WEBVTT
Kind: captions
Language: en
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What could the Derivative of a Meromorphic
function Relative the Spherical Metric Possibly
00:00:52.329 --> 00:00:53.329
be?
00:00:53.329 --> 00:00:58.710
Alright, so we have been seeing about meromorphic
functions and holomorphic functions converging
00:00:58.710 --> 00:01:06.070
normally, you know with respect to the spherical
metric, so I want to continue with a few examples,
00:01:06.070 --> 00:01:07.960
so that you get a feel for what we want.
00:01:07.960 --> 00:01:32.799
So let me write it down, examples of normal
convergence under the spherical metric.
00:01:32.799 --> 00:01:37.320
So this is what I want to talk about.
00:01:37.320 --> 00:01:46.090
So the 1st question, 1st question that we
ask is , we have already seen an example
00:01:46.090 --> 00:01:56.569
of a sequence of holomorphic functions which
tends identically to infinity, okay.
00:01:56.569 --> 00:02:03.420
That is basically it is a sequence of functions
it is that power n, okay and the domain is
00:02:03.420 --> 00:02:08.140
the exterior of the unit circle, okay.
00:02:08.140 --> 00:02:16.260
And z power n converges as n tends to infinity
to the function which is identical infinity,
00:02:16.260 --> 00:02:21.120
okay, point wise and this convergence is in
fact normal, okay, we have seen this.
00:02:21.120 --> 00:02:24.000
now you can answer, you can ask the following
question.
00:02:24.000 --> 00:02:29.930
Can you have a sequence of holomorphic functions
tend to a meromorphic function, function which
00:02:29.930 --> 00:02:34.500
is honestly meromorphic, which means it has
at least one pole?
00:02:34.500 --> 00:02:37.250
And the answer no, okay.
00:02:37.250 --> 00:02:41.880
A sequence of holomorphic function, if it
converges normally under the spherical metric,
00:02:41.880 --> 00:02:48.920
either it converges to the constant function
which is identical infinity or it converges
00:02:48.920 --> 00:02:53.510
to a good old holomorphic analytic function.
00:02:53.510 --> 00:02:59.310
You can get meromorphic functions, proper
meromorphic functions which are having some
00:02:59.310 --> 00:03:04.819
poles, you cannot, you cannot take normal
limit of analytic functions and expect a pole
00:03:04.819 --> 00:03:11.700
to pop-up, okay, that will not happen, that
is what the proof, okay. but we think of other
00:03:11.700 --> 00:03:18.090
possibilities, suppose we have sequence of
meromorphic functions, so can you think of
00:03:18.090 --> 00:03:26.650
a sequence of meromorphic functions which
which converge to say infinity?
00:03:26.650 --> 00:03:31.080
And then of course normally under the spherical
metric and can you think of a sequence of
00:03:31.080 --> 00:03:37.720
meromorphic function which converts normally
to a holomorphic function, okay.
00:03:37.720 --> 00:03:44.560
So they have already proved that if you take
a sequence of meromorphic functions and suppose
00:03:44.560 --> 00:03:48.440
it converges normally under spherical metric,
then the result is again a meromorphic function
00:03:48.440 --> 00:03:53.620
or it has to be the extreme case which is,
a function which is identical infinity, okay.
00:03:53.620 --> 00:04:00.040
And mind you, let me again stress the, this
is so important because you see in the limit,
00:04:00.040 --> 00:04:05.360
the limiting function that you get could have
developed bad singularities, it could have
00:04:05.360 --> 00:04:09.220
developed nonisolated singularity, okay.
00:04:09.220 --> 00:04:15.000
Or for example it could have had singularities
on a curve, it could have happened no, then
00:04:15.000 --> 00:04:16.370
it would have been terrible.
00:04:16.370 --> 00:04:21.500
for it could have developed an isolated singularity
which is essential and once you have an essential
00:04:21.500 --> 00:04:27.039
singularity, that function is out of our discussion
because our discussion we are trying to work
00:04:27.039 --> 00:04:28.310
only with meromorphic functions.
00:04:28.310 --> 00:04:34.590
And for meromorphic functions we allow the
poles as singularities, okay.
00:04:34.590 --> 00:04:43.530
So such bad thing to do happen, you do not
have an essential singularity popping up when
00:04:43.530 --> 00:04:47.220
you take the limit, normal limit with respect
to spherical metric.
00:04:47.220 --> 00:04:54.750
You do not have for example a nonisolated
singularity popping up when you take normal
00:04:54.750 --> 00:04:57.690
limit of meromorphic functions, okay.
00:04:57.690 --> 00:05:01.410
So this is, this is, I decide that this is
what happens.
00:05:01.410 --> 00:05:06.680
And that is what we proved the last 2 classes,
okay.
00:05:06.680 --> 00:05:13.349
now I just want to give you a couple of examples
where you can have a sequence of, I think
00:05:13.349 --> 00:05:18.520
you can have a sequence of meromorphic functions
that goes to holomorphic function, okay.
00:05:18.520 --> 00:05:27.319
And you can also have a sequence of
, so that, so that example is an example where,
00:05:27.319 --> 00:05:31.020
and of course you can also have a sequence
of meromorphic functions, honestly meromorphic
00:05:31.020 --> 00:05:33.169
functions go to infinity, okay.
00:05:33.169 --> 00:05:38.289
So you know, so let me write down those
2 examples.
00:05:38.289 --> 00:05:41.710
So here is example 1.
00:05:41.710 --> 00:05:49.770
So this this example 1 is as well, you know,
let me say, I have just recall that example
00:05:49.770 --> 00:05:57.270
of holomorphic functions that goes to infinity
and modify it carefully to make it into a
00:05:57.270 --> 00:06:01.349
sequence of meromorphic function that go to
infinity.
00:06:01.349 --> 00:06:10.509
So this, 1st example is just recalling, you
bought the domain D to be the, well the exterior
00:06:10.509 --> 00:06:20.229
of the unit disc, set of all z such that mod
of z greater than 1, mod z greater than 1,
00:06:20.229 --> 00:06:22.080
this is the exterior unit disc.
00:06:22.080 --> 00:06:28.160
Mind you this is a nice domain in the external
complex plane, okay.
00:06:28.160 --> 00:06:36.830
Well, well the fact is that you know if you
consider this as a domain in the complex plane,
00:06:36.830 --> 00:06:42.110
did not simply connected, okay, because it
has a hole in between.
