WEBVTT
Kind: captions
Language: en
00:00:43.460 --> 00:00:51.590
Alright so so what we did in the last lecture
was you know try to explain how the notion
00:00:51.590 --> 00:00:58.170
of normal family makes sense for a domain
which includes the point at infinity okay
00:00:58.170 --> 00:01:05.560
and the technique has that we always follow
is that you make a two-piece definition okay
00:01:05.560 --> 00:01:10.960
you make one definition for a domain punctured
at infinity which will be a domain in the
00:01:10.960 --> 00:01:15.990
usual plane and then to take care of a neighbourhood
of infinity what you do is that you invert
00:01:15.990 --> 00:01:22.890
the variable Z to 1 by w and instead of considering
Z equal to infinity you have to consider w
00:01:22.890 --> 00:01:30.170
equal to 0, so you take you consider a neighbourhood
of 0 okay which is again it is also a domain
00:01:30.170 --> 00:01:37.780
in the usual complex plane okay, so therefore
you are able to treat the case of normality
00:01:37.780 --> 00:01:39.410
at infinity okay.
00:01:39.410 --> 00:01:51.780
So with that we saw that both Montelís theorems
and Martyís theorem hold good for domain
00:01:51.780 --> 00:01:57.870
which probably include which may include the
point at infinity okay that is domain in the
00:01:57.870 --> 00:02:02.530
extended complex plane and of course I also
told you that it is a matter of terminology
00:02:02.530 --> 00:02:08.009
that a family being normally sequentially
compact that is sequentially compact with
00:02:08.009 --> 00:02:14.540
respect to normal convergence that is sometimes
abbreviated to simply normal family okay,
00:02:14.540 --> 00:02:20.840
so now what we are going to do is you know
as I told you we have toÖ We have come very
00:02:20.840 --> 00:02:25.480
close to the proof of the Picard theorem which
was the main aim of this all these lectures
00:02:25.480 --> 00:02:31.310
okay and but before that there is something
called there is a theorem which is called
00:02:31.310 --> 00:02:40.320
as Zalcman's Lemma which we have to use okay
and see this is got to do with characterisation
00:02:40.320 --> 00:02:50.000
of non-normality okay, so in order to you
know motivate this Zalcman's Lemma okay what
00:02:50.000 --> 00:02:56.299
we are going to do is we are going to try
to analyse what happens to a normal family
00:02:56.299 --> 00:02:59.090
at the point okay.
00:02:59.090 --> 00:03:04.209
So so let me make this definition we define
normality of a family at a single point to
00:03:04.209 --> 00:03:12.060
be normality in a small neighbourhood of that
point in an open disk containing that point
00:03:12.060 --> 00:03:18.799
okay, so this is just like definition of analyticity,
so you say a function is analytic if it is
00:03:18.799 --> 00:03:22.930
differentiable not only at a point but in
a small neighbourhood of that point, so in
00:03:22.930 --> 00:03:29.010
the same way normality is also defined at
a point by requiring the property in a small
00:03:29.010 --> 00:03:34.470
open disk surrounding that point or an nonempty
open set containing that point okay.
00:03:34.470 --> 00:03:41.480
Well of course any open set contains that
point is nonempty, so the reason for this
00:03:41.480 --> 00:03:49.470
definition is that normality is a local property
okay. If you have for example a cover of your
00:03:49.470 --> 00:03:57.459
space which is countable and if your domain
can be covered by countably many open sets
00:03:57.459 --> 00:04:03.570
and on each of these open sets if your family
is normal then on the whole domain also the
00:04:03.570 --> 00:04:07.859
family will be normal because you can useÖBecause
of accountability you can use a diagonalization
00:04:07.859 --> 00:04:12.680
argument okay, so normality is a local property
okay.
00:04:12.680 --> 00:04:20.050
So the 1st thing I am going to start with
is so I will put the heading as motivationÖlet
00:04:20.050 --> 00:04:36.340
me use a different color, motivation for Zalcman's
Lemma
00:04:36.340 --> 00:04:50.560
which is actually a characterisation
of non-normality, so what we will do is we
00:04:50.560 --> 00:05:10.590
will make the following definition, let script
F be a family of Meromorphic functions
00:05:10.590 --> 00:05:18.789
defined in a neighbourhood of a point Z naught
okay and of course this point Z naught could
00:05:18.789 --> 00:05:23.620
also be the point at infinity but let us assume
that Z naught is a point in the plane, the
00:05:23.620 --> 00:05:30.330
case of you know how to treat the case at
infinity okay, so suppose this is a family
00:05:30.330 --> 00:05:34.379
of Meromorphic function defined in a neighbourhood
of Z naught that means it is defined in an
00:05:34.379 --> 00:05:37.110
open disk containing Z naught okay.
