WEBVTT
Kind: captions
Language: en
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So see the point is that you know we are trying
to look at families of Meromorphic functions,
00:01:34.810 --> 00:01:41.090
we are trying to look at trying to look at
normally convergent families of Meromorphic
00:01:41.090 --> 00:01:45.020
functions, the reason is because you want
to do topology on the space of Meromorphic
00:01:45.020 --> 00:01:56.039
functions okay and because that is the kind
of you know set up that you need able to prove
00:01:56.039 --> 00:02:06.460
the big Picard theorem and the little Picard
theorem okay. So you know let me briefly remind
00:02:06.460 --> 00:02:13.270
you if you take inspiration from topology
alright, what is the topology that you will
00:02:13.270 --> 00:02:18.549
put on the space of functions, normally if
you have a topological space and you have
00:02:18.549 --> 00:02:23.080
you are looking at real valued or complex
valued functions then you will restrict yourself
00:02:23.080 --> 00:02:31.700
to continuous bounded real valued or complex
valued chance that is that is a banach algebra
00:02:31.700 --> 00:02:37.170
and it is also a topological space it is complete
as a metric space, the metric is induce by
00:02:37.170 --> 00:02:43.610
a norm and the norm is essentially convergence
in that norm is actually equivalent to uniform
00:02:43.610 --> 00:02:45.120
convergence okay.
00:02:45.120 --> 00:02:53.290
So the moral of the story is that if you are
going to inspiration from topology okay then
00:02:53.290 --> 00:03:00.350
trying to do topology on the space of functions
is the same as you studying functions under
00:03:00.350 --> 00:03:06.980
uniform convergence okay, now this is topological
so that means that you are only worried about
00:03:06.980 --> 00:03:12.230
continuous functions okay but now suppose
you come to complex analysis then you are
00:03:12.230 --> 00:03:18.100
worried about holomorphic functions or analytic
functions okay and we are worried about something
00:03:18.100 --> 00:03:22.620
even worse, we are worried about Meromorphic
functions which are actually which have the
00:03:22.620 --> 00:03:30.290
additional problem that they can have poles
at finitely many at a set of isolated points
00:03:30.290 --> 00:03:39.150
okay and of course finitely many if your domain
is compact is the whole Riemann sphere on
00:03:39.150 --> 00:03:40.620
the extended plane.
00:03:40.620 --> 00:03:48.799
So if you are looking at say holomorphic functions
on a domain, a domain in the complex plane
00:03:48.799 --> 00:03:51.430
or a domain in the extended complex plane
that does not matter suppose if you looking
00:03:51.430 --> 00:03:56.550
at the analytic functions or holomorphic functions
on the domain and you want to do topology
00:03:56.550 --> 00:04:03.640
on that set of functions okay. Mind you we
have body seen that the set of functions it
00:04:03.640 --> 00:04:10.879
is a ring in fact and then if you look at
Meromorphic functions it is a okay. It has
00:04:10.879 --> 00:04:16.500
algebraic structure but we are now worried
about the topology, so you want to do topology
00:04:16.500 --> 00:04:21.870
on the set of analytic functions on a domain
or you want to do more generally topology
00:04:21.870 --> 00:04:28.560
on the set of Meromorphic functions on a domain
which means analytic except for poles then
00:04:28.560 --> 00:04:34.040
you know if you try to draw inspiration from
ology you would just say that this is the
00:04:34.040 --> 00:04:42.500
same as studying them under uniform convergence
okay because topologically uniform convergence
00:04:42.500 --> 00:04:50.190
corresponds to convergence in the space of
functions okay but if you come to the case
00:04:50.190 --> 00:04:55.610
of analytic functions okay this is not the
right thing because you do not get uniform
00:04:55.610 --> 00:04:56.780
convergence.
00:04:56.780 --> 00:05:02.210
So I was trying to explain to you last time
that for example if you take the geometric
00:05:02.210 --> 00:05:08.010
series you take the functions that correspond
to partial sums of the geometric series okay
00:05:08.010 --> 00:05:14.900
then they are of course polynomials and they
converge absolutely in the unit disk to the
00:05:14.900 --> 00:05:20.110
sum of the geometric series which is one by
one minus the variable okay, so the geometric
00:05:20.110 --> 00:05:26.880
series is 1 plus Z plus Z square and so on
where Z is a variable and you are restricting
00:05:26.880 --> 00:05:31.640
Z to be in the unit disk that means you are
making mod Z is less than 1 then one plus
00:05:31.640 --> 00:05:39.210
Z plus Z square and so on that converges to
one by one minus Z that is the high school
00:05:39.210 --> 00:05:43.930
formula or geometry series. Now the point
is that this convergence is absolute on the
00:05:43.930 --> 00:05:49.830
unit disk there is no problem about that but
it is not uniform on the whole unit disk okay
00:05:49.830 --> 00:05:53.990
that was something that I told you I asked
you to check it as an exercise I hope you
00:05:53.990 --> 00:06:00.900
have done it, it is very easy to do convergence
is not uniform on the whole unit disk is only
00:06:00.900 --> 00:06:04.170
uniform on compact subset of the unit disk
okay.
