WEBVTT
Kind: captions
Language: en
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All right, so we continue with our discussion
of the spherical derivative, okay. So there
00:00:57.410 --> 00:01:03.830
are few things i wanted to point out with
regards to the spherical derivative, okay.
00:01:03.830 --> 00:01:08.320
So let me, so let me just recall.
00:01:08.320 --> 00:01:24.430
If f of z z in a domain d, z varying over
capital d which is a domain in c is a meromorphic
00:01:24.430 --> 00:01:41.890
function, function, that is f belongs to m
of d and mind you the set of meromorphic functions
00:01:41.890 --> 00:01:50.670
on d is consider now as a subset of, you know
in fact continuous maps from d to the extended
00:01:50.670 --> 00:01:58.600
complex plane which is c union infinity, okay.
So the script c denotes continuous maps and
00:01:58.600 --> 00:02:03.960
the point is that you make the meromorphic
function continuous even at the poles by defining
00:02:03.960 --> 00:02:13.360
the function value at the pole to be infinity,
okay. So you consider it like this and then
00:02:13.360 --> 00:02:29.549
the spherical derivative of f is f hash
of z, it is defined to be 2 times more f dash
00:02:29.549 --> 00:02:39.860
of z divided by 1plus mod of fz also. This
is the definition of the spherical derivative.
00:02:39.860 --> 00:02:48.530
And mind you why, why did we need the spherical
derivative is because of the following reason.
00:02:48.530 --> 00:02:54.840
So suppose the, you have this is the complex
plane with the variable z and you have this,
00:02:54.840 --> 00:03:01.760
suppose this is your this, the area inside
this dotted region is your domain d and suppose
00:03:01.760 --> 00:03:12.590
you had an arc gamma inside d, you take the
image of this arc gamma under f in the external
00:03:12.590 --> 00:03:17.690
complex plane, okay. So it means that you
know you are allowing also the value infinity,
00:03:17.690 --> 00:03:23.769
so for example you know gamma may pass through
your pole f. F is a meromorphic function,
00:03:23.769 --> 00:03:30.459
so f is meromorphic on d means f is holomorphic,
that is analytic on d, except for a subset
00:03:30.459 --> 00:03:34.260
of isolated points of d where f has poles.
00:03:34.260 --> 00:03:40.980
But at the poles also, the value of f has
defined to be infinity. So your gamma, your
00:03:40.980 --> 00:03:46.379
curve gamma pass through the poles and that
is the technical thing that i want to explain
00:03:46.379 --> 00:03:51.909
to you about. Now you identify this external
complex plane via the stereo graphic projection
00:03:51.909 --> 00:04:01.069
with the riemann sphere which i will briefly
draw like this so this is riemann sphere,
00:04:01.069 --> 00:04:12.010
which is s2, okay. And this, this isomorphism
is actually a homeomorphism given by the
00:04:12.010 --> 00:04:21.590
sphere graphic projection, this is the stereo
graphic projection with the, with the point
00:04:21.590 --> 00:04:24.860
infinity going to the north pole which is
this point here, all right.
00:04:24.860 --> 00:04:32.960
And the fact is that the image of gamma will,
see gamma will give you, you know, if you
00:04:32.960 --> 00:04:39.060
take the image of gamma, what will happen
is that you will get some curve here on the
00:04:39.060 --> 00:04:43.520
riemann sphere, okay. So it is a curve in
the external complex plane but you know, you
00:04:43.520 --> 00:04:47.641
are thinking of the external complex plane
as the riemann sphere when you think of, you
00:04:47.641 --> 00:04:52.710
may imagine that the image of gamma is a curve
on the riemann sphere itself, okay. And what
00:04:52.710 --> 00:04:59.960
is that curves, since this, this is just s
of gamma, okay, this is f of gamma and what
00:04:59.960 --> 00:05:04.520
is the big deal about the spherical derivative,
the big deal about the spherical derivative
00:05:04.520 --> 00:05:10.050
is that you can get the spherical length of
f of gamma, okay.