00:06:42.110 --> 00:06:46.879
It has a whole unit disc closed, unit disc
out as a hole inside.
00:06:46.879 --> 00:06:52.750
but if you consider this as a domain in the
extended complex plane, mind you it becomes
00:06:52.750 --> 00:06:54.770
simply connected.
00:06:54.770 --> 00:06:56.509
And how do you see that?
00:06:56.509 --> 00:07:02.219
It is actually, it is actually a neighbourhood
of infinity, okay.
00:07:02.219 --> 00:07:04.229
And how do you see that?
00:07:04.229 --> 00:07:08.539
If you, if you just take its image under a
stereographic projection, you will get a small
00:07:08.539 --> 00:07:14.930
disc surrounding the north pole on the Riemann
sphere.
00:07:14.930 --> 00:07:18.820
And that is clearly is simply connected set,
okay.
00:07:18.820 --> 00:07:27.539
So mind you including the point at infinity
done this from, you know multiplying connected
00:07:27.539 --> 00:07:30.069
domain to simply connected domain, okay.
00:07:30.069 --> 00:07:34.280
Anyway, that was just an observation, so you
take this domain D and then you take fn of
00:07:34.280 --> 00:07:40.059
z to be z power n, n greater than equal to1,
this is my sequence.
00:07:40.059 --> 00:07:46.499
Then you know that fn, then fn converges,
the fns are all holomorphic functions on D,
00:07:46.499 --> 00:07:50.229
okay, they are in H of D, these are all.
00:07:50.229 --> 00:07:55.569
Of course as fns are, in fact entire function,
that is polynomials, so they are holomorphic.
00:07:55.569 --> 00:08:07.469
And well, now but what happens, fn converges
to infinity normally, so when I say normally
00:08:07.469 --> 00:08:11.520
this is with respect to this spherical metric,
okay.
00:08:11.520 --> 00:08:15.580
And when I say fn converges to infinity, it
means, by infinity I mean the function which
00:08:15.580 --> 00:08:18.830
is constantly equal to infinity at all points.
00:08:18.830 --> 00:08:22.189
Mind you we have put that extra function there,
okay.
00:08:22.189 --> 00:08:26.689
And this is the extreme case, all right.
00:08:26.689 --> 00:08:31.619
And now what you can do is you can modify
this so that you can get a sequence of meromorphic
00:08:31.619 --> 00:08:38.029
functions which go to infinity for example
and you can also modify this to get a sequence
00:08:38.029 --> 00:08:42.390
of meromorphic functions which goes to holomorphic
functions, okay.
00:08:42.390 --> 00:08:43.760
You can do things like that.
00:08:43.760 --> 00:08:51.540
So for example I can make each of these functions
to have a pole, that is, for example lying
00:08:51.540 --> 00:08:55.529
on the real line, and it is approaching 1,
okay.
00:08:55.529 --> 00:09:01.580
And then if I take the limit, the poles are
moving towards 1.
00:09:01.580 --> 00:09:08.300
So what will happen is that the in the limit
I will get a function which will have, which
00:09:08.300 --> 00:09:14.600
will actually be holomorphic because I have
included mod z, I have excluded mod z equal
00:09:14.600 --> 00:09:16.800
to1 from my domain.
00:09:16.800 --> 00:09:28.380
Okay, so what you do, we we we said g of n
of z to be well, z power n plus let us add
00:09:28.380 --> 00:09:37.590
a pole at if you want, well 1
plus 1 by n which is a little to the right
00:09:37.590 --> 00:09:40.050
of the point 1, okay.
00:09:40.050 --> 00:09:51.640
So I put 1 by z dash 1 plus 1 by n and let
me make this n greater than equal to 1.
00:09:51.640 --> 00:09:58.400
So if you take these functions gns, they all
have poles at the points, the gn has hole
00:09:58.400 --> 00:10:04.130
at, it has only one pole, that is at 1plus1
by n and that is of course lying in mod z
00:10:04.130 --> 00:10:06.500
reference 1, all right.
00:10:06.500 --> 00:10:14.780
So for example z1 has a pole at 2, z, sorry
g1 has a pole at 2, g2 has a pole at 1plus
00:10:14.780 --> 00:10:23.680
half , 1plus half which is 3 by 2
and then and so on.
00:10:23.680 --> 00:10:29.150
You can see that as n tends to infinity, 1
by n tends to1, all right.
00:10:29.150 --> 00:10:38.510
And you can see that these gns are actually
meromorphic functions on D because they only,
00:10:38.510 --> 00:10:45.250
they are not holomorphic because they have
poles but and each has only one pole, each
00:10:45.250 --> 00:10:50.210
function has only one pole, so it is meromorphic
function by definition.
00:10:50.210 --> 00:10:57.300
These are all meromorphic functions and you
know if you if you let n to tend to
00:10:57.300 --> 00:11:07.800
infinity, then what will happen is that these
zn, of course these z power n will go to identically
00:11:07.800 --> 00:11:15.180
to infinity. for any fixed z with mod z greater
than 1, mod z power n is going to go to infinity
00:11:15.180 --> 00:11:16.400
as n tends to infinity.
00:11:16.400 --> 00:11:21.180
Therefore point wise these gns are going to
go to infinity, except we will have to worry
00:11:21.180 --> 00:11:23.450
about the poles.
00:11:23.450 --> 00:11:31.830
Mind you, in the neighbourhood, the poles
are all moving away, so if you if you if you
00:11:31.830 --> 00:11:38.660
take any particular value of z, even if is
one of these poles, okay, for example if you
00:11:38.660 --> 00:11:49.550
take 1plus1 by n, that is a pole of
zn but from gn plus1 onwards it is not a pole.
00:11:49.550 --> 00:11:57.410
And of course it is not equal for gn minus1
and functions before that, okay. but mind
00:11:57.410 --> 00:12:02.090
you the value at the pole is by definition
infinity. for a meromorphic function, we are
00:12:02.090 --> 00:12:04.430
allowing, mind you we are allowing the value
infinity, okay.
00:12:04.430 --> 00:12:10.650
So somehow what you will see is that, if you
now take the limit as n tends to infinity,
00:12:10.650 --> 00:12:15.850
okay, you will end up with you will end
up with the function which is identically
00:12:15.850 --> 00:12:17.250
infinity, okay.