00:05:37.110 --> 00:05:55.430
We say the family script F is normal at Z
naught if it is normal in a neighbourhood
00:05:55.430 --> 00:06:08.360
of Z naught in an open neighbourhood of Z
naught okay, so well that is the definition
00:06:08.360 --> 00:06:19.259
and so the question is suppose a familyÖ
And of course a family is normal on a domain
00:06:19.259 --> 00:06:37.560
if it is normal at every point okay. Further
family is normal on a domain if and only if
00:06:37.560 --> 00:06:53.990
it is normal at each point, so I should not
say further in fact this is a remark, so this
00:06:53.990 --> 00:07:02.800
is a remark that you can check. It just says
that the notion of normality is a local property,
00:07:02.800 --> 00:07:21.529
normality is a local property. Fine so here
is the question, the question as I now have
00:07:21.529 --> 00:07:25.889
a family defined in a neighbourhood of a point
okay it is a family of Meromorphic functions
00:07:25.889 --> 00:07:33.629
and it is given to me that this family is
normal okay and I want to analyse how the
00:07:33.629 --> 00:07:41.199
family behaves at that point okay, so the
clue to Zalcman's Lemma are at least an understanding
00:07:41.199 --> 00:07:44.999
or motivation for Zalcman's Lemma is trying
to understand this okay.
00:07:44.999 --> 00:08:05.289
So what we will do is let us so I will call
this as local analysis of normal families,
00:08:05.289 --> 00:08:27.800
so let script F be Meromorphic let script
F be a family of Meromorphic functions on
00:08:27.800 --> 00:08:36.700
mod Z minus Z naught less than or equal to
rho okay, so you say mod Z minus Z naught
00:08:36.700 --> 00:08:42.940
less than or equal to rho is actually a closed
set, it is a compact set okay that is provided
00:08:42.940 --> 00:08:46.971
Z naught is a point in the usual complex plane
in Z naught is infinity then you know you
00:08:46.971 --> 00:08:55.420
have to rewrite this asÖ You should not use
the Euclidean metric what you have to use
00:08:55.420 --> 00:09:00.370
the spherical metric okay but I am assuming
for simplicity that Z naught is a point in
00:09:00.370 --> 00:09:06.320
the plane okay and this is the closed end
bounded set I am taking a compact set because
00:09:06.320 --> 00:09:12.980
I can choose such rho certainly it is a family
of Meromorphic functions at the point Z naught
00:09:12.980 --> 00:09:16.480
so it is defined in the neighbourhood of Z
naught, so it is defined on a small enough
00:09:16.480 --> 00:09:21.660
disk which contains Z naught and if you take
a small enough radius then the closed disk
00:09:21.660 --> 00:09:27.480
centred at Z naught of that radius will also
be in the domain where the Meromorphic functions
00:09:27.480 --> 00:09:29.450
is defined where the family is defined.
00:09:29.450 --> 00:09:34.230
So I am taking without loss of this closed
set because the reason is I want to compact
00:09:34.230 --> 00:09:40.780
set so that I can you know I can get uniform
convergence, whenever I talk about convergence
00:09:40.780 --> 00:09:45.450
you know uniform convergence it happens only
on compact sets its normal convergence and
00:09:45.450 --> 00:09:49.690
whenever I am talking about uniform boundedness
it happens only on compact sets, so everything
00:09:49.690 --> 00:09:54.480
I need a compact set that is the reason why
I am doing this okay, so what it means is
00:09:54.480 --> 00:10:02.130
that a family is defined on bigger open set
which contains this closed set okay. Now you
00:10:02.130 --> 00:10:07.440
see what we want to do is we want to really
so you know let me draw a diagram, so here
00:10:07.440 --> 00:10:17.500
is Z naught alright and here is this there
is this disk centred at Z naught radius rho.
00:10:17.500 --> 00:10:23.890
So this is the this centred at Z naught radius
rho and the point is you give me function
00:10:23.890 --> 00:10:29.640
small f in the family script F okay and what
I want to really do is I want to really analyse
00:10:29.640 --> 00:10:36.570
this function in that small disk okay I want
to analyse this function, this function in
00:10:36.570 --> 00:10:44.000
this small disk and how do I do it? So what
I do is you know I have to look closer at
00:10:44.000 --> 00:10:49.261
the small disk, so what I actually do is zoom
in okay and how do I zoom in? So it is by
00:10:49.261 --> 00:10:57.900
scaling okay, so what I do is that I put I
define g for every function f in the family
00:10:57.900 --> 00:11:03.710
I define this g and what is this G? This g
is define with the new variable Zeta and this
00:11:03.710 --> 00:11:12.310
is defined as f of Z naught plus Epsilon times
Zeta I put this condition I make this
00:11:12.310 --> 00:11:21.390
definition, so what is happening is that this
is in the Z plane, so the variable here is
00:11:21.390 --> 00:11:28.570
Z so this is also the complex plane and then
I have another complex plane okay and this
00:11:28.570 --> 00:11:30.010
is the Zeta plane.