00:06:04.170 --> 00:06:11.720
So you do not get uniform convergence but
you get only normal convergence, so the moral
00:06:11.720 --> 00:06:19.069
of the story is that when you want to do topology
on a space of holomorphic functions you should
00:06:19.069 --> 00:06:23.940
not look at them under uniform convergence
you must look at them under normal convergence
00:06:23.940 --> 00:06:31.139
so that is the 1st moral 1st lesson to be
learned that is what you should keep at the
00:06:31.139 --> 00:06:37.790
back of your mind, so that is one thing so
you know more generally if you want to extend
00:06:37.790 --> 00:06:45.440
this to Meromorphic functions in serve more
complicated because now you have poles okay
00:06:45.440 --> 00:06:54.590
and the other thing is that of course you
know by going from uniform limits to normal
00:06:54.590 --> 00:07:01.270
limits things are going to be good okay because
normal convergence is just uniform convergence
00:07:01.270 --> 00:07:10.540
on compact sets okay it is weaker than uniform
convergence as it is okay but it is good enough
00:07:10.540 --> 00:07:17.789
for our purposes because as I told you in
the last lecture if we have sequence of holomorphic
00:07:17.789 --> 00:07:24.180
functions which converge normally to limit
function then that limit function is also
00:07:24.180 --> 00:07:27.160
holomorphic okay this is something that I
explained last time okay.
00:07:27.160 --> 00:07:33.900
Essentially it uses Clausius theorem and Morera's
theorem and the fact that analyticity is or
00:07:33.900 --> 00:07:41.710
holomorphicity is a local property. So therefore
there is no harm in relaxing the condition
00:07:41.710 --> 00:07:47.039
of uniform convergence with the condition
of normal convergence. It means uniform convergence
00:07:47.039 --> 00:07:51.229
only on compact sets okay and I also told
you philosophically why that is good enough
00:07:51.229 --> 00:07:55.319
for complex analysis because the moment you
say uniform convergence on compact sets you
00:07:55.319 --> 00:08:00.720
get uniform convergence on closed disk because
they are also compact and therefore you get
00:08:00.720 --> 00:08:07.150
uniform convergence on sufficiently small
open disk okay that is good enough for the
00:08:07.150 --> 00:08:12.780
analysis for the differentiation theory and
then for the integration theory also it is
00:08:12.780 --> 00:08:17.470
good because whenever you integrate on the
contour, the contour is a compact set therefore
00:08:17.470 --> 00:08:21.880
you will get uniform convergence on the contour
okay, so that helps in the integration theory,
00:08:21.880 --> 00:08:27.280
so for all practical purposes uniform convergence
on compact sets that is normal convergence
00:08:27.280 --> 00:08:31.620
is good enough okay, so that is what we have
to worry about.
00:08:31.620 --> 00:08:37.789
Now the other important thing that I want
to tell you is that you see at least if you
00:08:37.789 --> 00:08:43.350
are working with Meromorphic functions you
know the Meromorphic functions have poles
00:08:43.350 --> 00:08:49.560
okay, so at a pole a function is going to
behave in a bad way in the sense that the
00:08:49.560 --> 00:08:54.890
modulus of the function is going to blow up
to infinity okay, so for example that is one
00:08:54.890 --> 00:08:59.540
of the characterisations of a pole, the limit
of the function as you approach a poll is
00:08:59.540 --> 00:09:08.580
going to infinity and by that time limit goes
to the point at infinity okay and of course
00:09:08.580 --> 00:09:15.440
here you are using the topology on extended
complex plane namely the complex plane the
00:09:15.440 --> 00:09:21.650
point at infinity given by the one-point compactification
which makes it holomorphic to the Riemann
00:09:21.650 --> 00:09:23.800
sphere okay.
00:09:23.800 --> 00:09:30.260
Now is another pathology and that is the pathology
that I was trying to explain what is the and
00:09:30.260 --> 00:09:34.360
of the last lecture, so the pathology was
the following, you take the exterior of the
00:09:34.360 --> 00:09:41.970
unit disk or Z greater than 1 that is a variable,
we are on the Z plane the complex plane and
00:09:41.970 --> 00:09:46.070
you are taking the exterior of the unit disk
mod Z is greater than 1 and what you are doing
00:09:46.070 --> 00:09:52.730
as you are looking at these functions powers
of Z you are looking at one which is Z power
00:09:52.730 --> 00:09:58.990
zeros if you want then as Z square and Z cube
and so on okay. Now that is a sequence of
00:09:58.990 --> 00:10:04.579
functions and a point is that this sequence
of functions you can see point wise it will
00:10:04.579 --> 00:10:12.290
go to infinity because sense mod Z is greater
than 1 mod Z to the n to the power of n is
00:10:12.290 --> 00:10:17.060
going to go to infinity because it is for
a real number greater than 1 you know its
00:10:17.060 --> 00:10:25.190
higher powers will diverse to infinity, so
mod Zůso Z power n so this sequence of
00:10:25.190 --> 00:10:30.279
function is going to converge point wise to
the function with the constant function at
00:10:30.279 --> 00:10:31.279
infinity.
00:10:31.279 --> 00:10:36.850
Namely it is a function which associate every
point the value at infinity, so you have to
00:10:36.850 --> 00:10:42.730
worry about this crazy function okay so this
is a pathology that happens that you have
00:10:42.730 --> 00:10:49.310
to take care of and the point is that therefore
we are forced to introduce a function call
00:10:49.310 --> 00:10:54.839
infinity okay and this function infinity is
what it is just the constant function infinity,
00:10:54.839 --> 00:11:02.510
namely it is the function which maps every
value to infinity that is what it is and then
00:11:02.510 --> 00:11:06.790
if you think of that as a function I mean
it is a function of course theoretically if
00:11:06.790 --> 00:11:12.380
you want it is a function from your domain
to the extended complex plane because after
00:11:12.380 --> 00:11:18.860
all in the extended complex plane infinity
is a valid point okay it is a member of that
00:11:18.860 --> 00:11:25.600
site, so you can really think of the function
infinity as he constant function taking the
00:11:25.600 --> 00:11:31.450
value at infinity provided you extend your
values to not just complex values but also
00:11:31.450 --> 00:11:35.480
the extended plane you include the value at
infinity that is one thing.