00:05:10.050 --> 00:05:15.240
You can calculate the length of that image
curve, okay, and i have put subscript s for
00:05:15.240 --> 00:05:20.930
spherical lens because it is the length you
are computing the arc length on the sphere,
00:05:20.930 --> 00:05:25.720
okay. And how do you get it? You get it in
the following way, you simply integrate over
00:05:25.720 --> 00:05:35.669
gamma with the variable, see normally if you
now integrate over modern dz, if you integrate
00:05:35.669 --> 00:05:41.680
over mod dz simply on the plane over a curve
gamma, you simply get the arc length of the
00:05:41.680 --> 00:05:47.630
curve, okay. That is what integrating over
mod dz means because mod dz is infinitesimal
00:05:47.630 --> 00:05:54.639
arc length on the euclidean plane, on c common
complex plane, on the complex plane thought
00:05:54.639 --> 00:05:56.300
of as r2, okay.
00:05:56.300 --> 00:06:04.669
It is usual arc length, but you know if you
put, if instead of doing this, suppose
00:06:04.669 --> 00:06:11.039
i put, if i add the magnification factor given
by the, suppose i add the magnification factor
00:06:11.039 --> 00:06:18.289
given by the spherical derivative. So that
means i put f hash of z here and do this.
00:06:18.289 --> 00:06:24.780
Then what you will get is , i will get
actually the length of the image curve on
00:06:24.780 --> 00:06:31.449
the riemann sphere, okay. And so this is where
the spherical derivative is used, okay, the
00:06:31.449 --> 00:06:36.440
spherical derivative will give you, it is,
so without this if i do not, see if i remove
00:06:36.440 --> 00:06:41.750
this spherical derivative factor, okay, i
will get simply integral of us gamma mod dz
00:06:41.750 --> 00:06:43.759
and that is just length of gamma.
00:06:43.759 --> 00:06:49.590
But if you put the spherical derivative there,
okay, then i will not get the length of gamma
00:06:49.590 --> 00:06:54.130
but i will get the length of the image of
gamma under f. And mind you gamma can pass
00:06:54.130 --> 00:06:59.729
through, it can pass through a pole, the only
thing is it means that this image curve will
00:06:59.729 --> 00:07:04.400
pass through the north pole, that is all,
it is not going to create any problems. Because
00:07:04.400 --> 00:07:08.360
if it passes through a pole, the function
value there is infinity at infinity corresponds
00:07:08.360 --> 00:07:12.510
to the north pole on the riemann sphere under
the stereo graphic projection, okay. So the
00:07:12.510 --> 00:07:16.260
point is important that the spherical derivative
is that it gives you the spherical length,
00:07:16.260 --> 00:07:17.260
okay.
00:07:17.260 --> 00:07:26.759
But there are, there are a few technical things
about this , there are a few technical
00:07:26.759 --> 00:07:31.139
things about the spherical derivative which
i just indicated towards the end of my last
00:07:31.139 --> 00:07:38.400
lecture and i want to be more you know elaborate
about that. So you see, so i want to draw
00:07:38.400 --> 00:07:44.479
your attention to the, to this formula which
is a formula for the spherical derivative,
00:07:44.479 --> 00:07:51.169
okay. This is the formula for the spherical
derivative, there is something that is a little
00:07:51.169 --> 00:07:52.880
troublesome about this formula.
00:07:52.880 --> 00:07:58.880
See when i have defined and you go before
that, let me also tell you that here in this
00:07:58.880 --> 00:08:03.720
formula for length, spherical length of f
of gamma, you know if i replace the spherical
00:08:03.720 --> 00:08:10.080
derivative, if i instead of putting f hash
of z, suppose i put mod f dash of z, suppose
00:08:10.080 --> 00:08:16.569
i put modulus of the derivative of f, then
what i will get is actually the, and i assume
00:08:16.569 --> 00:08:22.910
that f is you know the holomorphic function.
Then i will get the image, the length of the
00:08:22.910 --> 00:08:29.819
image of gamma under f. Then f will map only
into the complex plane, if f is differentiable,
00:08:29.819 --> 00:08:35.680
okay, everywhere on gamma, okay, then it is
a meromorphic.