00:11:30.010 --> 00:11:40.310
So the variable here is Zeta alright and what
I do is I take this unit disk okay I take
00:11:40.310 --> 00:11:50.190
this unit disk, this is unit disk mod Zeta
less than 1 okay I will take this disk and
00:11:50.190 --> 00:11:56.230
now I am defining this function g as a function
of Zeta okay and what is it is actually you
00:11:56.230 --> 00:12:04.240
know you have just magnified the behaviour
of Ö The value of g at Zeta is the value
00:12:04.240 --> 00:12:10.680
of f at Z naught plus Epsilon times Zeta,
so what it means is that you know if you start
00:12:10.680 --> 00:12:23.080
withÖ And of course I am assuming that Epsilon
is less than rho okay, so you know if I take
00:12:23.080 --> 00:12:30.930
this disk of radius one centred at the origin
with variable Zeta then if I multiply by Epsilon,
00:12:30.930 --> 00:12:36.500
if I take Epsilon Zeta it will become disk
of radius Epsilon okay, so it will be smaller
00:12:36.500 --> 00:12:44.930
disk like this okay and this will have radius
Epsilon okay and then if I had Z naught to
00:12:44.930 --> 00:12:54.760
it I am just translating it to give me a small
disk of radius Epsilon centred at Z naught
00:12:54.760 --> 00:12:57.980
that is what this does okay.
00:12:57.980 --> 00:13:02.280
So you are just multiplying Epsilon to Zeta
then you are adding Z naught so that is a
00:13:02.280 --> 00:13:08.370
translation by Z naught okay. So basically
what you are doing is that when you do this
00:13:08.370 --> 00:13:15.180
you know you are actually looking at this
you are looking at this small disk centred
00:13:15.180 --> 00:13:23.750
at Z naught radius Epsilon and so you know
in principle for what values of Zeta is g
00:13:23.750 --> 00:13:32.140
defined, so you will see that g is defined
forÖshould be . See if mod Zeta is less than
00:13:32.140 --> 00:13:38.760
rho by Epsilon okay then mod Epsilon Zeta
will be less than rho and mod Z naught plus
00:13:38.760 --> 00:13:42.730
Epsilon Zeta will be less than therefore also
less than rho okay.
00:13:42.730 --> 00:13:52.540
See mod Zeta is less than rho by Epsilon means
that you know mod Epsilon Zeta is less than
00:13:52.540 --> 00:13:56.380
rho okay and what does this tell you? This
tells you that the distance of Epsilon Zeta
00:13:56.380 --> 00:14:02.560
from the origin is at most rho and therefore
if you add Z naught to Epsilon Zeta the distance
00:14:02.560 --> 00:14:09.480
of Z naught plus Epsilon Zeta to Z naught
is at most rho, so you see this g is defined
00:14:09.480 --> 00:14:21.140
on this disk and you see this disk is actuallyÖsee
this disk is larger see because you see rho
00:14:21.140 --> 00:14:25.960
the original disk centred at Z naught has
radius rho alright. Now the radius has become
00:14:25.960 --> 00:14:30.900
rho by Epsilon and Epsilon being smaller rho
by Epsilon should be larger if you think of
00:14:30.900 --> 00:14:36.150
Epsilon to be you know Epsilon is less than
rho, so rho by Epsilon is greater than 1 okay.
00:14:36.150 --> 00:14:42.490
So what you have done is by this transformation
this disk centred at Z naught radius rho okay
00:14:42.490 --> 00:14:47.980
if you want to study the function value the
values of the function f there the behaviour
00:14:47.980 --> 00:14:51.950
of the function f there, what you have done
is you have actually zoomed it by constructing
00:14:51.950 --> 00:14:57.150
this function G, so g is a kind of zooming
function okay it is like you have a microscope
00:14:57.150 --> 00:15:02.850
or telescope and you have 10 X zoom 20 X zoom
you see that also in camera these days okay
00:15:02.850 --> 00:15:11.090
you have 5X zoom and so on, so this is so
many X zoom and the zooming factor is 1 by
00:15:11.090 --> 00:15:16.710
Epsilon okay. Original disk of radius rho
has now become zoom to disk of radius rho
00:15:16.710 --> 00:15:26.020
by Epsilon okay and well the point is thatÖ
And of course rho by Epsilon will be slightly
00:15:26.020 --> 00:15:33.900
bigger okay you will get so this is Epsilon,
this thing will be rho by Epsilon and it is
00:15:33.900 --> 00:15:40.350
in this bigger disk open disk that the function
g is defined and studying this function g
00:15:40.350 --> 00:15:47.030
in this bigger disk is the same as studying
the functions f in the original disk okay.
00:15:47.030 --> 00:15:53.610
So you know I will write it like this I will
write g of Zeta is equal to zoom I will think
00:15:53.610 --> 00:16:01.170
of it as zooming, zoom the function f centred
at Z naught by a magnification factor 1 by
00:16:01.170 --> 00:16:09.480
Epsilon and use the variable Zeta okay I will
use this notation okay, so when I say zoom
00:16:09.480 --> 00:16:18.090
f Z naught 1 by Epsilon Zeta you are zooming
the function f centred at Z naught, so your
00:16:18.090 --> 00:16:22.850
zoom is centred at Z naught okay and 1 by
Epsilon is the magnification factor and Zeta
00:16:22.850 --> 00:16:33.050
is the variable of the zoom function okay.
Now wellÖ So this is how you zoom into the
00:16:33.050 --> 00:16:39.070
behaviour of function at a point okay, now
I am given that this function, I am given
00:16:39.070 --> 00:16:45.870
that this family script F is normal suppose
I am given it is normal at Z naught suppose
00:16:45.870 --> 00:16:51.741
I am given normal in the closed disk centred
at Z naught radius rho which means that it
00:16:51.741 --> 00:16:55.740
is actually normal in an open set which contains
that closed disk okay. Suppose that happens
00:16:55.740 --> 00:17:00.090
and let us see what it means for us.