00:12:17.250 --> 00:12:22.230
So this is just a modification of the original
example, okay.
00:12:22.230 --> 00:12:34.000
So gn, gn tends to, so gn tends to infinity,
right, so here is one example.
00:12:34.000 --> 00:12:41.400
So here is an example of Hurwitz meromorphic
functions which go to infinity, okay.
00:12:41.400 --> 00:12:55.670
And well, and in fact it would, if I forget
z power n and I took only the function 1 by
00:12:55.670 --> 00:12:58.640
z minus 1plus1 by n, okay.
00:12:58.640 --> 00:13:04.730
now that function if you see, if you look
at it, that will be the function which is
00:13:04.730 --> 00:13:11.510
again meromorphic with only one pole at 1plus1
by n. but now the point is that this function
00:13:11.510 --> 00:13:19.210
1 by z dash 1 plus 1 by n because z is in
the denominator, as you let z tend to infinity,
00:13:19.210 --> 00:13:23.380
this will go to 0, so this function is bounded
at infinity.
00:13:23.380 --> 00:13:32.960
So as a result, you know if you, if you put
Hn of z to be just 1 by z dash 1 plus 1 by
00:13:32.960 --> 00:13:36.310
n, then Hn of z, they will all be meromorphic.
00:13:36.310 --> 00:13:44.140
That gives you, you get an example of sequence
of Hurwitz meromorphic functions which tends
00:13:44.140 --> 00:13:48.141
finally function which is holomorphic and
it is also holomorphic at infinity, it is
00:13:48.141 --> 00:13:50.770
analytic at infinity because at infinity it
is bounded.
00:13:50.770 --> 00:13:55.680
So that will give me the next example, so
this is, this is, this was example 2 which
00:13:55.680 --> 00:13:57.630
was the modification of example 1.
00:13:57.630 --> 00:14:12.780
And the part of, and then we have example
3, so example 3 is put Hn, Hn of z to be simply
00:14:12.780 --> 00:14:20.690
z dash 1 plus1 by n and then you put it, put,
take the reciprocal so that you get the pole
00:14:20.690 --> 00:14:23.510
at 1plus1 by n for Hn.
00:14:23.510 --> 00:14:27.910
Then these are all of course in, they are
all meromorphic functions in D, so Hn tends
00:14:27.910 --> 00:14:31.690
to1 by 1 by z minus 1, okay, as n tends to
infinity.
00:14:31.690 --> 00:14:38.870
In 1 by z minus1 has a pole at 1 and that
is not a problem because one is not in our
00:14:38.870 --> 00:14:42.560
domain, all right, it is in the boundary of
our domain.
00:14:42.560 --> 00:14:45.260
fine, good.
00:14:45.260 --> 00:14:53.120
So, yah so now what I want to tell you is
that, what are we going towards next.
00:14:53.120 --> 00:15:00.060
See we need to understand several technical
concepts in order to get the proof of the
00:15:00.060 --> 00:15:02.110
Picard theorem which is our aim, okay.
00:15:02.110 --> 00:15:07.490
And one of these concepts, one of the technical
things that we have to worry about is the
00:15:07.490 --> 00:15:13.150
so-called spherical derivative, okay, the
absolute spherical derivative of a function,
00:15:13.150 --> 00:15:14.150
all right.
00:15:14.150 --> 00:15:21.760
now let me try to explain that and with reference
to what we already know.
00:15:21.760 --> 00:15:28.730
So the 1st thing is that, so I, I will tell,
I will put this heading as, so let me change
00:15:28.730 --> 00:15:39.590
colour and put this heading as discussion
of the spherical metric on the spherical metric
00:15:39.590 --> 00:15:50.550
and spherical derivative because this something
that we need, okay, we need to worry about
00:15:50.550 --> 00:15:53.590
this.
00:15:53.590 --> 00:16:05.110
So the 1st thing is, let go 1st Euclidean,
so that is on the complex plane, okay.
00:16:05.110 --> 00:16:14.960
So you know if you give me 2 points z1 and
z2 on the complex plane, then the distance,
00:16:14.960 --> 00:16:22.630
Euclidean distance between these 2 points
the z1 and z2, if well, mod z1 minus z 2,
00:16:22.630 --> 00:16:25.740
okay, this is the usual distance formula.
00:16:25.740 --> 00:16:36.590
And the infinitesimal distance, how the given,
you get the infinitesimal distance by putting
00:16:36.590 --> 00:16:47.680
z1 as say z and z2 as z plus Delta z, okay
and then mod z1 minus z2 becomes mod Delta
00:16:47.680 --> 00:16:49.200
z, okay.
00:16:49.200 --> 00:16:53.200
And then you change the Delta to D, okay,
to get the infinitesimal.
00:16:53.200 --> 00:17:11.459
So the infinitesimal distance , distance,
so this is arc length, this is mod ds, okay.
00:17:11.459 --> 00:17:24.709
Let me use, let me use the right rotation
dz, these are claimed small dz and roughly
00:17:24.709 --> 00:17:30.710
the way you think of it if you put z1 equal
to z and you put z2 equal to z plus Delta
00:17:30.710 --> 00:17:31.710
z.
00:17:31.710 --> 00:17:36.280
So the difference mod z1 minus z2 becomes
mod of Delta z and then change the Delta to
00:17:36.280 --> 00:17:37.280
D.
00:17:37.280 --> 00:17:43.530
The idea is you ignore, if you do such an
operation, you have to ignore higher powers
00:17:43.530 --> 00:17:47.570
of Delta z than the 1st.
00:17:47.570 --> 00:17:52.470
If we get Delta is at whole square, Delta
is at whole cube, okay, all these terms have
00:17:52.470 --> 00:17:53.929
to be ignored, okay.
00:17:53.929 --> 00:17:59.539
And then you replace Delta by the, so this
is a heuristic way of getting the infinitesimal
00:17:59.539 --> 00:18:00.539
distance.
00:18:00.539 --> 00:18:12.590
And, well and how do you , and
how do you for example check that this is,
00:18:12.590 --> 00:18:15.299
this infinitesimal distance is correct.
00:18:15.299 --> 00:18:25.820
So for example you know if I take, if I take
integral from z1 to z2 of mod dz, if I do
00:18:25.820 --> 00:18:37.200
this, I will get mod z1 minus z2, this is
what I will get.