00:11:35.480 --> 00:11:41.470
So in that sense you can say that this sequence
of functions f n of Z is equal to Z power
00:11:41.470 --> 00:11:49.250
n that converges to infinity you can say that
and when I say that converges to infinity
00:11:49.250 --> 00:11:55.320
I mean that it converges point wise in the
exterior of the unit circle to the function
00:11:55.320 --> 00:12:02.630
which is infinity okay. So you have this is
very nice situation, it is a very nice pathology
00:12:02.630 --> 00:12:09.230
you have these Z power n which are all holomorphic
functions in fact they are entire functions
00:12:09.230 --> 00:12:15.680
they are just polynomials and they converge
to the function infinity in the exterior of
00:12:15.680 --> 00:12:23.540
the unit disk the convergence is again a normal
convergence it is uniform on compact sets
00:12:23.540 --> 00:12:28.459
okay. It is still a normal convergence is
not just a point wise convergence but it is
00:12:28.459 --> 00:12:31.649
in fact even a normal convergence in a way
I will explain to you.
00:12:31.649 --> 00:12:38.360
So what is the moral of the story? The moral
of the story is you have sequence of holomorphic
00:12:38.360 --> 00:12:44.440
functions you have sequence of analytic functions
which is converging to the function infinity
00:12:44.440 --> 00:12:51.019
normally that also happens. You see this is
the extreme case that happens and this also
00:12:51.019 --> 00:12:59.700
has to be taken care of in our arguments okay
and mind you if this is happening for holomorphic
00:12:59.700 --> 00:13:03.500
functions will happen also for Meromorphic
functions because you know holomorphic functions
00:13:03.500 --> 00:13:09.829
are very good Meromorphic functions are worst
because they have poles, so even for a family
00:13:09.829 --> 00:13:15.160
of holomorphic functions even for a family
of analytic functions if you can get normal
00:13:15.160 --> 00:13:20.829
convergence the functions which is infinity
okay you should expect the same thing happen
00:13:20.829 --> 00:13:22.209
also for Meromorphic functions.
00:13:22.209 --> 00:13:29.709
So what I am trying to tell you is that if
you sum up all this if you want to study topology
00:13:29.709 --> 00:13:34.149
on the space of Meromorphic functions 1st
of all you must study with respect to normal
00:13:34.149 --> 00:13:40.329
convergence okay the 2nd thing is you have
to introduce this function keeping mind this
00:13:40.329 --> 00:13:48.430
function which is the function infinity okay
and then you have to justify this business
00:13:48.430 --> 00:14:02.690
of trying to make sense of normal convergence
okay and so let me begin by trying to you
00:14:02.690 --> 00:14:08.910
know explain at least in this particular case
where this normal convergence comes from okay,
00:14:08.910 --> 00:14:20.180
so let us do the following thing will worry
about metrics on the plane and metrics on
00:14:20.180 --> 00:14:26.600
the extended plane which are transported from
metrics on the Riemann sphere okay.
00:14:26.600 --> 00:14:34.720
So here is what I am going to do so let me
draw a diagram, I have this so this is my
00:14:34.720 --> 00:14:44.440
usual complex plane xy, so this is my usual
complex plane which is the xy plane and then
00:14:44.440 --> 00:14:56.440
I have the of course you know we are going
to compare everything with the Riemann sphere
00:14:56.440 --> 00:15:00.770
using the stereographic projection, so what
I am going to do let me draw this thing here
00:15:00.770 --> 00:15:10.079
which is Riemann sphere, so this is supposed
to be 1 this is minus 1 on the x axis and
00:15:10.079 --> 00:15:16.370
here is my sphere its cross-section on the
plane is the unit circle, so it is going to
00:15:16.370 --> 00:15:23.399
be so this is my Riemann sphere as it is and
then of course I have this 3rd axis which
00:15:23.399 --> 00:15:28.279
I will not call as Z I will call it as u because
Z is supposed to stand for x plus i y okay
00:15:28.279 --> 00:15:39.519
Z is supposed to be x plus i y and well and
here is the North pole okay suppose I start
00:15:39.519 --> 00:15:46.160
with 2 point Z 1 and Z 2 on the complex plane
okay.
00:15:46.160 --> 00:15:54.070
See what kind of the distances can I define
on these points, what kind of metric and I
00:15:54.070 --> 00:16:00.820
define on the complex plane that is the usual
metric which is D of Z 1, Z 2 so I will put
00:16:00.820 --> 00:16:06.040
d sub e for Euclidean metric and you know
what that is? It is simply modulus of Z 1
00:16:06.040 --> 00:16:11.570
minus Z 2 it is just the distance between
these 2 points okay. This is the good old
00:16:11.570 --> 00:16:16.800
metric that we use always in Euclidean space
right. It is actually the length of the line
00:16:16.800 --> 00:16:25.920
segment okay joining Z 1 and Z 2. Then the
other thing you can do is you can take the
00:16:25.920 --> 00:16:32.420
images of these points on the Riemann sphere
because of the stereographic projection and
00:16:32.420 --> 00:16:36.230
you can measure the distance between those
2 points and call that as the distance between
00:16:36.230 --> 00:16:40.920
these 2 points, so what you can do is so here
is the stereographic projection, so p 2 is
00:16:40.920 --> 00:16:45.970
this point here on the sphere which is the
unique point of intersection of the line joining
00:16:45.970 --> 00:16:53.029
n and Z 2 on the sphere okay and similarly
p 1 is unique point on the Riemann sphere
00:16:53.029 --> 00:17:00.570
which is the intersection of the surface of
the sphere with the line joining n and Z 1.
