WEBVTT
Kind: captions
Language: en
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Okay, so we have seen in the last lecture
the famous theorem of montel or normality
00:01:04.320 --> 00:01:09.310
of families and what it says is that suppose
we have a family of meromorphic functions
00:01:09.310 --> 00:01:14.439
defined on a domain, the domain can be extended
complex plane. To decide that the family is
00:01:14.439 --> 00:01:20.399
normal, all you have to do is to ensure that
all the functions in the family do not take
00:01:20.399 --> 00:01:27.330
3 distinct values in the extended plane, okay.
And you know because you are working with
00:01:27.330 --> 00:01:31.530
meromorphic functions, you have to allow the
value infinity because that is the value that
00:01:31.530 --> 00:01:34.299
a meromorphic function at a pole will take,
okay.
00:01:34.299 --> 00:01:40.909
And but of course you are looking at a family
of analytic functions, okay, then the condition
00:01:40.909 --> 00:01:46.029
is much more simpler, you have to just find
2 complex values which the functions in the
00:01:46.029 --> 00:01:52.590
family do not take and if that is true then
the family is normal, okay, there is this
00:01:52.590 --> 00:02:02.929
great theorem of montel. And you see
it is the key to proving the picard’s theorem
00:02:02.929 --> 00:02:06.970
which we will do, right.
00:02:06.970 --> 00:02:25.130
So, so, so here is picard’s big or great
theorem, okay. So you know what the theorem
00:02:25.130 --> 00:02:29.710
is. The theorem is that if you take a function
which has an isolated essential singularity,
00:02:29.710 --> 00:02:36.220
then the image of any neighbourhood of that
singularity is either the whole complex plane
00:02:36.220 --> 00:02:39.800
or it is a complex plane minus the single
value, okay. That means, it means, it means
00:02:39.800 --> 00:02:44.282
that it can at most omit one complex value,
all right. And what is the restatement, the
00:02:44.282 --> 00:02:55.230
restatement is that if it omits more than
one value, if it omits 2 values, that
00:02:55.230 --> 00:02:59.110
is something cannot be, that is not possible,
okay.
00:02:59.110 --> 00:03:05.940
So what we will do is we will assume that
it omits 2 values and then use the montel
00:03:05.940 --> 00:03:11.650
criterion that the resulting family of zoomed
functions is normal, okay. And examine the
00:03:11.650 --> 00:03:17.720
limit of the zoomed functions and that will
give you the proof, okay. So let me state
00:03:17.720 --> 00:03:35.250
it, let f of z, let f be, let f have an isolated,
let f of z have an isolated singularity at
00:03:35.250 --> 00:04:08.670
z equal to z0 , then the image under
f of any deleted neighbourhood of z0 is either
00:04:08.670 --> 00:04:23.750
all of complex plane or omits at most one
value in c. So this is the picard’s theorem,
00:04:23.750 --> 00:04:24.750
okay.
00:04:24.750 --> 00:04:31.280
And of course this is valid for every neighbourhood,
it means that you know it will take except
00:04:31.280 --> 00:04:36.090
for one value which it might omits, all other
complex values it will take infinitely many
00:04:36.090 --> 00:04:42.880
times because you, if you find the point where
it will take that value, then you can find
00:04:42.880 --> 00:04:47.700
a smaller neighbourhood, you can take a smaller
neighbourhood, deleted neighbourhood of z0
00:04:47.700 --> 00:04:51.910
and in that neighbourhood also again it has
to take that value and can go on like this,
00:04:51.910 --> 00:04:55.770
therefore it will take every value except
one value infinitely many times, okay. That
00:04:55.770 --> 00:05:04.080
is amazing behaviour of function, analytics
function around an isolated essential singularity,
00:05:04.080 --> 00:05:05.100
okay.
00:05:05.100 --> 00:05:19.300
So this is the theorem and, so what is the
proof? The proof is, by contradiction, by
00:05:19.300 --> 00:05:31.919
contradiction, by contradiction using the
zooming process. So you know the zooming process
00:05:31.919 --> 00:05:38.380
has been, we had been using it right from
zaltzmann lemma, okay and you know what is
00:05:38.380 --> 00:05:42.130
the main idea behind the zooming process?