00:18:37.200 --> 00:18:45.279
And that tells you that the infinitesimal
distance you have computed is correct.
00:18:45.279 --> 00:18:51.590
So how this calculation is done, so you have
for example, of course when I say this integral
00:18:51.590 --> 00:18:57.659
from z1 to z2, I am going along this, this
integral is along the straight line segment
00:18:57.659 --> 00:18:58.909
from z1 to z2.
00:18:58.909 --> 00:19:03.590
So whenever I write it, mind you whenever
you write an integral, there is always a path
00:19:03.590 --> 00:19:07.250
involved, without the path if you write it,
then it is very ambiguous.
00:19:07.250 --> 00:19:11.340
So here when I say integral from z1 to z2,
I mean the linear distance, the Euclidean
00:19:11.340 --> 00:19:12.340
distance between z1 and z2.
00:19:12.340 --> 00:19:16.440
That means I am taking straight line segment
from z1 to z2, okay.
00:19:16.440 --> 00:19:23.419
So what you have to do is, here is z1 and
earlier z2 and then you are taking this path,
00:19:23.419 --> 00:19:28.820
the straight line segment okay and then you
can parameterise this path.
00:19:28.820 --> 00:19:34.980
You can parameterise this path and then you
evaluate that integral, okay.
00:19:34.980 --> 00:19:39.559
And if you evaluate it, you will get mod z1
minus z2.
00:19:39.559 --> 00:19:47.600
So for example you know if you take point
t here, how will you, how will you write that,
00:19:47.600 --> 00:19:50.100
we write the parameter?
00:19:50.100 --> 00:19:55.360
If you take the parameter as t, so I should
write z of t, so what will happen to z of
00:19:55.360 --> 00:20:02.179
t, z of t will be, so you know z of t should
give me, for example t is the parameter varying
00:20:02.179 --> 00:20:08.440
from 0 to 1, then you know z of t has to be
a complex combination of z1 and z2.
00:20:08.440 --> 00:20:17.299
So when t is 0 I am supposed to get z1 and
when t is 1 I am supposed to get z2, okay.
00:20:17.299 --> 00:20:21.400
So I will get z1 plus I will get 1 minus t
times z2.
00:20:21.400 --> 00:20:26.269
So in this if I put t equal to1, I will get
z1, sorry I will, it should be the other way
00:20:26.269 --> 00:20:31.720
round, it should be the other way round, let
me change it.
00:20:31.720 --> 00:20:42.740
It is z2 plus1 minus t, I think I should put
t here also, so you put t equal to 0, I get
00:20:42.740 --> 00:20:50.220
z1 and you put t equal to 1 I get z2, this
is a parameter equation of this line segment,
00:20:50.220 --> 00:20:51.220
okay.
00:20:51.220 --> 00:20:54.179
And it is a function of t, it is a differentiable
function of t, in fact it is a linear function
00:20:54.179 --> 00:20:55.179
of t.
00:20:55.179 --> 00:21:05.600
And what is z dash of t, z dash of t gives
just z2 minus z1, okay, did the constant.
00:21:05.600 --> 00:21:10.110
And what is the, what does this integral come
out to, this integral comes out to, if you
00:21:10.110 --> 00:21:14.450
ballot it,, you have to, you know you have
to transform the integral in terms of the
00:21:14.450 --> 00:21:15.600
parameters.
00:21:15.600 --> 00:21:26.090
So you will put t equal to 0 to t equal to1
and you will plug-in this mod dz, so it will
00:21:26.090 --> 00:21:34.110
become, so dz is, z dash of t is dz of t by
dt, okay.
00:21:34.110 --> 00:21:41.749
So you will think of mod dz at, mod z dash
of t into mod dt.
00:21:41.749 --> 00:21:42.749
Okay.
00:21:42.749 --> 00:21:50.710
So I am going to get mod z dash of t into
mod dt and you know, well instead of mod z
00:21:50.710 --> 00:21:53.130
dash of the I am going to put mod z minus
z1, that is a constant, that will come out
00:21:53.130 --> 00:21:58.130
and I will simply get integrals 0 to 1 mod
dt, and that is going to be 1, so I will get
00:21:58.130 --> 00:22:04.289
mod z2 minus z1, okay.
00:22:04.289 --> 00:22:10.149
So that is how this formula is verified and
it is a very very simple calculations but
00:22:10.149 --> 00:22:16.789
I am just trying to justify that mod dz is
the correct infinitesimal version of the of
00:22:16.789 --> 00:22:18.789
the arc length, okay.
00:22:18.789 --> 00:22:25.539
And you know, you would have seen this also
in complex analysis, how do you measure the
00:22:25.539 --> 00:22:27.269
length of an arc on the plane?
00:22:27.269 --> 00:22:30.580
It is done by the same thing.
00:22:30.580 --> 00:22:36.070
So you know suppose you have an interval,
suppose you have this, say this interval,
00:22:36.070 --> 00:22:43.379
let me when would a, b, suppose this is closed
bounded interval in the real line and suppose
00:22:43.379 --> 00:22:51.610
I have I have a continuous map gamma
from that into the complex plane so you
00:22:51.610 --> 00:22:54.379
know it traces an arc.
00:22:54.379 --> 00:23:00.610
Sometimes the map is also called gamma, the
the image is also called gamma by abuse of
00:23:00.610 --> 00:23:01.610
notations.
00:23:01.610 --> 00:23:06.919
So this is, this point is that one which is
gamma of a and this point is z2 which is gamma
00:23:06.919 --> 00:23:11.299
of b and, well outdoor get the length of the
arc gamma.
00:23:11.299 --> 00:23:19.730
Well you get it by simply integrating from
a to b gamma dash of t, mod gamma dash of
00:23:19.730 --> 00:23:22.620
t into dt, okay.
00:23:22.620 --> 00:23:29.210
And this is the same as integrating over gamma
mod dz.
00:23:29.210 --> 00:23:35.129
And basically the integral over gamma mod
dz is transformed to the previous integral,
00:23:35.129 --> 00:23:41.440
integral from a to b gamma dash of t dt, mod,
mod gamma dash of t dt because you just plugging
00:23:41.440 --> 00:23:48.220
in z equal to z of t. because if you take
point on gamma, the point z there is given
00:23:48.220 --> 00:23:52.129
by that of t, it is a function t where c is
the variable here on the real line.