00:17:00.570 --> 00:17:05.910
Now you see p 1 and p 2 lie on the sphere,
now what I can do is that I can measure the
00:17:05.910 --> 00:17:12.000
distance between p 1 and p 2 okay. Now that
distance I measuring in 3 space because now
00:17:12.000 --> 00:17:18.060
everything once you draw the Riemann sphere
you are actually in R 3 and your R 2 which
00:17:18.060 --> 00:17:24.919
is xy plane corresponds to the complex plane
alright, so what you can do is you could define
00:17:24.919 --> 00:17:39.080
the following new distance also d let me call
this as d c is the chordal distance from Z
00:17:39.080 --> 00:17:51.860
1 and Z 2 it is just length
of the cord from p 1 to p 2 where p 1 the
00:17:51.860 --> 00:17:57.530
stereographic projection of p 1 is Z 1 and
the estimated projection of p 2 is Z 2 okay
00:17:57.530 --> 00:18:05.620
so this is another distance that I can define
it makesůsee what this distance does is that
00:18:05.620 --> 00:18:12.720
actually it is the metric in R3 after all
the cord joining p 1 p 2 is exactly the line
00:18:12.720 --> 00:18:19.420
segment p 1 to p 2 in 3 space and I am just
taking the length of that line segment, so
00:18:19.420 --> 00:18:27.220
it is actually the metric in R3 it is metric
in R3 and so it is a metric space you know
00:18:27.220 --> 00:18:31.440
whenever you have a metric on space and you
restrict to a substance then the subspace
00:18:31.440 --> 00:18:33.160
also becomes automatically a metric space.
00:18:33.160 --> 00:18:43.890
So this distance will make the Riemann sphere
into a metric subspace of R3 okay and what
00:18:43.890 --> 00:18:48.250
we are doing is that to the stereographic
projection you are transporting that metric
00:18:48.250 --> 00:18:54.690
to the complex plane because after all this
geographic projection is the bisection between
00:18:54.690 --> 00:18:58.810
the extended complex plane and the Riemann
sphere okay. The moment you have bisection
00:18:58.810 --> 00:19:03.970
of a set with a metric space it can transport
the metric on the metric space to the set,
00:19:03.970 --> 00:19:08.500
so I will just transported theů basically
what I have done is I have simply transported
00:19:08.500 --> 00:19:15.030
the metric on R3 restricted to the Riemann
sphere. I have simply unsupported it to the
00:19:15.030 --> 00:19:17.280
plane that is what this d c is.
00:19:17.280 --> 00:19:22.570
So this d c also will also d subsea that is
also a metric you can check that that also
00:19:22.570 --> 00:19:31.349
makes the complex plane into a metric space
okay and
00:19:31.349 --> 00:19:37.520
but the big deal is that all these metrics
are all equivalent okay namely the topologies
00:19:37.520 --> 00:19:42.369
that they induce on the complex plane there
are all the same that is the whole point okay
00:19:42.369 --> 00:19:48.690
and that is very important because what it
tells you is, it tells you the following thing
00:19:48.690 --> 00:19:54.440
if I want to study convergence of functions
you know as long as you are worried about
00:19:54.440 --> 00:20:02.020
continuous functions I can use any of these
metrics and the point is so let me tell that
00:20:02.020 --> 00:20:08.010
in advance, why I am worried about these extra
metrics is because I can also define the distance
00:20:08.010 --> 00:20:13.760
of a point on the complex plane to the point
at infinity okay because that the point at
00:20:13.760 --> 00:20:18.320
infinity will correspond to a finite point
namely the North pole on the Riemann sphere
00:20:18.320 --> 00:20:24.320
and distance to that is something that I can
measure okay, so that is the advantage.
00:20:24.320 --> 00:20:28.660
The advantage is you see I want to be able
to measure the distance the point at infinity
00:20:28.660 --> 00:20:32.810
R I cannot do it with the Euclidean metric
because 1st of all the point at infinity is
00:20:32.810 --> 00:20:38.960
not in my set okay it is an extra point I
have added for compactification and once I
00:20:38.960 --> 00:20:44.030
had these extra points I have to add this
extra topology, the topology of the one-point
00:20:44.030 --> 00:20:48.250
compactification but then that is not enough
I have to even make it a metric space and
00:20:48.250 --> 00:20:52.040
where will I get the metric structure? The
only way is I will have to get this metric
00:20:52.040 --> 00:20:58.310
structure from the Riemann sphere which is
what is holomorphic to the extended plane
00:20:58.310 --> 00:21:04.010
okay and therefore I am lead to look at the
metrics on the on the Riemann sphere, so this
00:21:04.010 --> 00:21:09.600
is the chordal metric okay, so d subsea is
the chordal metric and then here is the 3rd
00:21:09.600 --> 00:21:21.230
metric which is the spherical metric so d
sub s of Z 1, Z 2 this is the length of the
00:21:21.230 --> 00:21:37.630
arc of the minor R from p 1 to p 2 along is
a length of the minor arc from p 1 to p 2
00:21:37.630 --> 00:21:50.190
along the great circle through p 1 and p 2.