00:08:35.680 --> 00:08:39.699
So the image of gamma will lie in the plane
itself, it is not going to go to infinity,
00:08:39.699 --> 00:08:46.310
okay because there are no poles, okay. And
because f is, if you assume f to be an analytic
00:08:46.310 --> 00:08:52.130
function, okay. And then if you integrate
over gamma mod f dash of z into mod dz, what
00:08:52.130 --> 00:08:56.881
you will get is the length of the image under
f but this will be the euclidean length, it
00:08:56.881 --> 00:09:05.800
will be just length on the plane. But if you
integrate over gamma, mod dz with the coefficient
00:09:05.800 --> 00:09:13.660
f hash of z which is a spherical derivative
and in addition you allow also f to be meromorphic,
00:09:13.660 --> 00:09:19.980
you will actually get the length of the image
curve on the riemann sphere, part of the extended
00:09:19.980 --> 00:09:21.940
plane, that is what you understand.
00:09:21.940 --> 00:09:28.980
Okay, fine, you know there is a problem with
this at 1st sight with this definition of
00:09:28.980 --> 00:09:35.770
f hash because you see there is this f dash,
okay. F dash is the derivative of f at the
00:09:35.770 --> 00:09:41.110
point z but the problem is that if z is a
pole, then you are in trouble. At the pole
00:09:41.110 --> 00:09:45.000
the function is certainly not differentiable,
it is a single point, it is a pucca similar
00:09:45.000 --> 00:09:47.750
point, it is an honest singular point, it
is not a removable singularity, okay, the
00:09:47.750 --> 00:09:56.760
function is not differentiable. Alright. So
you are in trouble, so when i wrote this definition
00:09:56.760 --> 00:10:03.080
last time, you know i was only, you know i
was trying to heuristically tell you things
00:10:03.080 --> 00:10:07.330
but now i am going to tell you things more
seriously, so let us worry about, let us worry
00:10:07.330 --> 00:10:10.690
about this, this, this situation.
00:10:10.690 --> 00:10:19.090
So here is my, here is my domain d which is
interior of this dotted line, it is an open
00:10:19.090 --> 00:10:27.100
connected site, okay. This is inside the complex
plane and suppose i have a point z0 and of
00:10:27.100 --> 00:10:33.390
course i have this map f, f is a meromorphic
function of d and you know of course f is
00:10:33.390 --> 00:10:43.430
taking values in c union infinity and z0 is
a pole of f, of order let say, of order n,
00:10:43.430 --> 00:10:51.680
okay. And of course z0 will go to f of z0,
which is by definition infinity, okay, this
00:10:51.680 --> 00:10:58.040
is our definition. Now what about the spherical
derivative? Okay.
00:10:58.040 --> 00:11:06.790
See, what is, so the question is what this
f dash of z. So this is an, this is an issue,
00:11:06.790 --> 00:11:14.790
you see because what is f dash of z. If we
have to worry about this, the reason is because,
00:11:14.790 --> 00:11:20.750
see suppose i have a gamma, suppose i have
a path gamma passing through a pole, okay,
00:11:20.750 --> 00:11:25.320
then my formula for the length of gamma, the
spherical length of, the image of gamma under
00:11:25.320 --> 00:11:29.430
f on the riemann sphere which is identified
in the extended plane, what is the formula,
00:11:29.430 --> 00:11:36.440
it is l spherical of f of gamma, you know
that is what i have shown in the previous
00:11:36.440 --> 00:11:43.170
slide , the spherical length of f of gamma
is integral, you integrate over gamma.
00:11:43.170 --> 00:11:47.860
So i put, if i put just mod dz, i will get
just the length of gamma but i have put the
00:11:47.860 --> 00:11:52.950
magnification factor given by the spherical
derivative of f with respect to z. And what
00:11:52.950 --> 00:12:00.320
is a spherical derivative of f with respect
to z? It is, well it is f hash of z, you can
00:12:00.320 --> 00:12:09.350
write it, it is 2 times mod f dash of z divided
by 1plus more fz the whole square, this is
00:12:09.350 --> 00:12:18.020
what it is. It was there in the previous slide
also. Okay, now the point is, if i put z equal
00:12:18.020 --> 00:12:24.420
to z0, z0, see now gamma passes through gamma
passes through z0, okay.