00:23:52.129 --> 00:23:55.980
When t is a, you get gamma of a which is z1,
starting point.
00:23:55.980 --> 00:23:59.440
When t is b, you get gamma of b, which is
z2, the ending point.
00:23:59.440 --> 00:24:07.460
So basically if you want to, mod dz is the
infinitesimal distance and you integrate an
00:24:07.460 --> 00:24:12.330
arc integrate that over an arc, you will
get the arc length.
00:24:12.330 --> 00:24:16.899
And of course it is very important that you
are should be reasonably good, so that you
00:24:16.899 --> 00:24:19.830
get a finite number, okay.
00:24:19.830 --> 00:24:25.990
So arcs for which the arc length turns out
to be finite, they are called rectifiable
00:24:25.990 --> 00:24:28.230
arcs, okay, rectifiable arcs.
00:24:28.230 --> 00:24:32.310
And one has to worry about these things because
there are, if you just, of course whenever
00:24:32.310 --> 00:24:37.700
you say ask, it is always, this gamma is always
continuous, all right.
00:24:37.700 --> 00:24:43.929
but then you know, even to write that integral,
you see that integral actually means this,
00:24:43.929 --> 00:24:48.399
mod gamma dashed of t dt, so you know gamma
dash should exist.
00:24:48.399 --> 00:24:52.299
So gamma is not only continuous but actually
it is differentiable and you can relax it
00:24:52.299 --> 00:24:58.220
to be piecewise differentiable, piecewise
continuously differentiable, I, mind you whenever
00:24:58.220 --> 00:25:00.519
I integrate, it has to be continuous.
00:25:00.519 --> 00:25:06.280
So 1st for gamma dash has to exist and gamma
dash has to be continuous, only then I write
00:25:06.280 --> 00:25:07.280
that integral, okay.
00:25:07.280 --> 00:25:12.779
If I am being very naive, of course if I use
measure theory, I can do much more, I allowed
00:25:12.779 --> 00:25:15.200
is continuity on a measure, measured 0 set.
00:25:15.200 --> 00:25:19.320
but we are not going to that level of you
know complication.
00:25:19.320 --> 00:25:24.730
We are, we assume the naivest point of view
and even for the naivest point of view developed
00:25:24.730 --> 00:25:27.340
assume that gamma dash exist and it is continuous.
00:25:27.340 --> 00:25:32.169
And of course all Riemann integrals can be
evaluated even if there are a few finite,
00:25:32.169 --> 00:25:37.609
finitely many discontinuities which are with
finite jumps okay.
00:25:37.609 --> 00:25:41.899
So that is the reason we use the word contour,
we said that gamma should be a contour, which
00:25:41.899 --> 00:25:47.789
means gamma should be a piecewise continuously
differentiable function, okay.
00:25:47.789 --> 00:25:48.879
fine.
00:25:48.879 --> 00:25:54.960
So this is the length of gamma and of course
there is this important restriction that
00:25:54.960 --> 00:25:56.159
gamma should be a rectifiable arc.
00:25:56.159 --> 00:25:59.929
You have to worry about such thing because
there are there are there are strange things
00:25:59.929 --> 00:26:03.520
like which are called space filling curves
and these are curve which can fill out the
00:26:03.520 --> 00:26:05.250
whole region of space.
00:26:05.250 --> 00:26:08.750
And obviously the length of such a curve will
be infinite, okay.
00:26:08.750 --> 00:26:11.860
So you do not want to end up with such horrible
things.
00:26:11.860 --> 00:26:16.139
So that is the reason you have to worry about
computing this integral only for rectifiable
00:26:16.139 --> 00:26:18.539
arcs, okay.
00:26:18.539 --> 00:26:25.129
now, now what is it, how is this connected
with analytic functions?
00:26:25.129 --> 00:26:31.379
So you see, see the point is, suppose you
have, suppose you have complex plane and suppose
00:26:31.379 --> 00:26:38.139
you have domain here, there is some domain,
okay, so here is some domain D in the complex
00:26:38.139 --> 00:26:40.590
plane.
00:26:40.590 --> 00:26:45.039
And suppose there is a, there is an analysis
function f, analytic holomorphic function
00:26:45.039 --> 00:26:47.600
f which is defined this domain.
00:26:47.600 --> 00:26:54.929
If suppose I have, suppose I have a path,
I have an arc in the domain, okay, which means
00:26:54.929 --> 00:27:00.759
that, you know I have this continuous, piecewise
continuously differentiable function gamma
00:27:00.759 --> 00:27:06.179
from an open, I am in a closed bounded interval
in the real line.
00:27:06.179 --> 00:27:10.070
And it is giving me this arc, with this as
the starting point and this as the ending
00:27:10.070 --> 00:27:15.419
point and this is also, the labelled arc also
is gamma, where, actually it is an image of
00:27:15.419 --> 00:27:20.169
gamma but I will, we will use, in abused notation
we also call that gamma.
00:27:20.169 --> 00:27:25.929
So we use gamma for the image of the arc,
we also use gamma for the function that parameterises
00:27:25.929 --> 00:27:29.200
the arc, okay.
00:27:29.200 --> 00:27:35.269
So well, you know what is, what is length
of gamma, length of gamma is as a told you
00:27:35.269 --> 00:27:40.409
just now, it is integral over gamma mod dz,
this is the length of gamma, okay.
00:27:40.409 --> 00:27:52.430
now what will you get if you integrate over
f circle gamma mod f dash of, mod
00:27:52.430 --> 00:28:09.429
f dash of z , so let me write this
, so let me write something here, mod dw,
00:28:09.429 --> 00:28:10.809
okay.
00:28:10.809 --> 00:28:18.409
So you see, so you know I am writing this
mapping as W equal to f of z, which is similar
00:28:18.409 --> 00:28:22.019
to Y equal to f of X, if you are looking at
functions for a sphere variable.
00:28:22.019 --> 00:28:25.299
X is the independent, independent variable,
Y is the dependent variable.
00:28:25.299 --> 00:28:31.389
but now z is the independent variable and
W is the dependent variable, W depends on
00:28:31.389 --> 00:28:33.390
z by means of the function f.