00:21:50.190 --> 00:21:55.430
So you see this is a spherical distance, what
is a spherical distance? This is spherical
00:21:55.430 --> 00:22:01.900
distance actually am trying to measure distance
on the Riemann sphere on the surface of the
00:22:01.900 --> 00:22:07.129
sphere, so it is the curve distance okay and
I am trying to measure the shortest curve
00:22:07.129 --> 00:22:15.109
distance and you know you can imagine thisůbasically
what one is doing is that one is doing a kind
00:22:15.109 --> 00:22:19.970
of some kind of Riemann in geometry, what
is happening is that you have a surface okay
00:22:19.970 --> 00:22:25.190
you imagine some nice smooth surface you have
2 points okay then you can try to connect
00:22:25.190 --> 00:22:32.909
those 2 points by many arcs by many arcs on
the surface passing on the surface okay and
00:22:32.909 --> 00:22:39.330
then you can measure the lens of each of the
arcs and you can take the smallest length
00:22:39.330 --> 00:22:43.830
okay and the arc of smallest length is called
a geodesic okay.
00:22:43.830 --> 00:22:48.450
So now what is happening is that if you take
the sphere it is quite easy to see that the
00:22:48.450 --> 00:22:53.100
if you give me 2 points on this sphere,
on the surface of the sphere then the geodesic
00:22:53.100 --> 00:23:00.190
is exactly the following for those 2 points
you get a big circle, a great circle a circle
00:23:00.190 --> 00:23:05.859
of largest radius on the surface of the sphere
passing through those 2 points and you take
00:23:05.859 --> 00:23:12.859
the minor arc okay any 2 points of a circle
will split the circle into 2 arcs and you
00:23:12.859 --> 00:23:19.180
take the minor arc the one of smaller length
and take the length of that, that is exactly
00:23:19.180 --> 00:23:26.260
the spherical distance okay and that is what
I am denoting as d of s it is a geodesic for
00:23:26.260 --> 00:23:34.919
the sphere for any 2 points on the sphere,
so the great circle are the geodesic minor
00:23:34.919 --> 00:23:38.429
arcs of the great circle are the geodesic
for points on the sphere.
00:23:38.429 --> 00:23:43.389
So what is happening is that now I have all
these 3 metrics, the beautiful thing is that
00:23:43.389 --> 00:23:49.359
these 3 metrics give you metrics not only
on the complex plane the point is they give
00:23:49.359 --> 00:23:56.530
you metrics on the extended plane okay see
what I have drawn here is for Z 1 and Z 2
00:23:56.530 --> 00:24:01.560
imagining Z 1 and Z 2 as points on the complex
plane but I can very well make Z 1 or Z 2
00:24:01.560 --> 00:24:07.379
to be the point at infinity. Now when I say
I make the point Z 1 or Z 2 to be the point
00:24:07.379 --> 00:24:12.629
at infinity I cannot see it on the complex
plane okay but I can see its image on the
00:24:12.629 --> 00:24:18.460
Riemann sphere it is the North pole, so basically
what I am doing is I simply taking 2 points
00:24:18.460 --> 00:24:23.300
on the Riemann sphere and I am measuring their
distance, the distance between them either
00:24:23.300 --> 00:24:26.849
the chordal distance or I am measuring the
spherical distance, so the moral of the story
00:24:26.849 --> 00:24:37.480
is that these distances help you to give a
metric on the extended plane and the topology
00:24:37.480 --> 00:24:41.720
induced by all these metrics is one and the
same, all these metrics are equivalent okay.
00:24:41.720 --> 00:24:48.349
So this is the fact that you need to check
from you know this is very easy fact to check
00:24:48.349 --> 00:24:56.000
are logically let me tell you how to do it
how will check the 2 metrics are equivalent
00:24:56.000 --> 00:25:00.470
for a topological space it is very simple,
what you do is that you show that you take
00:25:00.470 --> 00:25:07.720
an for each point of the topological space
you take small open balls with respect to
00:25:07.720 --> 00:25:11.669
one metric and show that it contains a small
open ball with respect to the other metric
00:25:11.669 --> 00:25:17.360
and do this for both metrics symmetrically
and then you are done okay, so you can see
00:25:17.360 --> 00:25:24.230
pictorially you can see that it is true okay
if for all the 3 metrics, so therefore all
00:25:24.230 --> 00:25:29.670
these 3 metrics will give you one and the
same topological space structure on the extended
00:25:29.670 --> 00:25:34.790
complex plane which is the same as the one-point
compactification and that will be exactly
00:25:34.790 --> 00:25:43.240
holomorphic to the Riemann sphere by the stereographic
projection okay and the advantage of doing
00:25:43.240 --> 00:25:45.770
all this, why do all this? You can ask me
why do all this?
00:25:45.770 --> 00:25:50.490
The advantage of doing all this is that now
I can say, now I can make sense of the following
00:25:50.490 --> 00:25:59.020
statement. A sequence of f function f n converges
normally to infinity with a function infinity
00:25:59.020 --> 00:26:07.159
it makes sense now because I can say I can
say convergence with respect to this metric
00:26:07.159 --> 00:26:14.570
one of these metrics namely the 2nd and 3rd
one which are also defined for the point at
00:26:14.570 --> 00:26:20.410
infinity okay and that is the reason why we
need to use that okay, so let me write this
00:26:20.410 --> 00:26:36.370
down, all the so let me write somewhere here
may be use a different color. All the metrics
00:26:36.370 --> 00:26:54.160
below are equivalent on C alright and the
last 2 metrics are equivalent on the extended
00:26:54.160 --> 00:27:18.760
plane. The latter to so let me make some space
let me get rid of this, the latter two namely
00:27:18.760 --> 00:27:33.870
these 2 are equivalent on the extended plane
C union infinity okay and of course I rubbed
00:27:33.870 --> 00:27:41.849
of Z equal to x plus i y so let me write it
here okay and now here comes the here
00:27:41.849 --> 00:27:49.460
comes our the advantage of this.
