are able to integrate integrate it
00:34:10.600 --> 00:34:17.389
always even if your path of integration passes
through some poles of f that is very very
00:34:17.389 --> 00:34:19.429
serious, okay.
00:34:19.429 --> 00:34:25.869
But anyway what you see from here is that
because that is this mod f dash term here,
00:34:25.869 --> 00:34:32.480
okay what it will tell you is that if you
are looking at a family of analytic functions
00:34:32.480 --> 00:34:38.300
whose derivatives are normally uniformly bounded,
okay then these numerators are normally uniformly
00:34:38.300 --> 00:34:45.909
bounded, okay but then you know I can forget
the factor 1 plus mod f z squared denominator,
00:34:45.909 --> 00:34:52.560
okay because that is a factor greater than
or equal to 1 and its reciprocal is less than
00:34:52.560 --> 00:34:53.560
or equal to 1.
00:34:53.560 --> 00:35:03.880
So actually I can write f hash of z is actually
is less than or equal to 2 mod f dash of z
00:35:03.880 --> 00:35:11.810
I can write this this make sense, okay because
the denominator I can forget the denominator,
00:35:11.810 --> 00:35:17.210
okay and what does this tell you? This tells
you that whenever f dash is defined okay in
00:35:17.210 --> 00:35:20.710
particular if you are looking at a family
of analytic functions okay and the derivatives
00:35:20.710 --> 00:35:27.980
are make sense then if you know the derivatives
are bounded then it means that the spherical
00:35:27.980 --> 00:35:33.810
derivatives are bounded because the spherical
derivative is bounded by 2 times bound for
00:35:33.810 --> 00:35:35.910
the usual derivatives.
00:35:35.910 --> 00:35:44.520
So if you start with a family of analytic
functions such that the derivatives are bounded
00:35:44.520 --> 00:35:48.150
then the spherical derivatives are bounded
and Marty's Theorem will tell you that these
00:35:48.150 --> 00:35:53.700
family of analytic functions considered as
a family of meromorphic functions mind you
00:35:53.700 --> 00:35:57.360
analytic functions are also meromorphic functions
but when you considered it as a family of
00:35:57.360 --> 00:36:02.180
meromorphic functions you are allowing the
value infinity with that consideration this
00:36:02.180 --> 00:36:08.440
family becomes sequentially normally sequentially
compact that means given any sequence you
00:36:08.440 --> 00:36:13.540
can get hold of a subsequence which converges
normally.
00:36:13.540 --> 00:36:25.120
So finally what happens is that this box that
I wrote down here is the crucial condition
00:36:25.120 --> 00:36:31.320
that is crucial for both the original Montel
Theorem and also the meromorphic version of
00:36:31.320 --> 00:36:35.900
Montel's Theorem which is Marty's Theorem
so this is the crucial thing the boundedness
00:36:35.900 --> 00:36:48.310
of the derivatives, okay. So but the only
thing that can happen is that your sequence
00:36:48.310 --> 00:36:52.440
of analytic functions may go to infinity because
that is , okay.
00:36:52.440 --> 00:36:57.210
See if you go back and think we proved the
following thing you take a sequence of analytic
00:36:57.210 --> 00:37:04.390
functions on a domain, okay if you take convergence
with respect to the spherical metric either
00:37:04.390 --> 00:37:07.860
they will converge to an analytic function
or they will converge to a function which
00:37:07.860 --> 00:37:12.840
is identically infinity, okay and the same
kind the of thing happens in meromorphic functions
00:37:12.840 --> 00:37:19.030
you take a sequence of meromorphic functions
which converges normally on a domain, okay
00:37:19.030 --> 00:37:24.980
then either the limit is a meromorphic function
or it is the function which is you know identically
00:37:24.980 --> 00:37:30.440
infinity you do not get bad behaviour you
do not get a sequence of holomorphic functions
00:37:30.440 --> 00:37:33.630
or analytic functions going to a function
which is meromorphic strictly meromorphic
00:37:33.630 --> 00:37:38.280
or you do not get a sequence of meromorphic
functions which goes to a function which has
00:37:38.280 --> 00:37:43.100
funny singularities namely it may have non
isolated singularities or it may have isolated
00:37:43.100 --> 00:37:46.820
essential singularities such these kind of
horrible things do not happen, okay.
00:37:46.820 --> 00:37:55.120
So if you take this thing that I have put
down which I have now rounded in a long ellipse
00:37:55.120 --> 00:38:02.040
as the important condition, okay then that
is the condition for sequential compactness
00:38:02.040 --> 00:38:10.840
that is what I want to say and see these conditions
work if the functions are analytic, okay and
00:38:10.840 --> 00:38:16.990
the analogous condition namely the derivatives
replaced by the spherical derivatives that
00:38:16.990 --> 00:38:23.820
works if the functions are meromorphic, okay
so this is the very very important point in
00:38:23.820 --> 00:38:32.980
our theory that we are finally managed to
translate compactness of a family of meromorphic
00:38:32.980 --> 00:38:38.450
functions or analytic functions to just uniform
boundedness of derivatives that is all, okay.