00:28:33.390 --> 00:28:39.619
So this z varies over this copy which is the
source complex plane and in fact z is varying
00:28:39.619 --> 00:28:43.539
in D, that is where you are function is defined.
00:28:43.539 --> 00:28:48.850
And then you have, well here is the target
complex plane, okay.
00:28:48.850 --> 00:28:55.269
And W is in the target complex plane, and
mind you that the image of f, I am of course
00:28:55.269 --> 00:28:59.809
assuming that f is not a constant function,
I am assuming f is a nonconstant analytic
00:28:59.809 --> 00:29:00.809
function.
00:29:00.809 --> 00:29:05.720
We have this theorem, that is nonconstant
analytic map is an open map, okay.
00:29:05.720 --> 00:29:10.379
Therefore the image of f is an open set, okay,
that is a very deep fact.
00:29:10.379 --> 00:29:15.580
So this is the, this is f of D is E1, that
is open actually.
00:29:15.580 --> 00:29:19.879
And of course we are not looking at the case
when f is a constant function, okay, f is
00:29:19.879 --> 00:29:21.409
not constant.
00:29:21.409 --> 00:29:27.090
I do not want to worry about that case, okay
because the image is only one point, okay.
00:29:27.090 --> 00:29:34.129
So f of d open and, well, what happens to,
what is f of, what is the image of gamma,
00:29:34.129 --> 00:29:37.279
well I will get another curve here.
00:29:37.279 --> 00:29:50.600
So I get this, I get this as f of gamma, okay,
this is just the image of the curve gamma
00:29:50.600 --> 00:29:52.970
under f.
00:29:52.970 --> 00:29:59.389
And you know if I now take the variable in
the target, if I take the variable in the
00:29:59.389 --> 00:30:05.470
target plane which is W and if I integrate
f circle gamma, okay, f circle gamma is just
00:30:05.470 --> 00:30:10.200
mind you f of gamma, it is just, what is f
circle gamma, you 1st apply gamma and then
00:30:10.200 --> 00:30:11.200
apply f.
00:30:11.200 --> 00:30:15.289
Which is the same as taking the image of gamma
under f, okay.
00:30:15.289 --> 00:30:20.519
So f circle gamma is in principle f gamma,
f of gamma, okay.
00:30:20.519 --> 00:30:27.369
And if you integrate f of gamma over f of
gamma dw, then you should get the length of
00:30:27.369 --> 00:30:28.369
f of gamma.
00:30:28.369 --> 00:30:34.820
So this is going to give you length of f of
gamma, okay, by what we have seen, right.
00:30:34.820 --> 00:30:38.460
1st what is, but what is, now you plug-in
what dw is.
00:30:38.460 --> 00:30:45.470
W is fz, so mod dw will become mod f dash
of z into mod dz.
00:30:45.470 --> 00:30:49.440
Okay, so what will happen, so let me write
that down, I will get, I need a little space,
00:30:49.440 --> 00:30:53.109
so let me rub this off and write it on the
open side.
00:30:53.109 --> 00:31:01.029
So this will be integral over f circle gamma
of mod f dash of z into mod dz, this is what
00:31:01.029 --> 00:31:02.029
I get.
00:31:02.029 --> 00:31:10.029
Okay, so if you look at it very carefully
, you see that if you normally integrate
00:31:10.029 --> 00:31:15.830
over dz, okay and of course you know since
I change the variable from W to z, I should
00:31:15.830 --> 00:31:22.820
change the curves from f circle gamma to gamma
itself because now my z is in the source and
00:31:22.820 --> 00:31:27.150
what is in the source is gamma, not f circle
gamma, f circle gamma is in the target, okay.
00:31:27.150 --> 00:31:30.440
So I should change this to gamma, so now,
now look at this.
00:31:30.440 --> 00:31:36.830
now compare the 1st equation which says that
L of gamma is gotten by integrating over gamma
00:31:36.830 --> 00:31:42.359
just mod dz and look at the 2nd equation,
what you have done is instead of simply integrating
00:31:42.359 --> 00:31:48.210
mod dz over gamma, you have multiplied it
by this mod f dash of z, okay.
00:31:48.210 --> 00:31:49.539
And what does it give you?
00:31:49.539 --> 00:31:57.950
It gives you the, it gives you the length
of the image of gamma under f, okay.
00:31:57.950 --> 00:32:04.419
If you simply integrate over mod dz, you get
the length but if you integrate over mod dz,
00:32:04.419 --> 00:32:13.090
the derivative of your function, okay, then
you get the length of the image of the curve
00:32:13.090 --> 00:32:15.769
under the function, okay.
00:32:15.769 --> 00:32:25.139
So this, so this is what I want to tell you,
the, the, whatever you put, whenever you integrate
00:32:25.139 --> 00:32:32.340
over gamma, mod dz and you put something in
the integrand, that something in the scaling
00:32:32.340 --> 00:32:33.340
factor, okay.
00:32:33.340 --> 00:32:37.570
It is a, it is a factor of scaling, all right.
00:32:37.570 --> 00:32:45.009
So if you for example you know if I put 2,
if I instead of integrating over gamma, integrating
00:32:45.009 --> 00:32:50.899
over gamma mod dz, suppose I integrate 2 mod
dz, I will get by the length of the curve.
00:32:50.899 --> 00:32:51.929
Okay.
00:32:51.929 --> 00:33:01.629
So and similarly if I integrate it with some
constant times mod dz, I will get, of course
00:33:01.629 --> 00:33:06.229
the constant should be positive constant
because you are worried about length, then
00:33:06.229 --> 00:33:08.710
you will get positive constant times that
length, okay.
00:33:08.710 --> 00:33:15.090
If I integrate, if I integrate let us say
K mod dz, I will get K times the length of
00:33:15.090 --> 00:33:16.179
the curve.
00:33:16.179 --> 00:33:21.279
That is because mod integrating just over
mod dz is going to give me the length of the
00:33:21.279 --> 00:33:23.999
curve as it is and it is multiplied by K,
the K will come out.
00:33:23.999 --> 00:33:27.740
but then this is all right if I am putting
a constant but instead of a constant I can
00:33:27.740 --> 00:33:35.669
actually put a function which varies as the
point z varies on the curve, okay.
00:33:35.669 --> 00:33:40.900
And then what will happen is that you are
scaling the, you are just scaling the, you
00:33:40.900 --> 00:33:45.749
are getting a scalar version of the length
of the curve.