00:12:24.420 --> 00:12:30.170
So when i calculate this integral on the right
side gamma i have, when you do, when you do
00:12:30.170 --> 00:12:34.430
and integration the variable of integration
will lie on the region of integration. In
00:12:34.430 --> 00:12:41.079
this case region of integration is the path
gamma. So z will pass through z0, it will
00:12:41.079 --> 00:12:46.240
bury, at some point z will become z0. But
when z become z0, there is this integrand
00:12:46.240 --> 00:12:51.209
which is f dash, f hash of z, the spherical
derivative, that is in trouble. Because you
00:12:51.209 --> 00:12:58.260
know f hash of z depends on f dash of z0 in
the numerator. F hash of z0 will be, will
00:12:58.260 --> 00:13:02.070
involve f dash of z0 but f dash of z0 does
not make sense.
00:13:02.070 --> 00:13:08.769
Why, because z0 is a pole, i cannot differentiate
at a pole, i just cannot find derivative at
00:13:08.769 --> 00:13:12.639
a pole. So what is it, what is the big deal?
So there is, so you see this formula as we
00:13:12.639 --> 00:13:17.839
have written it last time has this issue that
has to be fixed. And the reason is because,
00:13:17.839 --> 00:13:24.930
the fact is that, as i was telling you last
time, even at z0, this f dash of z0 is not
00:13:24.930 --> 00:13:31.019
defined but this spherical derivative is defined
as it finite quantity. That is the beauty,
00:13:31.019 --> 00:13:36.990
that is the reason why this integral works,
okay, that is what i want to explain to you.
00:13:36.990 --> 00:13:43.420
So you see, so let me, so let me say that,
so you see, let us assume that z0 is a pole
00:13:43.420 --> 00:13:44.550
of order n.
00:13:44.550 --> 00:13:48.980
So then what happens is that you know you
will get a small disc surrounding z0, so let
00:13:48.980 --> 00:13:56.100
me use a different colour, see i will get
a small disc surrounding z0, okay, i can find
00:13:56.100 --> 00:14:00.630
a small disc surrounding z0 where z0 is the
only pole, okay. Because you know the poles
00:14:00.630 --> 00:14:05.610
of an analytic function are isolated in any
case. And in fact our meromorphic functions
00:14:05.610 --> 00:14:10.480
are supposed to be having only pole singularities,
okay. These are the only singularity that
00:14:10.480 --> 00:14:15.690
are allowed, okay. So i can find a small disc
surrounding z0 where z0 is the only pole and
00:14:15.690 --> 00:14:20.560
well, you know, if you, if you call this,
if you call the radius of this disc as say
00:14:20.560 --> 00:14:28.060
epsilon, okay, then mod z minus z0 less than
epsilon, which is interior of that small disc.
00:14:28.060 --> 00:14:36.350
You see you can write f of z, you can write
f of that as you know g of z divided by z
00:14:36.350 --> 00:14:51.740
minus z0 to the power of n where g of
, where g of z0 is not 0, okay. So you can,
00:14:51.740 --> 00:15:01.329
you can write it like this. And of course
g analytic in mod z minus z0 less
00:15:01.329 --> 00:15:08.990
than epsilon, okay. So, okay, i will give
a little more space, let me rewrite that.
00:15:08.990 --> 00:15:18.649
Let me just write g analytic. I can do this
because this is how the function looks near
00:15:18.649 --> 00:15:25.300
a pole of order capital n, okay. Now, now
watch carefully. Let us calculate the derivative
00:15:25.300 --> 00:15:29.480
of f, not at z0 because at z0 you cannot calculate
the derivative.