00:38:38.450 --> 00:38:46.510
And if they are usual analytic functions use
the usual derivatives, if they are meromorphic
00:38:46.510 --> 00:38:51.040
functions use spherical derivatives that is
all, okay. So bringing in the derivatives
00:38:51.040 --> 00:38:57.620
is the big deal here, okay. So we will now
need to see a look at a proof of this theorem
00:38:57.620 --> 00:39:05.330
and the proof is pretty except that you will
have to worry about all these issues there
00:39:05.330 --> 00:39:09.080
are little little things that need to be checked,
okay.
00:39:09.080 --> 00:39:19.110
So I will try to write down the proof pretty
short steps and I will ask you to do some
00:39:19.110 --> 00:39:30.370
small verifications so let me say the following
thing so you see let me write the so here
00:39:30.370 --> 00:39:34.850
is what I have to proof I have domain in the
complex plane, I have a family of meromorphic
00:39:34.850 --> 00:39:41.380
functions I assume if I have to assume first
that the family is normally sequentially compact
00:39:41.380 --> 00:39:47.390
and I have to show that the spherical derivatives
are bounded, okay normally uniformly bounded
00:39:47.390 --> 00:39:49.870
and I have to do the other way round.
00:39:49.870 --> 00:40:01.220
And what does the proof actually involve it
involves the few simple results
00:40:01.220 --> 00:40:10.800
so let me write this so here the few lemmas
that I want to worry about or rather let me
00:40:10.800 --> 00:40:22.520
say lemma
if a sequence f k of meromorphic functions
00:40:22.520 --> 00:40:38.100
converges to f, okay in m D okay then the
same thing happens to the sequence of spherical
00:40:38.100 --> 00:40:50.300
derivatives, okay ofcourse here again I must
so I have written in very very simple words
00:40:50.300 --> 00:40:56.350
but I must again insist when I say converges
I mean converges normally, okay it means it
00:40:56.350 --> 00:40:58.320
is uniform of compact subsets.
00:40:58.320 --> 00:41:03.760
So if f k is a sequence of meromorphic functions
on D it converges uniformly on compact subsets
00:41:03.760 --> 00:41:09.560
to a function f we have already seen that
this f can either be meromorphic or it can
00:41:09.560 --> 00:41:15.130
be the function which is identically infinity
that we have already seen then this normal
00:41:15.130 --> 00:41:21.630
convergence preserves derivatives, okay. So
this is something that we have seen with analytic
00:41:21.630 --> 00:41:25.520
functions if a sequence of analytic functions
converges normally to a given function then
00:41:25.520 --> 00:41:32.620
the limit function is also analytic and you
know you can the nth order derivatives of
00:41:32.620 --> 00:41:38.390
the original sequence of functions will converge
normally to the nth order derivative of the
00:41:38.390 --> 00:41:42.920
limit function this is all just because of
normal convergence uniform convergence of
00:41:42.920 --> 00:41:45.260
compact subsets, okay.
00:41:45.260 --> 00:41:49.720
So the same thing happens with spherical derivatives
this is one fact that we will have to use,
00:41:49.720 --> 00:42:02.300
okay. So you can check this and well if you
want to go back to the let me tell you about
00:42:02.300 --> 00:42:07.590
the let me tell you atleast in words about
the proof of this theorem ya atleast one way
00:42:07.590 --> 00:42:15.850
is very clear, okay. Suppose your family is
normally sequentially compact, okay and suppose
00:42:15.850 --> 00:42:23.850
contrary to what is required the family of
derivatives spherical derivatives is not normally
00:42:23.850 --> 00:42:27.920
uniformly bounded then you know there is a
compact subset on which these derivatives
00:42:27.920 --> 00:42:31.180
will go to infinity, okay.
00:42:31.180 --> 00:42:37.670
And so there is a compact subset and a sequence
of functions where the spherical derivatives
00:42:37.670 --> 00:42:44.260
will go to infinity, okay that is what you
get if you contradict a normal uniform boundedness,
00:42:44.260 --> 00:42:49.830
okay of the spherical derivatives but if that
happens then the original family could not
00:42:49.830 --> 00:42:56.130
have be normal because if the original family
were normal then what would happen is that
00:42:56.130 --> 00:43:04.170
the original from every sequence you can pick
out a normally convergent subsequence if the
00:43:04.170 --> 00:43:09.330
subsequence is normally convergent then the
spherical derivative is also normally convergent,
00:43:09.330 --> 00:43:20.250
okay but then we have already obtained a sequence
of spherical derivatives which does not converge,
00:43:20.250 --> 00:43:21.330
okay.
00:43:21.330 --> 00:43:27.140
So the point you will have to remember here
is that when you are considering spherical
00:43:27.140 --> 00:43:31.280
derivatives the convergence is with respect
to the usual distance function on the real
00:43:31.280 --> 00:43:37.780
line, okay mind you that is another important
point the spherical derivative is a positive
00:43:37.780 --> 00:43:42.890
non-negative real valued function, okay and
whenever you talk about convergence of the
00:43:42.890 --> 00:43:50.140
spherical derivatives you are working with
convergence on the real line that is something
00:43:50.140 --> 00:43:54.760
that you should not forget, okay and therefore
you get a contradiction. So this is one way
00:43:54.760 --> 00:43:58.630
of the theorem the other way I will prove
in the next lecture, okay.