The idea behind the zooming process is, as
00:05:42.130 --> 00:05:47.800
you zoom into a normal point, then all those
zoomed functions will converge normally to
00:05:47.800 --> 00:05:54.440
a constant function. And if you zoom to a
non-normal point, then all the zoomed functions
00:05:54.440 --> 00:05:57.540
will converge to a nonconstant meromorphic
function. Okay that is the, basically the
00:05:57.540 --> 00:06:00.160
principle, all right.
00:06:00.160 --> 00:06:23.590
So what we will do is, so assume f omits 2
values in f omits 2 values in c in some deleted
00:06:23.590 --> 00:06:36.370
neighbourhood 0 less than mod z minus
z0 less than say rho, okay. So you have to
00:06:36.370 --> 00:06:47.639
show that f takes either take all values or
it will take all values except 1. So if you
00:06:47.639 --> 00:06:53.210
want to contradict that, you have to assume
that it omits at least 2 values, okay. So
00:06:53.210 --> 00:07:03.240
let us assume that, all right. So you, what
you do is, so here is my diagram. So i have,
00:07:03.240 --> 00:07:10.850
so this is the, this is the complex plane,
this is z plane and i have this point z0.
00:07:10.850 --> 00:07:22.670
And you know i, there is this small disc surrounding
z0, radius rho, this disc with radius rho
00:07:22.670 --> 00:07:31.169
and on this disc f does not take 2 values,
2 complex values.
00:07:31.169 --> 00:07:35.720
That means there are 2 distinct complex values
which f will take, it may not take many more
00:07:35.720 --> 00:07:42.140
values also but at least 2 values of it misses,
okay. And what am i going to do, i am going
00:07:42.140 --> 00:07:45.920
to construct a sequence of zoomed functions.
So what is the sequence of zoomed functions,
00:07:45.920 --> 00:07:56.350
what you do is, well take any, take any, so
before that let me write it ideologically.
00:07:56.350 --> 00:08:06.000
Zoom into the function f at z0 itself, okay.
Mind you the function in, z0 is an isolated
00:08:06.000 --> 00:08:09.050
singular point, it is an essential single
point, therefore the deleted neighbourhood
00:08:09.050 --> 00:08:16.919
of z0 in this deleted disc, this punctured
disc centred at z0 and radius rho, you throw
00:08:16.919 --> 00:08:19.590
out the point z0.
00:08:19.590 --> 00:08:24.180
In the punctured disc it is the function is
analytic, mind you, okay. And what i am going
00:08:24.180 --> 00:08:29.211
to do, i am actually going to zoom in to z0,
okay, i am just going to zoom in to z0 and
00:08:29.211 --> 00:08:33.460
how do i zoom in to z0, by taking smaller
and smaller, discs of smaller and smaller
00:08:33.460 --> 00:08:39.010
radii which are centred at z0. And of course
i have to exclude z0 because z0 is a point
00:08:39.010 --> 00:08:51.950
of singularity of f. So what you do is, zoom
in to z0 and so let me say that is take a
00:08:51.950 --> 00:09:08.540
sequence epsilon n tending to 0, 0 less than
epsilon n lesser than rho, okay. So you take
00:09:08.540 --> 00:09:12.270
a sequence of smaller and smaller radii, okay.
00:09:12.270 --> 00:09:25.430
And let gn of zeta be the zooming of f the
same function f centred at z0 , the scaling
00:09:25.430 --> 00:09:30.540
factor is 1 by epsilon n and the variable
is that, okay. So this means that you are
00:09:30.540 --> 00:09:39.030
just taking f of z0 plus epsilon n times theta,
this is a function. So this is what my gn
00:09:39.030 --> 00:09:44.520
is, so i am using that single function and
constructing a family of functions. Using
00:09:44.520 --> 00:09:50.170
the single function f i am constructing a
family of functions gn, okay. And where are
00:09:50.170 --> 00:09:56.710
these gns defined? So you see, this is the
z plane and then if you look at correspondingly,
00:09:56.710 --> 00:10:02.620
we have the, you have, this is a complex which
is the z plane, then i also have complex thing
00:10:02.620 --> 00:10:04.459
which is zeta plane, okay.