00:15:29.480 --> 00:15:35.881
But in the deleted neighbourhood of z0, let
us calculate f of z. And you also when i,
00:15:35.881 --> 00:15:42.600
when i write this in mod z minus z0 less than
epsilon, of course z should not be z0. I mean
00:15:42.600 --> 00:15:49.089
i cannot literally plug-in z equal to z0 because
z equal to z0 is a for land z minus z0, denominator
00:15:49.089 --> 00:15:53.900
vanishes, i cannot write that. Of course we
have agreed to put it as z equal to z0 and
00:15:53.900 --> 00:15:58.600
equate it to infinity, that is when you consider
f to be a function with values in c union
00:15:58.600 --> 00:16:03.560
infinity but nevertheless if you think of
it and the usual function, then you do not
00:16:03.560 --> 00:16:07.300
plug-in z equal to z0 because you do not divide
by 0, okay.
00:16:07.300 --> 00:16:11.210
And that is the situation you must be, if
you want, if you want really differentiate
00:16:11.210 --> 00:16:20.320
this, okay. So you see, so now let us calculate,
let us do this calculation. See what is, so
00:16:20.320 --> 00:16:27.870
let me write this, in mod z minus z0 less
than epsilon, z not equal to z0, what is f
00:16:27.870 --> 00:16:35.339
dash of z. F dash of z is just d by dz of
f of z which is now g of z by z minus z0 to
00:16:35.339 --> 00:16:44.110
the power of n, okay. And you can calculate
this by if you want quotient rule from basic
00:16:44.110 --> 00:16:50.300
calculus. What will be, i will get z minus
z0 to the power of 2n and i will get, what
00:16:50.300 --> 00:17:00.420
will i get here, z minus z0 to the power of
ng dash of z plus, minus g of z ,n z minus
00:17:00.420 --> 00:17:08.130
z0 to the power of n minus1, okay, this is
what i get, if i do this computation.
00:17:08.130 --> 00:17:16.179
And you see now what you do is, see this
is all right for z not equal to z0, all right.
00:17:16.179 --> 00:17:23.470
And you will certainly have a problem if you
let z tend to z0, okay. Limit, if you actually
00:17:23.470 --> 00:17:31.659
see in the usual sense if i let z tends to
z0, then what will happen is that in the numerator
00:17:31.659 --> 00:17:39.670
the 1st term will go, all right, the 2nd term,
well it will go, provided n is greater than
00:17:39.670 --> 00:17:45.650
1, okay. If n equal to1, i will get gz0, okay
but the problem will be the denominator. As
00:17:45.650 --> 00:17:51.790
z tends to z0, i will end up with, essentially
i will, what will happen is that because f
00:17:51.790 --> 00:17:56.960
has a pole of order capital n at z0, its derivative
will have a pole of capital, order capital
00:17:56.960 --> 00:18:00.120
n plus1 at z0, okay.
00:18:00.120 --> 00:18:04.179
That is what could happen, so it is going
to only get worse, limit z tend to z0, f dash
00:18:04.179 --> 00:18:10.030
of z will not exist. And if, in fact the worst-case
you want to make it exists is you can define
00:18:10.030 --> 00:18:14.710
it to be infinity by thinking of f dash also
is a meromorphic function but now with values
00:18:14.710 --> 00:18:18.980
in c union infinity, you can do that. But
in any case it is not a finite, it is not,
00:18:18.980 --> 00:18:24.250
it is not a proper limit in the usual sense,
okay, you have to include the value infinity.
00:18:24.250 --> 00:18:34.840
But then, so i will say limit z tends to z0,
f dash of z does not, it is not a complex
00:18:34.840 --> 00:18:38.700
number, okay. It is, if you, if you include
c union infinity, then you can call it as
00:18:38.700 --> 00:18:40.650
infinity, that is that, but that is not the
case.