00:27:49.460 --> 00:28:07.070
We can now say that f n of Z equal to Z power
N, n greater than or equal to 1 converges
00:28:07.070 --> 00:28:20.179
normally to infinity on mod Z greater than
1 you can say this you can make this statement
00:28:20.179 --> 00:28:31.090
it makes sense okay and and why is that correct?
You have to do a little bit of you know
00:28:31.090 --> 00:28:35.669
why I am spending so much time on this is
because you see this normal convergence
00:28:35.669 --> 00:28:41.770
the function which is infinity is something
that is hard toůas it is if you do not analyse
00:28:41.770 --> 00:28:46.659
it is very hard to digester, so it is very
important that you understand what is happening
00:28:46.659 --> 00:28:53.020
here and you must understand that therefore
even if you are looking at the normal convergence
00:28:53.020 --> 00:29:00.339
of holomorphic analytic functions can still
end up with a function which is infinity okay.
00:29:00.339 --> 00:29:03.930
So you see what is happening?
00:29:03.930 --> 00:29:12.859
So again let me draw another diagram, so you
see here is the here is your complex plane
00:29:12.859 --> 00:29:20.119
as it is in R 2 and this is the origin and
of course I have, so let me draw the Riemann
00:29:20.119 --> 00:29:29.661
sphere here okay so this is the situation
and you see of course mod Z greater than 1
00:29:29.661 --> 00:29:38.910
is this is the region of the plane exterior
of the unit circle okay, so basically so this
00:29:38.910 --> 00:29:49.090
is mod Z greater than 1 yes it is very hard
to draw that in three-dimensional diagram,
00:29:49.090 --> 00:29:59.630
so let me do the following thing let me use
another color you know I just looking at so
00:29:59.630 --> 00:30:07.150
it is all these, so it is this exterior of
this unit circle on the complex plane which
00:30:07.150 --> 00:30:15.270
is thought of as xy plane okay and of course
you know the 3rd axis is I am calling it as
00:30:15.270 --> 00:30:16.270
u.
00:30:16.270 --> 00:30:27.060
Now you see, so let me change color again
so I have this let me draw this also so that
00:30:27.060 --> 00:30:37.359
this is u and I have the North pole here okay,
now you see these red lines that I have
00:30:37.359 --> 00:30:43.340
drawn they are supposed to extend outside
unit circle the whole exterior of the unit
00:30:43.340 --> 00:30:48.330
circle and what they are going to give me?
They are going to give me this domain mod
00:30:48.330 --> 00:30:52.630
Z greater than 1 if you consider it as a domain
in the extended plane it is a neighbourhood
00:30:52.630 --> 00:30:58.610
of infinity okay and what is its image under
the stereographic projection it is exactly
00:30:58.610 --> 00:31:05.730
the upper hemisphere okay it is the whole
upper hemisphere which you can see clearly
00:31:05.730 --> 00:31:12.220
is neighbourhood of the North pole okay that
is the reason you are looking at alright.
00:31:12.220 --> 00:31:19.820
Now you see take a compact subset of mod Z
greater than 1 take a compact subset is a
00:31:19.820 --> 00:31:26.720
close and bounded subset of mod Z greater
than 1 okay and so you know if you are looking
00:31:26.720 --> 00:31:33.500
at a compact subset on the plane okay then
any compact subset in mod Z greater than 1
00:31:33.500 --> 00:31:42.150
on the plane to be has to lie within a
sufficiently well-chosen annulus okay it should
00:31:42.150 --> 00:31:47.850
lie within an annular region consisting of
an inner circle and an ouster circle centred
00:31:47.850 --> 00:31:54.160
at the origin sufficiently small inner radius
greater than 1 and sufficiently large radius
00:31:54.160 --> 00:32:00.910
greater than of course the inner radius. So
you see if I take some compact set here, so
00:32:00.910 --> 00:32:09.410
here is some compact set, k compact set in
the complex plane then you see this is k of
00:32:09.410 --> 00:32:15.619
course lies inside suitable annulus, so it
is going to look like this you know I am going
00:32:15.619 --> 00:32:26.169
to get something like this, I am going to
get this annulus here so I am going to
00:32:26.169 --> 00:32:34.349
get this annular region mind you this is annular
region on the complex plane consisting of
00:32:34.349 --> 00:32:41.860
the region between these 2 circles and am
also including the boundaries go make it compact
00:32:41.860 --> 00:32:43.639
okay.
00:32:43.639 --> 00:32:48.460
So it is a closed and bounded set it is compact
and this is a compact set and the point I
00:32:48.460 --> 00:32:54.790
want to make is that instead of just considering
any compact set k in the complex plane which
00:32:54.790 --> 00:33:01.560
is lying in this domain mod Z greater than
1 it is enough to just consider such annuli
00:33:01.560 --> 00:33:08.770
which lie in the exterior of that circle okay
the exterior of the unit circle and you see
00:33:08.770 --> 00:33:14.090
if you watch carefully how is this annulus
going to be given by well this annulus is
00:33:14.090 --> 00:33:20.030
going to be given by mod Z less than small
r I mean less than capital R less than or
00:33:20.030 --> 00:33:23.320
equal to capital R less than or equal to small
r which is greater than 1 this is how it is
00:33:23.320 --> 00:33:28.570
going to be where small r is the radius of
the inner circle capital R is the radius
00:33:28.570 --> 00:33:29.780
of the outer circle okay.