00:33:45.749 --> 00:33:53.649
And the fact here is that multiplying by mod
f dash of z gives you the image, the length
00:33:53.649 --> 00:33:55.049
of the image curve.
00:33:55.049 --> 00:34:00.940
And the image curve is part of the original
curve scaled by the map f.
00:34:00.940 --> 00:34:08.119
The map f maps gamma, the original curve onto
the image f gamma, it scales it.
00:34:08.119 --> 00:34:12.860
And that scaling factor is mod f dash of z
and that is what appears in this integral,
00:34:12.860 --> 00:34:13.860
okay.
00:34:13.860 --> 00:34:18.970
So why I am saying all this is, I am trying
to say that, you know, you look at the coefficient
00:34:18.970 --> 00:34:27.840
of mod dz in the integral, that should give
you the modulus of the derivative, okay.
00:34:27.840 --> 00:34:33.690
The modulus of the derivative mod f dash of
z is the coefficient in the, it is the coefficient
00:34:33.690 --> 00:34:37.899
of more dz in the expression for the length
of the curve, okay.
00:34:37.899 --> 00:34:47.310
now what you have to do is, if you try to
mimic this with the following modifications,
00:34:47.310 --> 00:34:53.830
namely you take again a function f, defined
on a domain in the complex plane.
00:34:53.830 --> 00:34:58.220
but now I assume that the function is taking
values on the Riemann sphere, I mean which
00:34:58.220 --> 00:35:03.600
is essentially taking values of the external
complex plane, okay.
00:35:03.600 --> 00:35:12.860
And on the external complex plane you use
the spherical metric, then you know you would
00:35:12.860 --> 00:35:19.390
like to have an idea of what is the derivative
of the function with respect to the spherical
00:35:19.390 --> 00:35:23.820
metric, okay.
00:35:23.820 --> 00:35:29.070
What we are looking at is trying to get an
idea, we are trying to guess at what is the
00:35:29.070 --> 00:35:33.900
derivative of a function with respect to the
spherical metric, okay.
00:35:33.900 --> 00:35:35.630
And what does it mean?
00:35:35.630 --> 00:35:39.570
You are taking a function which is defined
on the domain of the complex plane, it is
00:35:39.570 --> 00:35:42.820
taking values in the external complex plane
because that is where the spherical metric
00:35:42.820 --> 00:35:47.160
makes sense and then you want to take the
derivative of the function with respect to
00:35:47.160 --> 00:35:48.490
the spherical metric, okay.
00:35:48.490 --> 00:35:53.680
It will be ideal if we get the, if we can
guess formal for that.
00:35:53.680 --> 00:35:59.900
but then more importantly what this argument
tells you that, if you take a curve in the
00:35:59.900 --> 00:36:08.340
domain, you take its image under the function
in the extended plane, that means you are
00:36:08.340 --> 00:36:11.620
actually looking at its image on the Riemann
sphere, okay.
00:36:11.620 --> 00:36:17.940
And if you find the length of that curve in
the image, in the Riemann sphere, that is
00:36:17.940 --> 00:36:24.760
with respect to the spherical metric and in
that if you take out the coefficient of, the
00:36:24.760 --> 00:36:31.460
coefficient of mod dz, okay or if you want
mod d of whatever variable you are using then
00:36:31.460 --> 00:36:35.450
that coefficient should give you the modulus
of the spherical derivatives, okay.
00:36:35.450 --> 00:36:39.750
So I am just trying to tell you how this argument
tells you how to guess what the modulus of
00:36:39.750 --> 00:36:43.060
spherical derivative is, okay.
00:36:43.060 --> 00:36:47.440
The only difference is you are not looking
at an analytic functions, you are looking
00:36:47.440 --> 00:36:51.750
at a function which is, which is having values
in the, not in the complex plane but in the
00:36:51.750 --> 00:36:53.110
extended complex plane.
00:36:53.110 --> 00:37:00.500
And mind you the advantage is that, because
you are allowing external complex plane, your
00:37:00.500 --> 00:37:04.080
function can take the value infinity, okay.
00:37:04.080 --> 00:37:08.390
And mind you once you do that, you can also
deal with meromorphic functions, okay.
00:37:08.390 --> 00:37:13.610
The whole purpose of allowing the value at
infinity is to deal with meromorphic functions.
00:37:13.610 --> 00:37:22.800
So you can now generalise all this to taking
the image under a meromorphic function of
00:37:22.800 --> 00:37:27.900
a curve in a domain and taking its image in
the external complex plane thought of as a
00:37:27.900 --> 00:37:30.960
Riemann sphere and trying to measure the spherical
distance.
00:37:30.960 --> 00:37:34.290
So whatever we did with the Euclidean distance,
you have to do with the spherical distance.
00:37:34.290 --> 00:37:36.930
So let me, let me try to quickly tell you
how that goes.
00:37:36.930 --> 00:37:53.560
So you see if you take the, so now, with respect
to spherical distance, so you see I want to
00:37:53.560 --> 00:37:55.010
do 1st check that.
00:37:55.010 --> 00:38:02.360
Well so if you take the, if you take the Riemann
sphere, which we draw like this, it is just
00:38:02.360 --> 00:38:09.550
a unit sphere in 3 space, real 3 space and
where they XY plane is thought of as the complex
00:38:09.550 --> 00:38:13.990
plane, the 3rd axis is called the U axis and
this is the north pole which corresponds to
00:38:13.990 --> 00:38:15.840
the point at infinity, okay.
00:38:15.840 --> 00:38:23.780
now you know you give me 2 points on the,
on the Riemann sphere.
00:38:23.780 --> 00:38:30.450
Suppose these are 2 points that corresponds
to 2 points on the plane, so, well, so there
00:38:30.450 --> 00:38:39.620
is a point z1 and there is a point, well
, well if I go by the diagram, then I
00:38:39.620 --> 00:38:41.470
will have to be careful.
00:38:41.470 --> 00:38:46.640
So let me, let me redraw this little bit.
00:38:46.640 --> 00:38:55.570
So let me have 2 points, so this is z1, let
say this is z2 and you know by the stereographic
00:38:55.570 --> 00:39:00.240
projection z1 will correspond to a point here
which is P1 and z2 will correspond to say
00:39:00.240 --> 00:39:05.510
another point here which is P2, okay.