00:18:40.650 --> 00:18:46.020
We do not want to include the value infinity
when we are talking about derivative. But,
00:18:46.020 --> 00:18:55.760
you see, what look at what on the other hand
you look at what is f hash of z. Look at the
00:18:55.760 --> 00:18:59.830
spherical derivative, if you look at the spherical
derivative, what i will guess, i will get,
00:18:59.830 --> 00:19:07.270
c will get 2 times modulus of f dash of z,
okay, so i will get 2 times modulus of this
00:19:07.270 --> 00:19:17.920
whole quantity, okay, divided by 1plus mod
f the whole square and mod f the whole square
00:19:17.920 --> 00:19:24.730
will be, mod f is the modulus of this quantity,
so it will be 1plus modulus of g of z by z
00:19:24.730 --> 00:19:29.900
minus z0 to the power of n the whole square,
this is what i will get.
00:19:29.900 --> 00:19:36.140
Okay, this is what i will get and now you
take limit z tends to z0. Now you take limit
00:19:36.140 --> 00:19:47.440
z tends to z0, of f hash of z you take, you
calculate this limit. What will happen if
00:19:47.440 --> 00:19:54.720
you see the, in the denominator you have 1plus
mod gz the whole squared divided by mod z
00:19:54.720 --> 00:20:02.510
minus z0 to the 2n, okay. So denominator will
go through infinity, the denominator will
00:20:02.510 --> 00:20:08.020
go to infinity faster than the numerator,
so this, so the whole quantity will be bounded
00:20:08.020 --> 00:20:15.770
as z tends to z0, that is the whole point.
So you see if you calculate it, okay, if you
00:20:15.770 --> 00:20:24.510
calculate it, what will happen, so, so let,
so this exists. So if you write it out, you
00:20:24.510 --> 00:20:30.280
know i am going to get, so i will, to simplify
things i will multiply both numerator and
00:20:30.280 --> 00:20:36.990
denominator by z minus, mod of z minus z0
to 2n, which is what is the common denominator.
00:20:36.990 --> 00:20:42.310
So what i will get is that i will get, well
let me write it here, f dash of, sorry f hash
00:20:42.310 --> 00:20:49.809
of z is going to be 2 times, i will get the
numerator of this, which is more z minus z0
00:20:49.809 --> 00:21:04.440
to the n, so i will get this, mod z minus
z0 to the power n gz minus gz, oops, i think
00:21:04.440 --> 00:21:15.790
that must have been, that is g dash of z minus
g of zn into z minus z0 to the n minus1 mod
00:21:15.790 --> 00:21:24.110
divided by, okay, i have multiplied by, multiplied
by this modulus of this quantity, mod z minus
00:21:24.110 --> 00:21:29.850
z0 to the 2n, okay. So that is gone, so in
the denominator i will get more z minus z0
00:21:29.850 --> 00:21:36.820
to the 2n plus mod gz the whole square.
00:21:36.820 --> 00:21:44.120
This is what i will get if i multiply it by
mod z minus z0 to the 2n, okay. And mind you
00:21:44.120 --> 00:21:47.659
the spherical derivative is an absolute derivative,
so it is only absolute value, so it is a nonnegative
00:21:47.659 --> 00:21:53.990
real valued by the way. Now you, now you do,
if you take limit at z tends to z0, what is
00:21:53.990 --> 00:22:01.970
going to happen. You see as z tends to z0,
this term will vanish because z minus z0 power
00:22:01.970 --> 00:22:06.649
n is there and of course this n is of course
greater than or equal to1. It is the order
00:22:06.649 --> 00:22:11.890
of the pole, so this whole of order 1 or higher,
okay. So this is going to vanish and this
00:22:11.890 --> 00:22:18.669
fellow here, what will happen here, depends
on whether n equal to1 or n is greater than
00:22:18.669 --> 00:22:19.760
1, okay.
00:22:19.760 --> 00:22:24.790
See, if, if n is equal to1, what is going
to happen, if n is equal to1 then this term
00:22:24.790 --> 00:22:35.330
does not exist, okay. And let z tend to z0,
i will get, i will get 2 times mod gz 0, g
00:22:35.330 --> 00:22:41.780
is anyway mind you in analytic, discontinuous,
so z tends to z0 gz, g z0 and modulus also
00:22:41.780 --> 00:22:46.929
a continuous function, so i can push the limit
inside the variable, okay, inside the argument
00:22:46.929 --> 00:22:51.730
of the function. And then that is what i said
in the numerator, so this is, this is if n
00:22:51.730 --> 00:22:57.480
is 1, okay, this is if n is 1. And in the
denominator what i am going to get, this term
00:22:57.480 --> 00:23:02.720
is going to vanish as z tends to z0, i am
going to simply get mod g, again i will get
00:23:02.720 --> 00:23:08.580
mod gz0 the whole squared, i will get divided
by mod gz 0 the whole square, which is just
00:23:08.580 --> 00:23:12.150
able to by mod g z0, this is what i will get.