00:10:04.459 --> 00:10:11.760
And in the zeta plane what happens is that,
you know if i take, if i take this disc centred
00:10:11.760 --> 00:10:21.470
at the origin and radius, if i take the radius
to be rho by epsilon n, okay, that is for
00:10:21.470 --> 00:10:27.010
mod zeta less than epsilon n, mod, epsilon
and theta will be less than rho. And therefore
00:10:27.010 --> 00:10:32.700
this is the, so this is the disc in which
gn is defined by the only thing is it is not
00:10:32.700 --> 00:10:37.800
defined as the origin because at the origin
the origin corresponds to z0. And f at z0
00:10:37.800 --> 00:10:43.190
is not defined because z0 is a singularity
of f, okay.
00:10:43.190 --> 00:10:50.330
So let us write that down, gn is defined,
so let me write this here, gn of zeta is defined
00:10:50.330 --> 00:11:02.209
in a 0 less than mod zeta less than rho by
epsilon n. And of course you know the point
00:11:02.209 --> 00:11:08.589
about this whole business is that this epsilon
n tends to 0plus, rho by epsilon n tends to
00:11:08.589 --> 00:11:15.960
infinity and therefore gn zeta is going to
be, you can talk about convergence, normal
00:11:15.960 --> 00:11:20.680
convergence of gn zeta on the whole punctured
plane, okay, except that, the whole plane
00:11:20.680 --> 00:11:25.630
except the origin. Because eventually any
compact subset of the punctured plane other
00:11:25.630 --> 00:11:30.510
than which does not contain the origin is
going to be, is going to be contained in the
00:11:30.510 --> 00:11:33.630
domain of gn zeta for n sufficiently large,
okay.
00:11:33.630 --> 00:11:48.480
So, so let me write this down, note that
any compact subset
00:11:48.480 --> 00:12:13.920
of c not containing the origin, the origin
is going to be in the domain of definition
00:12:13.920 --> 00:12:25.420
of gn zeta for n sufficiently large, okay.
So now, now i want you to just watch. See
00:12:25.420 --> 00:12:34.890
after all, you know the, the values of gn
in this punctured disc centred at the origin
00:12:34.890 --> 00:12:41.330
o and radius rho by epsilon n correspondence
to exactly the values of f in the punctured
00:12:41.330 --> 00:12:48.720
disc centred at z0, the values of f inside
this whole disc, punctured disc centred at
00:12:48.720 --> 00:12:54.110
z0 radius rho, they are exactly the values
of gn, in the punctured this centred at the
00:12:54.110 --> 00:12:59.860
origin, radius row by epsilon n, because this
is just a scaling and data solution, okay.
00:12:59.860 --> 00:13:08.480
Now you see now you know f, but what is,
what have we assumed about f, we assumed that
00:13:08.480 --> 00:13:17.560
f omits 2 complex values, f is analytic function
and it omits at least 2 complex values in
00:13:17.560 --> 00:13:23.860
this disc, punctured disc centred at z0 radius
rho. Therefore each of the gns will also omit
00:13:23.860 --> 00:13:31.400
those 2 values in their domains, okay. And
of course each of the gns are also analytic
00:13:31.400 --> 00:13:37.380
functions because they differ from f only
by bilinear transformation consisting of scaling
00:13:37.380 --> 00:13:43.910
under and incarceration. But now you know
we are in good shape because what we have
00:13:43.910 --> 00:13:48.570
done you know, we have been able to get a
family of, we have been able to get a family
00:13:48.570 --> 00:13:53.840
of functions gns which are analytic and which
omit 2 values.
00:13:53.840 --> 00:13:58.510
Now immediately montel’s great theorem will
tell you that there has to be a normal and
00:13:58.510 --> 00:14:02.721
you get a convergence of sequence and then
you have to examine the limit, okay. And the
00:14:02.721 --> 00:14:08.000
limit will give you contradiction, all right.
So basically the contradiction will be set
00:14:08.000 --> 00:14:13.860
the limit function at the origin will, examining
the limit function at the origin will tell
00:14:13.860 --> 00:14:18.430
you that the origin has to be either pole
or removable singularity for f which is not
00:14:18.430 --> 00:14:23.800
true. I mean analysis of limit function will
tell you that the, you take this limit function,
00:14:23.800 --> 00:14:28.410
this limit function will also be defined on
the punctured plane, okay.