00:17:00.090 --> 00:17:13.620
So let me continue like this suppose now that
you know script F is normal in mod Z minus
00:17:13.620 --> 00:17:25.900
Z naught less than or equal to rho okay. By
that I mean it is normal in a slightly bigger
00:17:25.900 --> 00:17:35.110
open disk okay. Then so what does normality
means, it means actually normally sequentially
00:17:35.110 --> 00:17:41.289
compact is the correct notion of compactness
for as when we do complex analysis okay, so
00:17:41.289 --> 00:17:45.810
what it means is that you give me any sequence
of functions in script F I can always find
00:17:45.810 --> 00:17:52.460
a subsequence which converges normally okay
it is normal sequential compactness, it is
00:17:52.460 --> 00:18:03.379
sequential compactness with respect to normal
convergence. So given any sequence f n in
00:18:03.379 --> 00:18:21.129
script F we can find a subsequence let me
call that as f n k that converges normally
00:18:21.129 --> 00:18:31.030
in mod Z minus Z naught less than rho okay
I can put less than or equal to okay.
00:18:31.030 --> 00:18:37.110
In fact mod Z minus Z naught less than or
equal to rho is actually compact therefore
00:18:37.110 --> 00:18:42.149
it will converge even uniformly there okay.
So it converges normally so what happens is
00:18:42.149 --> 00:18:52.480
that you will get this subsequence of f n
k of Z it will go to some function f Z a and
00:18:52.480 --> 00:18:57.130
f is a normal limit of Meromorphic functions
so you know what is going to happen you have
00:18:57.130 --> 00:19:00.900
body seen this a normal limit of Meromorphic
function is either identically infinity or
00:19:00.900 --> 00:19:07.009
it is also a Meromorphic function it could
even be analytic okay, so then of course then
00:19:07.009 --> 00:19:21.009
as we have seen either f is identically infinity
or f is Meromorphic this is something that
00:19:21.009 --> 00:19:30.600
we have seen but the point I want to emphasise
is that is the following issue, you see you
00:19:30.600 --> 00:19:39.279
have these functions f n k they are converging
to f alright and what is happening is that
00:19:39.279 --> 00:19:46.320
now you know mind you I have the zoomed function
okay I have these zoomed functions by zooming
00:19:46.320 --> 00:19:48.630
factor Epsilon is than rho.
00:19:48.630 --> 00:19:56.360
So look at the zoom functions, so what happens
is that I take g N k which is actually the
00:19:56.360 --> 00:20:06.580
zooming ofÖso g n k of Zeta is the zooming
of f n k centred at Z naught with the magnification
00:20:06.580 --> 00:20:14.539
factor 1 by Epsilon and I am using the variable
Zeta and similarly I get zoomed function for
00:20:14.539 --> 00:20:24.149
the limit function also, so I get g also is
equal to zoomed function of f centred at Z
00:20:24.149 --> 00:20:32.380
naught magnification factor 1 by Epsilon my
variable is Zeta okay, so this is g of Zeta
00:20:32.380 --> 00:20:42.580
so I have that and of course you know if f
n k converges to f then the zoomed function
00:20:42.580 --> 00:20:47.619
of the f n k namely the g n k will converge
to the zoomed function of f which is g and
00:20:47.619 --> 00:20:53.530
this is just because the zooming is just the
scaling followed by a translation okay.
00:20:53.530 --> 00:21:00.909
So after all what is the zooming? The zooming
is you multiply, you multiply Zeta by Epsilon
00:21:00.909 --> 00:21:06.460
and then you add Z naught to it okay that
is just a in fact it is actually a bilinear
00:21:06.460 --> 00:21:11.910
transformation it is a linear fractional transformation,
it is a Mobius transformation okay so it will
00:21:11.910 --> 00:21:22.240
preserve all properties convergence everything
okay. So you get this, now you see so let
00:21:22.240 --> 00:21:28.611
me write this, this is normally so here also
I will get so this will imply that this will
00:21:28.611 --> 00:21:38.169
also converge normally okay and the point
I want to make is that you see I have assumed
00:21:38.169 --> 00:21:46.619
that the original family is normal okay and
we have seen namely Martyís theorem that
00:21:46.619 --> 00:21:54.029
normality is equivalent to spherical derivatives
being normally bounded okay.
00:21:54.029 --> 00:22:01.039
So if you recall you know Martyís theorem
says that a family of Meromorphic functions
00:22:01.039 --> 00:22:04.429
on a domain, the domain could even be a domain
in the extended complex plane it could contain
00:22:04.429 --> 00:22:11.669
the point at infinity. Such a family is normally
if and only if the spherical derivatives of
00:22:11.669 --> 00:22:15.610
those functions, the family of spherical derivatives
is normally uniformly bounded so it should
00:22:15.610 --> 00:22:19.940
be on every compact subset of the domain the
family of spherical derivatives should have
00:22:19.940 --> 00:22:29.629
uniform bound okay. So by Martyís theorem
you know that the family script F itself has
00:22:29.629 --> 00:22:33.460
if you take the spherical derivatives than
that has uniform bound.