00:23:12.150 --> 00:23:22.710
And mind you gz0 is not 0 because g z0 is
, g is the, you know if you want, g is
00:23:22.710 --> 00:23:29.480
analytic function divided by z minus z0 power
n which is equal to f in the neighbourhood
00:23:29.480 --> 00:23:35.799
of f. In fact you know g z0 is, if you check
very carefully, g z0 is, if the coefficient
00:23:35.799 --> 00:23:44.170
of the, of 1 by z minus z0 power n, if you
write down the lagrange expansion, okay and
00:23:44.170 --> 00:23:50.820
that is not supposed to be 0, okay, because
g, f has a pole of capital n, right. So this
00:23:50.820 --> 00:23:52.240
is what you will get.
00:23:52.240 --> 00:24:00.460
And you see, and mind you in the case that
n equal to1, g z0 is actually the coefficient
00:24:00.460 --> 00:24:06.370
of 1 by z minus z0 power n which is 1 by z
minus z0. But you know what is the efficient
00:24:06.370 --> 00:24:11.470
of 1 by z minus z0 called, it is called the
residue. So actually this is 2 divided by
00:24:11.470 --> 00:24:20.340
your residue of f at z0, that is what it is.
This is nothing but 2 divided by modulus of
00:24:20.340 --> 00:24:27.980
residue of f at z0, this is what happens if
you get, if f is a simple pole, capital n
00:24:27.980 --> 00:24:34.510
equal to1, all right. And the point is that,
now if n is greater than 1, everything is
00:24:34.510 --> 00:24:39.250
gone, because you see is n is greater than
1, there is no following the denominator,
00:24:39.250 --> 00:24:42.929
i will get mod g the whole squared, this term
is anyways going to vanish.
00:24:42.929 --> 00:24:48.310
And the numerator will also go now, numerator
has z minus z0 term common, so it is going
00:24:48.310 --> 00:24:55.799
to go. So i will get 0 if n is greater than
1. So here is the, so here is the, so of course
00:24:55.799 --> 00:25:05.290
this is on the, i forgot to write f hash of
z. So here is a nice thing, f hash of z0,
00:25:05.290 --> 00:25:13.100
you can now call, see you can define f hash
of z0 by continuity to be equal to limit z
00:25:13.100 --> 00:25:20.600
tends to z0 f hash of z, okay. If you think
of f hash as a continuous function, okay,
00:25:20.600 --> 00:25:24.110
if you want to think of the spherical derivative
as a continuous function, then it is natural
00:25:24.110 --> 00:25:30.290
define f hash at z0 to be the limit as z tends
to z0 as f hash of z, okay.
00:25:30.290 --> 00:25:35.720
And you see, this, what this does is that
it makes the spherical derivative continuous
00:25:35.720 --> 00:25:42.250
even at z0 and mind you z0 is a pole. So what
this tells you is that the spherical derivative
00:25:42.250 --> 00:25:49.340
f hash of z is continuous at all poles. So
it is continuous throughout domain and therefore
00:25:49.340 --> 00:25:57.091
because it is continuous at all, throughout
the domain, this formula is valid, okay. What
00:25:57.091 --> 00:26:02.850
i really meant here is, what is f hash of
z0, okay. Of course f dash of z0 does not
00:26:02.850 --> 00:26:11.840
make sense, so the question is what is f hash
of z0, all right. So , so now you know
00:26:11.840 --> 00:26:20.760
f hash makes sense even at poles, so this
integral is well-defined, there is no issue.