00:33:29.780 --> 00:33:37.530
This is what this annulus is going to be given
by and well what is its image going to be
00:33:37.530 --> 00:33:42.980
on the Riemann sphere stereographic projection
I am going to get this, see I am going to
00:33:42.980 --> 00:33:52.540
get something like this here this is what
I will get, I will get a curved annular region
00:33:52.540 --> 00:34:07.149
centred at the North pole alright and now
you see that
00:34:07.149 --> 00:34:19.940
now you can see why the f n converges to the
function infinity normally okay because our
00:34:19.940 --> 00:34:31.510
definition of convergence is in the following
sense okay. So the definition of convergence
00:34:31.510 --> 00:34:41.260
will be of course point wise convergence okay
but then point wise convergence if you try
00:34:41.260 --> 00:34:44.710
to write it in the metric it will create a
problem when you put the point at infinity
00:34:44.710 --> 00:34:55.010
okay, so I cannot say that or each point Z
naught f n of Z naught converges to infinity
00:34:55.010 --> 00:35:07.510
as n tends to infinity I cannot say that.
I can say that in a topological sense but
00:35:07.510 --> 00:35:11.531
I cannot say that in a metric sense but if
I use Euclidean metric but then if I use a
00:35:11.531 --> 00:35:14.380
spherical metric I can say that okay.
00:35:14.380 --> 00:35:25.410
So the moral of the story is that if you look
at the distance the spherical distance between
00:35:25.410 --> 00:35:34.000
a point Z I take the point Z in K my compact
set okay. If I look at the spherical distance
00:35:34.000 --> 00:35:46.900
between f n of Z which is in this case Z power
n okay and the point at infinity okay and
00:35:46.900 --> 00:35:54.460
here you see what I mean here is infinity
of Z. See infinity of Z I am thinking of the
00:35:54.460 --> 00:35:59.480
function which is constant function which
gives the value infinity to every point so
00:35:59.480 --> 00:36:03.260
what I have written there is actually infinity
of Z and I am saying that it is spherical
00:36:03.260 --> 00:36:10.670
distance between f n of Z and infinity of
Z that can be made uniformly less than Epsilon
00:36:10.670 --> 00:36:18.020
irrespective of Z if I choose n sufficiently
large, so I can make this less than Epsilon
00:36:18.020 --> 00:36:25.700
okay for an a grade than or equal to capital
n irrespective ofů
00:36:25.700 --> 00:36:34.250
So given if I start with an Epsilon greater
than 0 okay for Z in K I can make the spherical
00:36:34.250 --> 00:36:38.840
distance between f n of Z which is Z power
n and infinity I can make it less than this
00:36:38.840 --> 00:36:45.540
Epsilon for n whenever small n is greater
than or equal to a large enough n such a large
00:36:45.540 --> 00:36:50.090
enough n exist the point is that this large
enough n does not have anything to do with
00:36:50.090 --> 00:36:56.190
the Z. It will work for all Z in the compact
set that is the uniformness that is uniformness
00:36:56.190 --> 00:37:02.450
of the convergence on the compact set, so
n this n is so there exit this n and this
00:37:02.450 --> 00:37:14.220
is independent of Z and you see this fact
is true this fact is very. See suppose I give
00:37:14.220 --> 00:37:21.270
you an Epsilon okay what is the spherical
distance between f n of Z and infinity it
00:37:21.270 --> 00:37:26.580
is actually the spherical distance between
Z power n and infinity alright and Z power
00:37:26.580 --> 00:37:35.390
n is going to lie where, it is going to lie
in mod Z power n is going to lie in this annulus
00:37:35.390 --> 00:37:46.020
okay and you know if you see if Iů
00:37:46.020 --> 00:37:50.420
The inner radius of this annulus is r power
n small r power n the outer radius is capital
00:37:50.420 --> 00:37:58.870
R power n and you know if I and since
R is greater than 1 if I increase n R power
00:37:58.870 --> 00:38:06.680
n is small R power n itself is going to shoot
up okay, so moral of the story is that this
00:38:06.680 --> 00:38:14.900
region if I have its image in the on the Riemann
sphere, what I am going to get is sufficiently
00:38:14.900 --> 00:38:22.930
small annular region surrounding the North
pole and clearly the spherical distance can
00:38:22.930 --> 00:38:28.650
be made less than Epsilon for any point in
that region okay, so that is you know that
00:38:28.650 --> 00:38:32.100
is pictorial justification or the statement.
00:38:32.100 --> 00:38:38.620
So the moral of the story is now you know
you are able to justify that this sequence
00:38:38.620 --> 00:38:48.680
Z power n converges normally the function
infinity okay on the exterior of the unit
00:38:48.680 --> 00:38:54.900
disk in the extended plane okay and the point
is that you are using the spherical metrics
00:38:54.900 --> 00:38:59.760
okay that is the advantage you are using the
spherical metrics because it allows you to
00:38:59.760 --> 00:39:07.270
give you to measure the distance even to the
point at infinity from a finite point in the
00:39:07.270 --> 00:39:13.450
complex plane which you cannot do with the
Euclidean method and the way all this is done
00:39:13.450 --> 00:39:20.610
it will also extend the usual definition of
normal convergence. Suppose you have a sequence
00:39:20.610 --> 00:39:27.560
of analytic function which converges to a
analytic function itself okay a finite valued
00:39:27.560 --> 00:39:28.560
function.