00:14:28.410 --> 00:14:33.709
Because all the original gns are all defined
outside 0, okay, so if you analyse the limit
00:14:33.709 --> 00:14:41.890
function. See the limit function is like zooming
into f at z0 infinitely many times, okay.
00:14:41.890 --> 00:14:47.769
So behaviour of the limit function at the
origin which is an isolated singularity will
00:14:47.769 --> 00:14:55.450
reflect upon the behaviour of f at z0. And
by analysis we will show that if you analyse
00:14:55.450 --> 00:15:01.959
the limit function, there are only 2 possibilities
z0 should either be removable singularity
00:15:01.959 --> 00:15:07.490
or it has to be a pole and both of these contradictions
because i have assumed z0 at f to be a essential
00:15:07.490 --> 00:15:12.730
singularity, okay. And that is how the proof
goes, so it becomes as simple as that.
00:15:12.730 --> 00:15:35.250
So let me write this down note that gn is
a normal family as it also does not assume
00:15:35.250 --> 00:15:51.779
the values that f omits, okay. So, so what
does it mean, it is a normal form means, it
00:15:51.779 --> 00:16:00.149
means that you know, and it is a normal family
and mind you for gn because it is a zoomed
00:16:00.149 --> 00:16:06.470
function, whose domains are becoming bigger
and bigger and bigger, you can think of them
00:16:06.470 --> 00:16:22.170
as a normal family you know in with a
limit in the punctured plane, all right. So
00:16:22.170 --> 00:16:26.920
what you must understand is that if you take,
for example if you take the punctured unit
00:16:26.920 --> 00:16:34.190
disc, okay, to take the punctured unit disc
then for n sufficiently large, all the gns
00:16:34.190 --> 00:16:38.010
are going to be defined, their domains are
going to become bigger and bigger and they
00:16:38.010 --> 00:16:40.040
are going to contain the punctured unit disc,
okay.
00:16:40.040 --> 00:16:47.931
So you can see that if you want to take the,
if you want to talk about the domain of the
00:16:47.931 --> 00:16:54.190
gns, okay, you can assume that for n, p on,
for n sufficiently large the domain contains
00:16:54.190 --> 00:16:59.720
unit disc if you want. Or for that matter,
any finite disc with of course the origin
00:16:59.720 --> 00:17:06.530
omitted, okay. And when you take the limit
function that is because gn is a normal, is
00:17:06.530 --> 00:17:10.040
normal, if you take the limit function, the
limit function will be defined on the whole
00:17:10.040 --> 00:17:15.800
punctured plane because it will make sense,
because you are covering every point in the
00:17:15.800 --> 00:17:21.650
plane literally. Because for every point in
the plane if you take epsilon n sufficiently
00:17:21.650 --> 00:17:25.970
small, rho by epsilon n becomes sufficiently
large and gns beyond a certain state will
00:17:25.970 --> 00:17:27.660
be defined with that point.
00:17:27.660 --> 00:17:33.780
And therefore the limit of all the, if you
take convergence of sequence of gns, the limit
00:17:33.780 --> 00:17:44.740
will also be defined with that point, okay.
So let me write this, thus gn k, thus there
00:17:44.740 --> 00:18:04.400
exists gn k case of sequence that converges
normally as g on, g of zeta if you want on
00:18:04.400 --> 00:18:10.570
c minus the origin, okay. So this happens,
because what is the meaning of normal family?
00:18:10.570 --> 00:18:17.390
Normal means that, normally sequentially compact,
that is give me any sequence, there is a normal
00:18:17.390 --> 00:18:24.220
convergence subsequence. So when gn itself
is a normal family, it is already a sequence,
00:18:24.220 --> 00:18:28.880
so it will have a normally convergence of
sequence. So you have a sub sequence gn k
00:18:28.880 --> 00:18:32.020
which converges normally to g on c minus origin,
okay.
00:18:32.020 --> 00:18:37.080
But the point is that it is not, it is not,
what is important is that it is g lives in
00:18:37.080 --> 00:18:47.290
a neighbourhood of 0, okay and now, now try
to understand each gn is analytic on, on the
00:18:47.290 --> 00:18:51.760
punctured disc, on the punctured disc centred
at the origin, okay. Therefore this limit
00:18:51.760 --> 00:18:56.179
function g is a normal limit of analytic function.