00:22:33.460 --> 00:22:48.340
So let me write that down by Martyís theorem
script F has uniform bound, script F hash,
00:22:48.340 --> 00:22:53.289
so when I put script F upper hash means the
family of spherical derivatives of the functions
00:22:53.289 --> 00:23:02.740
in f okay. This has a uniform bound in mod
Z minus Z naught less than or equal to rho
00:23:02.740 --> 00:23:07.250
I have this okay because this mod Z minus
Z naught less than or equal to rho is a compact
00:23:07.250 --> 00:23:11.539
set alright and on a compact set I have a
uniform bound for spherical derivatives and
00:23:11.539 --> 00:23:19.520
that is equivalent to normality okay this
is something that I have this is Martyís
00:23:19.520 --> 00:23:20.520
theorem.
00:23:20.520 --> 00:23:31.490
So what does it mean? It means that if you
take any function f so let me not
00:23:31.490 --> 00:23:38.899
use f let me use h because I have already
use f for the limit, if you take a h in f
00:23:38.899 --> 00:23:56.860
then h hash this is the spherical derivatives
of h than m for all h and same m right, so
00:23:56.860 --> 00:24:02.639
you have this you have uniform bound alright
and this bound applies to all members of spherical
00:24:02.639 --> 00:24:08.940
derivatives of all members of the family f
script F so it applies also to the f n case
00:24:08.940 --> 00:24:22.070
okay, so in particular what you will have
is you know, you will have that f n k hash
00:24:22.070 --> 00:24:28.670
the spherical derivatives you see these things
are going to be bounded by m it is going to
00:24:28.670 --> 00:24:34.019
happen because all these f n case are anyway
in the family okay but you know we have also
00:24:34.019 --> 00:24:39.429
seen this earlier if a family of Meromorphic
functions converges normally to limit function
00:24:39.429 --> 00:24:41.559
then the spherical derivatives will also converge
okay.
00:24:41.559 --> 00:24:53.700
Taking the spherical derivative will respect
convergence okay, so you also have from here
00:24:53.700 --> 00:25:03.779
if you want, so I need to go like this that
f N k if I take the spherical derivatives
00:25:03.779 --> 00:25:14.739
that will go to f hash okay you have this
okay and of course you know if I by the same
00:25:14.739 --> 00:25:20.129
token if I do it to the g n k the spherical
derivatives of g n k will go to the spherical
00:25:20.129 --> 00:25:27.850
derivatives of g okay mind you g n k is just
f n k translated and scaled, so when you translate
00:25:27.850 --> 00:25:31.960
and scale a function its nature does not change,
so if you take a Meromorphic function and
00:25:31.960 --> 00:25:35.049
translate and scale it you will again get
a Meromorphic function okay.
00:25:35.049 --> 00:25:40.649
After all translating and scaling is not going
to change the nature of singularities okay
00:25:40.649 --> 00:25:44.730
and if you translate and scale an analytic
function you will again get an analytic function
00:25:44.730 --> 00:25:52.529
okay and so on and so forth, so g n k are
also Meromorphic functions and mind you you
00:25:52.529 --> 00:25:57.889
have to remember that I am considering the
f n k in mod Z minus Z naught less than or
00:25:57.889 --> 00:26:07.619
equal to rho but I am actually considering
the g n k in mod Zeta less than rho by Epsilon
00:26:07.619 --> 00:26:13.489
okay that is the zoomed disk centred at the
origin where the zoom functions are being
00:26:13.489 --> 00:26:24.179
looked at, so let me write this here on mod
Zeta less than rho by Epsilon okay this is
00:26:24.179 --> 00:26:29.119
the Epsilon less than rho alright so this
is a quantity greater than 1, fine.
00:26:29.119 --> 00:26:39.019
So I have this and I shall have the same thing
for these guys I will also have g n k hash
00:26:39.019 --> 00:26:50.690
going to g hash alright and of course it is
again normal
00:26:50.690 --> 00:26:59.470
this also normal okay and the point is that
but there is an inequality coming out, see
00:26:59.470 --> 00:27:09.499
what is g n k? g n k of Zeta is by definition
g n k of Z naught is Epsilon Zeta this is
00:27:09.499 --> 00:27:18.139
the zoomed function, so what is g n k hash
of Zeta, what is it? Say it is by definition
00:27:18.139 --> 00:27:24.499
it is supposed to be 2 times the modulus of
the derivative of g n k with respect to Zeta
00:27:24.499 --> 00:27:31.070
divided by 1 plus modulus of g n k of Zeta
the whole square, this is the definition of
00:27:31.070 --> 00:27:37.669
spherical derivative okay and but then this
is what is the g n k dash Zeta in the numerator?