00:39:28.560 --> 00:39:34.280
A function that does not take values infinity
then what is the usual convergence that we
00:39:34.280 --> 00:39:37.590
talked about, the usual convergence that we
talk about when you for example when you do
00:39:37.590 --> 00:39:40.750
a 1st course in complex analysis the usual
convergence that you talking about is with
00:39:40.750 --> 00:39:46.080
respect to Euclidean metric okay you have
to only worry about the Euclidean metric,
00:39:46.080 --> 00:39:56.960
nobody is worried about the point at infinity
to begin with okay. Now if you take a usual
00:39:56.960 --> 00:40:01.430
sequence of holomorphic functions on a domain,
analytic functions on a domain suppose it
00:40:01.430 --> 00:40:10.890
is converging normally again to an analytic
function on the domain okay. Then suppose
00:40:10.890 --> 00:40:16.450
this convergence is in the usual sense I am
saying this convergence is also correct with
00:40:16.450 --> 00:40:19.500
respect to the spherical metric.
00:40:19.500 --> 00:40:24.240
The reason is because the spherical metric
when you restrict it to the usual complex
00:40:24.240 --> 00:40:29.510
plane it is equivalent to the usual Euclidean
metrics, so you do not lose anything. So what
00:40:29.510 --> 00:40:36.140
I am saying is that this definition of sequence
of functions converging to another function
00:40:36.140 --> 00:40:45.970
normally on a domain that works irrespective
of whether you are using the Euclidean metric
00:40:45.970 --> 00:40:53.600
or whether you are using the spherical metric
but the point is it helps you when infinity
00:40:53.600 --> 00:41:00.060
values are taken it helps you because when
infinity values are taken you cannot use Euclidean
00:41:00.060 --> 00:41:05.010
metric you can use only the spherical metric
okay. So the normal convergence under the
00:41:05.010 --> 00:41:10.450
spherical metric is just an extension of the
normal convergence under the Euclidean metric
00:41:10.450 --> 00:41:15.930
as far as subset of the complex plane is concerned
convergence under this spherical metric is
00:41:15.930 --> 00:41:19.120
same as convergence under the Euclidean metric.
00:41:19.120 --> 00:41:22.640
Normal convergence under the spherical metric
is same as normal convergence under the Euclidean
00:41:22.640 --> 00:41:30.320
metric because they are equivalent okay, so
so this is 1 point that you need to understand,
00:41:30.320 --> 00:41:41.010
so you know so let me say this so let me write
this specifically we need to therefore worry
00:41:41.010 --> 00:41:57.800
about by the time it include we need to worry
about that this include the function infinity
00:41:57.800 --> 00:42:19.660
that takes the value infinity at every point
of your domain. Then we may define normal
00:42:19.660 --> 00:42:40.380
convergence as follows; let f n be a sequence
of holomorphic functions or analytic holomorphic
00:42:40.380 --> 00:43:00.990
is same as analytic functions on a domain
D in the complex plane. We say f n converges
00:43:00.990 --> 00:43:24.460
to f on D normally okay. If the spherical
distance between f n of Z and f of Z goes
00:43:24.460 --> 00:43:34.360
to 0 normally which means uniformly on compact
sets
00:43:34.360 --> 00:44:01.620
on D where we allow f to be the function infinity
this is an exceptional case okay, so you define
00:44:01.620 --> 00:44:04.580
normal convergence in the following way.
00:44:04.580 --> 00:44:08.850
So what is the normal convergence you have
sequence of functions on a domain in the complex
00:44:08.850 --> 00:44:22.440
plane you say f n converges to f on the domain
okay. If the spherical distance okay that
00:44:22.440 --> 00:44:31.830
converges to 0 okay and you this spherical
distance is a function of Z, so Z is varying
00:44:31.830 --> 00:44:43.510
on the domain, so I wanted to understand this
Z is varying on the domain. This quantity
00:44:43.510 --> 00:44:52.200
here is also a function of Z it measures for
each Z it measures the spherical distance
00:44:52.200 --> 00:45:00.330
between f n of Z and f of Z okay and what
I want is that a function of Z should go to
00:45:00.330 --> 00:45:08.650
0 the constant function 0 uniformly in Z on
compact subset of set of the domain, so I
00:45:08.650 --> 00:45:15.150
wanted to go to 0 normally on the that is
my definition and now the beautiful thing
00:45:15.150 --> 00:45:20.210
is you know we need this definition because
if you take the domain as I told you if you
00:45:20.210 --> 00:45:23.950
take the domain to be mod Z greater than 1
and you take f n of Z to be Z power n such
00:45:23.950 --> 00:45:25.770
a definition is necessary.
00:45:25.770 --> 00:45:30.330
So what it tells you is that now you have
to also worry you are not worried only about
00:45:30.330 --> 00:45:35.910
functions which takes complex values you have
also allow functions in the value infinity
00:45:35.910 --> 00:45:41.060
but then notice if you take the value infinity
if you allow function take the value infinity
00:45:41.060 --> 00:45:45.600
then you can include Meromorphic functions
because you can define the value or the Meromorphic
00:45:45.600 --> 00:45:50.350
functions at a pole to be infinity and the
beautiful thing is the very same definition
00:45:50.350 --> 00:45:57.960
this very same definition works absolutely
well if you change holomorphic by Meromorphic.
00:45:57.960 --> 00:46:06.440
So that is because of a circle and cemetery
rotational symmetry that is the about for
00:46:06.440 --> 00:46:11.140
this spherical metric that I will explain
in the next lecture at the point is so the
00:46:11.140 --> 00:46:18.170
important observation is that this same definition
works with holomorphic replaced by Meromorphic
00:46:18.170 --> 00:46:23.420
and that is all that we need to do all the
analysis we want okay. So I will stop here.