We have already seen such a normal limit can
00:18:56.179 --> 00:19:01.390
have only 2 possibilities, either the, either
the normal limit can completely be analytics
00:19:01.390 --> 00:19:06.679
or it can be identically infinity, these are
the only 2 possibilities. So let us, so let
00:19:06.679 --> 00:19:21.850
us write that down. Thus g is identically
infinity or g is analytic in c minus0, okay.
00:19:21.850 --> 00:19:34.910
Now let us look at both of these cases. Suppose,
g is identically infinity, okay, suppose g
00:19:34.910 --> 00:19:47.100
is identically infinity, so what does it mean,
what this will mean, see, think of it heuristically,
00:19:47.100 --> 00:19:55.070
g is f zoomed you know infinitely at z0 and
if g is identically infinity, what you are
00:19:55.070 --> 00:20:00.230
actually saying is that f is infinity in the
neighbourhood of, f is going to be infinity
00:20:00.230 --> 00:20:06.450
in the neighbourhood of z0, right, because
the values of g are just limits of values
00:20:06.450 --> 00:20:11.490
of gns and the values of gns are just values
of f in smaller and smaller neighbourhood.
00:20:11.490 --> 00:20:19.100
So if g is identically infinity, okay, that
means that the value, the gns are getting
00:20:19.100 --> 00:20:22.380
larger and larger in modulus, okay.
00:20:22.380 --> 00:20:28.270
And that means that the values of f are getting
closer and closer to infinity as you approach
00:20:28.270 --> 00:20:32.850
z0. But that means z0 is a pole, but that
is not possible because z0 is an essential
00:20:32.850 --> 00:20:37.799
singularity, so this is not possible, so you
ruled out this case, okay. So let me write
00:20:37.799 --> 00:20:53.500
this down. This means that g of zeta is infinity
for all zeta in c minus infinity. I will have
00:20:53.500 --> 00:21:02.500
to make use of the fact that you know gn converges
to g, okay, g is not just, it is not simply
00:21:02.500 --> 00:21:05.850
limit of gns. Of course it is point wise limit
of gns but it is not, it is more than that,
00:21:05.850 --> 00:21:08.280
it is not just a point wise limit, it is just
a normal limit.
00:21:08.280 --> 00:21:18.120
So the convergence is uniform complexion since,
okay. So you know gn of zeta converges to
00:21:18.120 --> 00:21:35.179
g of zeta uniformly on mod zeta is equal to
say r for any r greater than 0. Because you
00:21:35.179 --> 00:21:40.690
see mod theta equal to r is a circle in the
case of plane centred at the origin, radius
00:21:40.690 --> 00:21:46.350
r and that is certainly a compact circuit,
it is closed and bounded. And g is a normal
00:21:46.350 --> 00:21:50.919
limit, therefore the convergence gn to g should
be uniform on any compact subset, so it has
00:21:50.919 --> 00:21:57.840
to be uniform on mod theta equal to r, okay.
But, but of course what is, but what is g
00:21:57.840 --> 00:22:05.059
zeta? G zeta is infinity, if g is identically
infinity. So what this and and and what does
00:22:05.059 --> 00:22:13.340
uniform convergence mean, it means that all
the gns, they will come to within an epsilon
00:22:13.340 --> 00:22:20.419
of g zeta, okay, if you take n sufficiently
large irrespective of zeta, right, therefore
00:22:20.419 --> 00:22:21.760
that means, okay.
00:22:21.760 --> 00:22:27.419
But of course you want to come to an epsilon
of infinity so you will have to be careful
00:22:27.419 --> 00:22:35.630
and you have to use a spherical metric. So
let me write this down, thus given epsilon
00:22:35.630 --> 00:22:41.620
greater than 0, the spherical distance between
gn zeta and g of the zeta which is actually
00:22:41.620 --> 00:22:55.549
infinity can be made less than epsilon for
n sufficiently large. And for all
00:22:55.549 --> 00:23:07.350
theta with mod theta equal to r, okay but
now, but what is gn zeta? You see gn zeta,
00:23:07.350 --> 00:23:15.360
our definition is just f of z0 plus epsilon
n theta, this is what it is. Okay.