00:27:37.669 --> 00:27:42.820
It is the derivative of g n k with respect
to Zeta it is derivative with respect to Zeta
00:27:42.820 --> 00:27:48.139
okay and mind you this formula for spherical
derivative also works when Zeta is a pole
00:27:48.139 --> 00:27:53.531
okay we have seen that you know by continuity
if the pole is of higher-order then the spherical
00:27:53.531 --> 00:27:57.940
derivative at the pole is 0, if the pole is
a simple pole then spherical derivatives at
00:27:57.940 --> 00:28:01.889
the pole is actually 2 divided by modulus
of the residue of the pole which is something
00:28:01.889 --> 00:28:07.149
that we have seen already okay, so there is
no problem if Zeta is a pole okay this formula
00:28:07.149 --> 00:28:13.779
works but then you know if I differentiate
g n k with respect to Zeta it is the same
00:28:13.779 --> 00:28:19.169
as differentiating f n k and then you know
by the chain rule differentiating the argument
00:28:19.169 --> 00:28:22.830
of the f n k with respect to Zeta and that
will bring me a multiple of Epsilon.
00:28:22.830 --> 00:28:30.559
So what I will get is, I will get Epsilon
times f n k hash of Z this is what I will
00:28:30.559 --> 00:28:36.299
get okay not Z in fact I have to plug-in the
right substitution scale to variable Z naught
00:28:36.299 --> 00:28:44.870
plus Epsilon so this is what I will get okay
but the f n k are all bounded by what? The
00:28:44.870 --> 00:28:52.820
f n k are bounded by m alright so this is
bounded by Epsilon m okay so the moral of
00:28:52.820 --> 00:29:00.951
the story is that because you because of your
scaling okay. See what you must understand
00:29:00.951 --> 00:29:09.620
is that you zoomed the small disk centered
at Z naught radius rho to a larger disk okay
00:29:09.620 --> 00:29:14.159
you scaled and you got the zoom function but
what has happened is that the bound for the
00:29:14.159 --> 00:29:19.539
spherical derivative have become smaller because
it has got multiplied by this the inverse
00:29:19.539 --> 00:29:27.109
of the zooming factor. Zooming factors 1 by
Epsilon where Epsilon is very small okay then
00:29:27.109 --> 00:29:31.359
bound for the zoom function becomes much more
smaller okay.
00:29:31.359 --> 00:29:38.859
Now that is what it says, now this is true
for every m K but you know g n k hash converges
00:29:38.859 --> 00:29:44.980
normally to g hash and each of the g n k is
bounded by Epsilon m therefore g hash will
00:29:44.980 --> 00:29:52.100
also be bounded by Epsilon M, so these 2 put
together will tell you that g hash will also
00:29:52.100 --> 00:30:04.299
be bounded by Epsilon m okay so this will
happen alright, so that is because of normal
00:30:04.299 --> 00:30:10.890
convergence okay. So the moral of the story
is that so this is what is happening, so what
00:30:10.890 --> 00:30:18.960
have we shown so far? You take a point Z naught
a point where a family is normal okay and
00:30:18.960 --> 00:30:24.639
then you take any sequence in the family you
will get a convergence of sequence, normally
00:30:24.639 --> 00:30:30.460
convergence of sequence you look at those
functions in the subsequence okay and zoomed
00:30:30.460 --> 00:30:31.460
them.
00:30:31.460 --> 00:30:40.299
Then the zoomed functions their spherical
derivatives becomes very small and in fact
00:30:40.299 --> 00:30:48.519
the zoom functions if you take the limit of
the zoom functions, the limit function will
00:30:48.519 --> 00:30:53.549
be a Meromorphic function whose spherical
derivatives is very small that is what it
00:30:53.549 --> 00:30:59.320
says okay and now you know, so this is the
first step of the argument. Now what I am
00:30:59.320 --> 00:31:08.169
going to do is I am going to introduce a level
of complication by doing the following thing,
00:31:08.169 --> 00:31:12.300
what you do is instead of considering a single
Epsilon you consider sequence of Epsilon going
00:31:12.300 --> 00:31:22.279
to 0. Imagine you are considering I am using
the same Epsilon here for all the functions
00:31:22.279 --> 00:31:29.559
okay but suppose for g n I use an Epsilon
n okay and such that the Epsilon n are going
00:31:29.559 --> 00:31:35.669
to go to 0 okay that means I am doing ultra-zooming,
I am zooming into smaller and smaller and
00:31:35.669 --> 00:31:38.269
smaller and smaller neighbourhood of Z naught
okay.
00:31:38.269 --> 00:31:42.730
If I do that then what will happen is that
you can guess what is going to happen? This
00:31:42.730 --> 00:31:52.889
g hash will be 0 because see g hash will
the thing on the left side g n k hash that
00:31:52.889 --> 00:31:59.440
is going to be bounded by Epsilon n k m okay
and if I take limit as N k tends to 0 Epsilon
00:31:59.440 --> 00:32:04.669
NK is going to go to 0, so Epsilon NK m is
going to go to 0 because m is anyway finite
00:32:04.669 --> 00:32:09.230
quantity therefore g hash is going to go to
0 and what does it mean if g hash goes to
00:32:09.230 --> 00:32:14.409
0? It means that g is a constant. A spherical
derivative of function cannot be 0 unless
00:32:14.409 --> 00:32:15.909
it is a constant.