00:39:05.510 --> 00:39:16.330
And, well and you know now on the Riemann
sphere, you know if I join this, if I join
00:39:16.330 --> 00:39:25.800
this cord, so let me use a different colour,
if I this cord from P1 to P2, so the cordal
00:39:25.800 --> 00:39:35.040
distance between P1 and P2 which you can define
to be either cordal distance between z1 and
00:39:35.040 --> 00:39:41.470
z2, okay, you can define it like that because
you just transported distances on the Riemann
00:39:41.470 --> 00:39:46.040
sphere to the external complex plane because
of the stereo graphic projection.
00:39:46.040 --> 00:39:49.650
And what is the cordal distance between between
P1 and P2, I want to do check this, it is
00:39:49.650 --> 00:39:50.930
a very simple exercise.
00:39:50.930 --> 00:39:58.180
It is actually, this is actually equal to
2 times modulus of z1 minus z2 by root of
00:39:58.180 --> 00:40:06.710
1plus mod z1 square into root of 1plus mod
z2 square, okay, this is what it is.
00:40:06.710 --> 00:40:12.260
I want, this is just analytic geometry, okay,
I want you to calculate this, okay.
00:40:12.260 --> 00:40:20.000
And then what happens is that, you see, as,
this is the cordal distance between 2 points
00:40:20.000 --> 00:40:24.330
and what it is, is actually, this is actually
the distance of those 2 points P1 and P2 in
00:40:24.330 --> 00:40:26.210
R3, okay.
00:40:26.210 --> 00:40:30.920
Mind you this whole diagram is in R3, they
XY plane in the complex plane, okay.
00:40:30.920 --> 00:40:36.590
You can check that this is the distance in
R3, it is a very simple analytic geometry
00:40:36.590 --> 00:40:37.590
calculation, right.
00:40:37.590 --> 00:40:43.890
now what is the, what is the spherical distance
between P1 and P2?
00:40:43.890 --> 00:40:51.970
It is the length of the arc, minor arc from
P1 to P2 on the major circle that passes through
00:40:51.970 --> 00:40:54.710
P1 and P2 on the Riemann sphere.
00:40:54.710 --> 00:40:59.460
So let me use a different colour for that
and that is going to be, let me use something
00:40:59.460 --> 00:41:01.430
else, let me use green, okay.
00:41:01.430 --> 00:41:09.930
So here is the, so here is this, so I also
have this d sub, so there is this d sub s,
00:41:09.930 --> 00:41:17.060
this is the spherical distance between z1
and z2, okay and how do, how do you get the
00:41:17.060 --> 00:41:19.000
spherical distance?
00:41:19.000 --> 00:41:25.600
Mind you it is the length of an arc, it is
the length of this arc, the circular arc along
00:41:25.600 --> 00:41:27.690
the great circle passing through P1 and P2.
00:41:27.690 --> 00:41:31.770
And how do you get the length of the arc,
how do you get the length of an arc, you get
00:41:31.770 --> 00:41:37.750
the length of the arc by integrating an element
of the infinitesimal distance, okay.
00:41:37.750 --> 00:41:43.650
now the cordal distance, as P1 and P2 comes
closer, the cordal distance infinitesimally
00:41:43.650 --> 00:41:47.290
becomes a spherical distance, okay.
00:41:47.290 --> 00:41:53.350
Therefore and you know if you make z1 and
z2 closer, if you put z1 equal to that and
00:41:53.350 --> 00:41:58.230
put that vehicle to z plus Delta z, you know
what you will get is that you will get the
00:41:58.230 --> 00:42:04.860
element of spherical distance, you will get
this, you will get this, you will get this
00:42:04.860 --> 00:42:05.860
expression.
00:42:05.860 --> 00:42:14.610
You will simply get mod dz by 1plus mod z
the whole square, okay. because you see in
00:42:14.610 --> 00:42:22.110
this in this expression here, in this expression
you put the z2 equal to z1 plus Delta z, okay
00:42:22.110 --> 00:42:27.890
and you look at only Delta z terms and you
change the Delta to d, you will get this,
00:42:27.890 --> 00:42:29.040
all right.
00:42:29.040 --> 00:42:33.030
And of course, of course there is 2, I forgot
2, that is 2 here.
00:42:33.030 --> 00:42:38.750
And therefore the spherical distance between
z1 and z2 is simply given by integrating,
00:42:38.750 --> 00:42:46.600
okay along the arc of the great circle
from b1 to P2, that is the same as integrating
00:42:46.600 --> 00:42:53.230
along this line segment L, because we image
of this line segment L on the Riemann sphere
00:42:53.230 --> 00:42:57.400
is exactly that great, that minor arc of the
great circle passing through P1 and P2.
00:42:57.400 --> 00:43:08.420
So this integral over L, 2 times mod dz by
1plus mod z the whole square, okay.
00:43:08.420 --> 00:43:15.070
So this is the spherical distance between
2 points on the, on the complex plane.
00:43:15.070 --> 00:43:21.560
And this works even if you assume one
of the points to be the point at infinity.
00:43:21.560 --> 00:43:27.090
Mind you because your points can vary on the
external complex plane, so both points could
00:43:27.090 --> 00:43:31.640
be the point the north pole itself, okay.
00:43:31.640 --> 00:43:37.770
fine, so now what we can do is that we can
use this to guess what the spherical derivative
00:43:37.770 --> 00:43:38.770
should be.
00:43:38.770 --> 00:43:42.510
We can use this and the previous argument
to guess what the spherical distance should
00:43:42.510 --> 00:43:43.510
be.
00:43:43.510 --> 00:43:49.540
And I will just state it and stop.
00:43:49.540 --> 00:44:11.830
The spherical derivative is, of f is f hash
of z, it is 2 into mod f dash of z 2 by
00:44:11.830 --> 00:44:19.010
1plus f mod fz the whole square, this
is what it will be.
00:44:19.010 --> 00:44:24.770
This is the, this is the, this is absolute
value with respect to derivative.
00:44:24.770 --> 00:44:29.180
The spherical derivative, so I should say
absolute value.
00:44:29.180 --> 00:44:34.970
This is what you will get if you think about
it and I will explain this in the next lecture,
00:44:34.970 --> 00:44:35.540
okay.