00:23:15.360 --> 00:23:28.730
And as you know, you see even if mod zeta
is r, if your, your epsilon ns are becoming
00:23:28.730 --> 00:23:34.309
smaller, so you are covering smaller and smaller
disc, you are covering smaller and smaller
00:23:34.309 --> 00:23:42.350
circles centred at z0, okay. And therefore
what you are saying is that the function values
00:23:42.350 --> 00:23:49.070
of f on smaller and smaller circles centred
at z0 are getting close to infinity, okay.
00:23:49.070 --> 00:23:59.030
And and and that is enough to tell you that
f, the limit of f let as z tends to z0 is
00:23:59.030 --> 00:24:03.700
actually infinity, which means that z0 has
to be a pole. But that is a contradiction
00:24:03.700 --> 00:24:08.659
to our assumption that z0 is actually an isolated
essential singularity, okay.
00:24:08.659 --> 00:24:22.159
So let me write this down, this means that
the spherical distance between f of z and
00:24:22.159 --> 00:24:38.549
infinity can be made less than epsilon uniformly
on compact subsets
00:24:38.549 --> 00:25:04.650
of mod z minus z0 lesser than epsilon n for
n sufficiently large, okay. So, you know because,
00:25:04.650 --> 00:25:12.180
what you must understand is that this, this
r is, this capital r is at our disposal. You
00:25:12.180 --> 00:25:16.669
can make this capital r as small as you want,
close to 0, you can make it as large as you
00:25:16.669 --> 00:25:23.910
want, okay. So you can, you can literally
cover all the , you can cover all the
00:25:23.910 --> 00:25:31.400
circles centred at z0 of fixed radius, okay,
below a certain positive value.
00:25:31.400 --> 00:25:36.710
So in some sense you are, therefore you are
able to cover complete deleted neighbourhood
00:25:36.710 --> 00:25:52.270
of z0, okay, that is the whole point. So,
well, so , so this implies z0 is a pole
00:25:52.270 --> 00:25:59.760
of f, which is a contradiction. Because you
will assume z0 to be an essential singularity,
00:25:59.760 --> 00:26:07.260
okay. So does, you know you, the limit, the
zoomed limit function g cannot be identically
00:26:07.260 --> 00:26:12.570
infinity, okay. Therefore what is the other
possibility, it has to be only be an analytic
00:26:12.570 --> 00:26:27.770
function in the, with, in the punctured plane,
punctured r, okay. Thus g is analytic in the
00:26:27.770 --> 00:26:34.900
punctured plane, okay. And what does that
mean, it means of course the origin is a singularity,
00:26:34.900 --> 00:26:39.600
origin is an isolated singularity for g, okay.
And now you can ask what kind of singularity
00:26:39.600 --> 00:26:40.660
it is.
00:26:40.660 --> 00:26:48.480
But you know the point is that again you should
not try to study the singularity of g at the
00:26:48.480 --> 00:26:55.640
origin, if not do that. Because after all
g at the origin is going to reflect f at z0,
00:26:55.640 --> 00:27:03.289
okay. So, mind you the gns are all zoomings
of f at z0 and their limit is g, okay. So
00:27:03.289 --> 00:27:09.730
g is some kind of infinite zooming of f at
z0. G at the origin is infinite zooming of
00:27:09.730 --> 00:27:15.730
f at z0, okay. Therefore you should not study
g at the origin but you must make use of the
00:27:15.730 --> 00:27:22.330
fact that g is analytic in the outside the
origin. So if you again take this mod theta
00:27:22.330 --> 00:27:28.809
equal to r, it would take circles centred
at the origin, this is the plane, radius r.
00:27:28.809 --> 00:27:35.320
Mind you again that is a complex set and g
being analytic, if continuous, then on the
00:27:35.320 --> 00:27:39.789
compact that it will be bounded, okay. And
now this bound will apply to f in the neighbourhood
00:27:39.789 --> 00:27:46.530
of z0, okay. And we will tell you that f is
bounded in the neighbourhood of z0 but then
00:27:46.530 --> 00:27:50.669
riemann’s removable singularity will tell
you that z0 has to be removable singularity
00:27:50.669 --> 00:27:55.970
and again that is a contradiction. And that
is the proof of the theorem, okay, proof of
00:27:55.970 --> 00:28:04.360
the theorem is so simple. So let us go to
the other case, again there exists an m greater
00:28:04.360 --> 00:28:11.360
than 0 such that again on mod z, mod zeta
equal to r greater than 0, there exists an
00:28:11.360 --> 00:28:18.510
m greater than 0 such that mod g is less than
or equal to m, okay.