00:32:15.909 --> 00:32:20.030
It can be even be constant function that is
uniformly infinity mind you for the constant
00:32:20.030 --> 00:32:25.299
function which is uniformly infinity also
the spherical derivatives is 0 okay in line
00:32:25.299 --> 00:32:28.139
with the fact that the philosophy that whenever
you have a constant function the derivative
00:32:28.139 --> 00:32:41.669
of any type should be okay. So the moral of
the story is that if you zoom in a family
00:32:41.669 --> 00:32:51.190
of convergent, family of functions at a point
okay then the limit function is going to be
00:32:51.190 --> 00:32:57.409
constant that is the full point okay, so let
me add that level of complication but now
00:32:57.409 --> 00:32:59.159
there is something very nice.
00:32:59.159 --> 00:33:07.409
What is happening is that so let me put this
in a different color here whatever is happening
00:33:07.409 --> 00:33:19.899
happens in mod Zeta less than rho by Epsilon
okay this was the disk where everything is
00:33:19.899 --> 00:33:27.649
happening okay but now you know if I make
these Epsilon goes to 0 by choosing Epsilon
00:33:27.649 --> 00:33:34.840
N which go to 0 something nice is going to
happen as Epsilon N go to 0 rho by Epsilon
00:33:34.840 --> 00:33:40.299
N go to infinity, so your disks are becoming
bigger and bigger and bigger and the beautiful
00:33:40.299 --> 00:33:44.869
thing is that any compact subset of the complex
plane, the Zeta plane will be contained in
00:33:44.869 --> 00:33:52.869
a sufficiently large disk and therefore you
get a family of functions for which you can
00:33:52.869 --> 00:33:59.350
talk about normal convergence on the whole
plane okay, so this g will be defined
00:33:59.350 --> 00:34:06.779
on the whole plane and you will get a constant
function okay that is the whole point, so
00:34:06.779 --> 00:34:10.139
let me write this down.
00:34:10.139 --> 00:34:28.109
Now let us consider a sequence of Epsilon
say Epsilon n going to 0, so Epsilon going
00:34:28.109 --> 00:34:40.369
to 0 plus so they are all positive and you
know let me say even I mean say even decreasing
00:34:40.369 --> 00:34:55.599
okay anyway sequence going to 0 eventually
it is going to be a decreasing sequence alright
00:34:55.599 --> 00:35:00.589
strictly decreasing sequence you can think
of it as smaller and smaller neighbourhoods
00:35:00.589 --> 00:35:13.460
then what happens is that you we get for you
know so here is the funny thing you define
00:35:13.460 --> 00:35:23.960
g m all Zeta to be you zoom f so you take
f you write Z naught plus Epsilon n Zeta,
00:35:23.960 --> 00:35:30.120
this is actually the zooming of the function
f centred at Z naught scaling factor is now
00:35:30.120 --> 00:35:37.180
1 by Epsilon n and the variable I am using
a Zeta okay and mind you this 1 by Epsilon
00:35:37.180 --> 00:35:41.819
n is going to go to infinity because Epsilon
n is going to 0 plus 1 by Epsilon n is going
00:35:41.819 --> 00:35:47.590
to infinity that means I am actually I am
doing ultra-zooming you know at the point
00:35:47.590 --> 00:35:50.520
Z naught and I am looking at the zoom functions
alright.
00:35:50.520 --> 00:35:59.349
Now see this is defined as I told you this
is defined in mod Zeta less than rho by Epsilon
00:35:59.349 --> 00:36:09.089
n and the point is that this goes to infinity
this goes to plus infinity as N tends to infinity
00:36:09.089 --> 00:36:19.059
because Epsilon N goes to 0 rho is a fixed
quantity alright and note that what will happen
00:36:19.059 --> 00:36:31.369
is that g n k hash will go to g hash as before
okay this will be normal but the beautiful
00:36:31.369 --> 00:36:39.410
thing is this will be normally on C on the
whole of C that is the beautiful part. Earlier
00:36:39.410 --> 00:36:49.020
this g n k hash going to g hash was on only
on this bounded domain it was on mod Zeta
00:36:49.020 --> 00:36:54.299
less than rho by Epsilon, it was normal on
the convergence g n k hash going to g hash
00:36:54.299 --> 00:36:59.069
wasÖit was normal convergence on this bounded
domain but now since this Epsilons are becoming
00:36:59.069 --> 00:37:07.079
smaller the g n are you being defined on bigger
and bigger domains okay and that is an increasing
00:37:07.079 --> 00:37:13.200
sequence of open disk centred at the origin
that will eventually cover the whole plane.
00:37:13.200 --> 00:37:19.170
So it will eventually cover any compact subsets
of the plane and therefore on any compact
00:37:19.170 --> 00:37:24.350
subset of the plane I can say that this sequence
is going to converge normally because I will
00:37:24.350 --> 00:37:28.130
have to consider the sequence only beyond
a certain stage okay. See whenever you are
00:37:28.130 --> 00:37:33.240
looking at convergence the 1st finitely no
matter how large, the 1st finitely many terms
00:37:33.240 --> 00:37:42.809
do not matter okay, so the moral of the story
is that the advantage of taking smaller and
00:37:42.809 --> 00:37:48.619
smaller Epsilons is that you are zooming into
the point, you are zooming into the behaviour
00:37:48.619 --> 00:37:53.980
of the functions at that point but then you
get a limit function which is defined on the
00:37:53.980 --> 00:37:59.609
whole plane, so you get a Meromorphic function
on the plane, so this g it will be a Meromorphic
00:37:59.609 --> 00:38:05.320
function on the whole plane okay g will be
a Meromorphic function on the whole plane
00:38:05.320 --> 00:38:16.109
and the nice thing is that as I was telling
you the bound for the spherical derivatives
00:38:16.109 --> 00:38:18.040
of g n will be what?