00:28:18.510 --> 00:28:28.250
And, but after all, since gn converges to
g normally, in fact uniformly on mod zeta
00:28:28.250 --> 00:28:42.710
equal to r, because, again because it is a
compact set we have fn f is bounded
00:28:42.710 --> 00:28:54.690
in a deleted neighbourhood of z0, this again
implies by the riemann’s removable singularity
00:28:54.690 --> 00:29:09.070
theorem
00:29:09.070 --> 00:29:34.090
that z0 is removable singularity of f,
again a contradiction. And you know that finishes
00:29:34.090 --> 00:29:41.750
the proof. There are only 2 choices for g
and both choices lead to contradictions. So
00:29:41.750 --> 00:29:49.200
that is the of the famous big picard’s theorem.
And we can, as a corollary we can deduce the
00:29:49.200 --> 00:29:50.200
little picard’s theorem.
00:29:50.200 --> 00:29:53.789
What is the little picard’s theorem? It
tells you that the image of the complex plane
00:29:53.789 --> 00:30:00.120
under entire function is again the whole plane
or the plane minus the point. And now what
00:30:00.120 --> 00:30:06.140
is the proof, the proof is very simple, taken
entire function, of course we should take
00:30:06.140 --> 00:30:09.641
a nonconstant entire function, okay because
if you take a constant entire function, the
00:30:09.641 --> 00:30:15.210
image of a constant function is always only
one point. So you must be careful, i must
00:30:15.210 --> 00:30:20.870
have been carefully saying that statement.
If you take a nonconstant entire function,
00:30:20.870 --> 00:30:24.580
then you know the image of the complex plane
should be either the whole complex plane or
00:30:24.580 --> 00:30:28.820
complex plane minus a point, it can omit only
one value at most.
00:30:28.820 --> 00:30:34.840
What is the proof, very simple, take the,
take the entire function and look at the point
00:30:34.840 --> 00:30:41.410
at infinity, okay. The point at infinity becomes
the isolated singularity because the complex
00:30:41.410 --> 00:30:47.730
plane is a deleted neighbourhood of infinity
in the external complex plane, okay. So your
00:30:47.730 --> 00:30:54.900
function f, your entire function f has infinity
is an isolated singularity, okay. Now for
00:30:54.900 --> 00:30:59.190
an isolated singularity what are the possibilities?
It can be removable, it can be pole or it
00:30:59.190 --> 00:31:06.549
can be essential. If it is removable, it means
that, if infinity is a removable singularity,
00:31:06.549 --> 00:31:10.679
it means that f is bounded at infinity and
that means that by louisville’s theorem,
00:31:10.679 --> 00:31:12.440
f has to be a constant.
00:31:12.440 --> 00:31:16.669
So if we take f to be a nonconstant entire
function, okay, infinity cannot be removable
00:31:16.669 --> 00:31:21.210
singularity, all right. So it can only be
a pole or essential singularity. If infinity
00:31:21.210 --> 00:31:25.570
is a pole, then you have already seen that
f has to be polynomial, okay. And you know
00:31:25.570 --> 00:31:29.840
a polynomial will take all values because
of the fundamental theorem of algebra. So
00:31:29.840 --> 00:31:34.250
if f is a, f has infinity as a pole, it is
a polynomial, the image is a complex plane
00:31:34.250 --> 00:31:37.820
under f is the whole complex plane, using
the fundamental theorem of algebra. So the
00:31:37.820 --> 00:31:41.080
only thing is infinity is an essential singularity.
00:31:41.080 --> 00:31:48.770
If infinity is an essential singularity for
f, apply the great picard’s theorem, okay.