00:38:18.040 --> 00:38:26.400
It will be Epsilon n k times m, so you will
get g n k hash is bounded by Epsilon n k times
00:38:26.400 --> 00:38:35.240
m okay because you know earlier we got the
bound as Epsilon m for g n k okay but now
00:38:35.240 --> 00:38:41.510
the Epsilon is Epsilon n k in this case so
I will get this okay and this goes to g hash
00:38:41.510 --> 00:38:53.980
and as k tends to infinity and well this was
to 0 as k tends to infinity. So the moral
00:38:53.980 --> 00:39:03.020
of the story is g hash 0 and this implies
that g is a constant
00:39:03.020 --> 00:39:06.360
and when I say g is a constant mind you g
could be the constant function which is identically
00:39:06.360 --> 00:39:12.690
infinity that is also allowed okay. So you
know how do you argue that g is a constant
00:39:12.690 --> 00:39:18.750
mind you g becomes GÖ anyway the limit function
g is Meromorphic on the whole plane okay.
00:39:18.750 --> 00:39:27.599
So it has only poles on the plane okay and
outside the poles it is analytic function
00:39:27.599 --> 00:39:33.829
okay but outside the poles the spherical derivatives
is 0 means ordinary derivatives is 0 okay
00:39:33.829 --> 00:39:39.980
and if ordinary derivative is 0 the function
is constant, so except for these isolated
00:39:39.980 --> 00:39:43.890
set of points which are poles everywhere else
the function is constant and those points
00:39:43.890 --> 00:39:49.850
cannot be there you cannot have on a single
pole because if you have a pole as the points
00:39:49.850 --> 00:39:54.070
approach the pole the function goes, the modulus
of the function value goes to infinity. How
00:39:54.070 --> 00:39:58.109
can it be constant in a deleted neighbourhood
of a pole? So that forces the function has
00:39:58.109 --> 00:40:01.299
to be identically constant that is how you
get g is identically constant g is only a
00:40:01.299 --> 00:40:07.490
Meromorphic function okay. So this is the
important thing, the important thing is that
00:40:07.490 --> 00:40:11.600
if you take a normal family, so what have
we prove?
00:40:11.600 --> 00:40:18.740
You take a point where a family is normal
okay and you take any sequence in the normal
00:40:18.740 --> 00:40:23.630
family you can find a normally convergence
subsequence. If you take those sequence of
00:40:23.630 --> 00:40:30.610
functions and do zoom in at the point you
are able to constructÖsee the zoom functions
00:40:30.610 --> 00:40:40.910
will converge to a constant function that
is what it says okay. If you look at normally
00:40:40.910 --> 00:40:47.610
convergent family of Meromorphic function
zoomed at a point you will get only a constant
00:40:47.610 --> 00:40:51.980
function limit that is what you are saying
okay this is the behaviour of a normal family
00:40:51.980 --> 00:41:00.279
at a point okay and what I want to do next
is introduce another level of complication
00:41:00.279 --> 00:41:05.039
okay and this next level of complication is
to vary Z naught also okay.
00:41:05.039 --> 00:41:11.400
So instead of considering a point Z naught
all this time I had fixed Z naught but now
00:41:11.400 --> 00:41:16.950
what you do is you take a sequence point Z
N which goes to Z naught okay and at each
00:41:16.950 --> 00:41:24.520
Z n you zoom to Epsilon N or the function
f n and do this process okay and you will
00:41:24.520 --> 00:41:30.940
still get the same result okay and the beautiful
thing is that whatever you get there okay
00:41:30.940 --> 00:41:39.500
that behaviour is good enough to characterise
normal families okay and the point about Zalcman's
00:41:39.500 --> 00:41:46.980
Lemma is the negativity of that statement
okay, so roughly Zalcman's Lemma is a Lemma
00:41:46.980 --> 00:41:52.510
that is actually a theorem which will explain
when family is not normal it will give you
00:41:52.510 --> 00:41:56.720
a sufficient conditions for a family to be
not normal and guess what the condition will
00:41:56.720 --> 00:42:01.789
be, the condition will be at you can do all
this but the limit function that you get g
00:42:01.789 --> 00:42:07.730
it will be non-constant that will be the only
difference.
00:42:07.730 --> 00:42:12.220
You will get a limit function which will be
a non-constant Meromorphic function, so its
00:42:12.220 --> 00:42:18.130
spherical derivatives will not be 0 okay and
that is the characterisation of non-normality
00:42:18.130 --> 00:42:27.049
okay. See this is the motivation for Zalcman's
Lemma okay, so in my next lecture I will you
00:42:27.049 --> 00:42:32.940
how to instead of considering Z naught consider
a sequence ZN which goes to Z naught and have
00:42:32.940 --> 00:42:37.610
the same argument okay and continue with Zalcman's
Lemma alright, so I will stop here.