00:31:48.770 --> 00:31:54.029
F can, any neighbourhood of infinity has to
be mapped by f into the whole complex plane
00:31:54.029 --> 00:31:58.370
or the complex plane minus the point. At the
complex plane itself is a deleted neighbourhood
00:31:58.370 --> 00:32:06.470
of infinity, so f has to map the whole complex
plane or the complex plane minus the point,
00:32:06.470 --> 00:32:20.039
that is it. Okay, so i will just write this
down.
00:32:20.039 --> 00:32:35.919
So corollary, picard’s little theorem, is
f is a nonconstant entire function, then
00:32:35.919 --> 00:33:11.630
f c is equal to c or c minus z0, okay,
for some z0 in c. So proof is infinity
00:33:11.630 --> 00:33:21.720
is an isolated singular, is an isolated singular
point
00:33:21.720 --> 00:33:40.370
for f in c union infinity as c is a deleted
neighbourhood of infinity where f is analytics,
00:33:40.370 --> 00:34:04.370
okay. Thus infinity is either removable pole
or essential. If infinity is it removable,
00:34:04.370 --> 00:34:21.309
f is bounded at infinity, so by louisville
f is constant, a contradiction. Because i
00:34:21.309 --> 00:34:32.190
am assuming f is a nonconstant fashion, okay,
nonconstant entire function. If infinity is
00:34:32.190 --> 00:34:55.549
a pole, we have seen earlier that f has to
be polynomial, be a nonconstant polynomial,
00:34:55.549 --> 00:35:22.109
which assumes all complex values with the
fundamental theorem of algebra, okay.
00:35:22.109 --> 00:35:28.279
So the only other case is if infinity is a
pole, i mean if infinity is an essential singularity,
00:35:28.279 --> 00:35:41.499
infinity is an essential singularity, plus
the big picard’s theorem, by the big picard’s
00:35:41.499 --> 00:36:09.440
theorem or the great picard’s theorem, f
of c is c or c minus point z0, okay. So that
00:36:09.440 --> 00:36:19.269
finishes the proof of the little picard’s
theorem. And therefore you see you are
00:36:19.269 --> 00:36:26.289
able to prove the picard’s theorem very
easily. And the key to all this is, this this
00:36:26.289 --> 00:36:31.930
really great theorem of montel, it says, this
is the criterion for normality of a family,
00:36:31.930 --> 00:36:32.930
okay.
00:36:32.930 --> 00:36:38.839
And it is a very very simple criteria, in
the sense that if you know is family functions,
00:36:38.839 --> 00:36:45.079
if it is a family of meromorphic functions,
if you know that it omits 3 values, okay,
00:36:45.079 --> 00:36:50.890
in the extended plane, then you what is normal.
If it is a family of analytics functions,
00:36:50.890 --> 00:36:56.569
that means it omits 2 complex values, then
you know again it is normal. And the advantage
00:36:56.569 --> 00:37:02.289
of normality is that it is a kind of compactness.
Namely it is normal sequential compactness
00:37:02.289 --> 00:37:07.779
which allows you to extract from any sequence
is of sequence which converges normally, that
00:37:07.779 --> 00:37:10.630
is which converges uniformly at compact subsets,
okay.
00:37:10.630 --> 00:37:18.319
So that finishes the proof of the picard theorem
which was the main aim of this course, okay.
00:37:18.319 --> 00:37:27.609
What i would like to next do is to tell you
that, to tell you how, how powerful zaltzmann’s
00:37:27.609 --> 00:37:36.440
lemma is in several other contexts, okay.
Mind you that this reasonably simplified
00:37:36.440 --> 00:37:41.990
proof of the great picard theorem was possible
because of the montel’s theorem on common
00:37:41.990 --> 00:37:49.430
normality, okay. And that in turn was proved
by zaltzmann lemma, okay.
00:37:49.430 --> 00:37:54.170
So these are all actually all the simplifications
are because of zaltzmann lemma, that is the
00:37:54.170 --> 00:38:00.229
most important thing. But the zaltzmann is,
so similarly in the proofs of various other
00:38:00.229 --> 00:38:06.269
theorems and various other theories of complex
functions, zaltzmann lemma provides us with
00:38:06.269 --> 00:38:14.059
easier proofs of some very deep results and
also provides us with new results. And i will
00:38:14.059 --> 00:38:18.560
try to outline those results in the coming
lectures, okay. So i will stop here.