00:26:20.760 --> 00:26:26.540
You can blindly integrate f hash, you can,
you cannot blindly integrate f dash because
00:26:26.540 --> 00:26:33.350
f dash will not exist at a pole. You can integrate
f dash, mod f dash only where so long
00:26:33.350 --> 00:26:38.080
as you are an apart which is not going through
any poles. And if it is going through a pole,
00:26:38.080 --> 00:26:43.860
you cannot integrate f dash but you can integrate
f hash always, even if you are passing through
00:26:43.860 --> 00:26:47.700
a pole, that is a big deal, that is a big
deal. So that is the reason why this formula
00:26:47.700 --> 00:26:53.710
works. And what this calculation we did just
tells you is that the spherical derivative
00:26:53.710 --> 00:27:08.200
is actually 2 divided by modulus of the residue
of f at simple pole z0 if z0 is a simple pole
00:27:08.200 --> 00:27:24.299
and it is 0 if not, this is not means, i mean
pole of higher-order, all right.
00:27:24.299 --> 00:27:31.500
So the moral the story is that you know, you
are in, you are in good shape. F hash spherical
00:27:31.500 --> 00:27:37.480
derivative is a very nice thing, okay. And
therefore when you, whenever you want to find
00:27:37.480 --> 00:27:48.799
the arc length, you can integrate mod dz over,
multiplied with, with the, you know integrand
00:27:48.799 --> 00:27:55.980
as f hash and that is pretty important. Now,
and you know again, i will tell you why we
00:27:55.980 --> 00:28:03.010
are doing all this, we are doing all this
because you know somehow the kind of analysis
00:28:03.010 --> 00:28:07.100
that is required to prove picard’s theorem
is involves montel’s theorem, okay.
00:28:07.100 --> 00:28:12.490
And this, i will tell you roughly the idea
is that you know there are, there are these,
00:28:12.490 --> 00:28:21.270
there is a, there is a very close relationship
as i told you between compactness and sequential
00:28:21.270 --> 00:28:30.020
compactness and equi-continuity and normal,
normal convergence, okay and bounded mass
00:28:30.020 --> 00:28:35.230
of the derivatives, okay. So this is the,
this is a bunch of results same analysis which
00:28:35.230 --> 00:28:43.960
is usually covered by the arzela ascoli theorem,
okay. And that is a, there is a, the montel’s
00:28:43.960 --> 00:28:50.289
theorem is something that comes out of that,
okay. And why we are doing all this is because
00:28:50.289 --> 00:28:56.169
you know you, basically you know the idea
is that you want to look at the space of meromorphic
00:28:56.169 --> 00:28:58.529
functions on a domain, okay.
00:28:58.529 --> 00:29:03.640
So you have some domain, all right, this is
a domain in extended plane, it could include
00:29:03.640 --> 00:29:08.880
infinity also, okay, on that domain you are
looking as meromorphic functions, all right.
00:29:08.880 --> 00:29:12.610
And you are looking at, you want to think
of them at least as continuous functions,
00:29:12.610 --> 00:29:17.809
so you are allowing the value infinity at
a pole. So you are looking at that pace of
00:29:17.809 --> 00:29:23.710
meromorphic functions and you see the convergence
that you are worried about is normal convergence,
00:29:23.710 --> 00:29:27.520
okay, it is not uniform convergence everywhere,
it is only uniform convergence restricted
00:29:27.520 --> 00:29:29.200
to compact sets, which for normal convergence.
00:29:29.200 --> 00:29:36.300
And with this convergence idea you want to
study apology of this space of functions.
00:29:36.300 --> 00:29:40.309
And explicitly what kind of topology you want
to study compactness, you want to study compactness,
00:29:40.309 --> 00:29:44.429
okay. You want to study compactness and you
know if you have studied, for example in euclidean
00:29:44.429 --> 00:29:50.290
space, compactness is as good as sequential
compactness which is the same as saying that
00:29:50.290 --> 00:29:55.149
you know every sequence, if you have an infinite
sequence, there is always a convergence of
00:29:55.149 --> 00:30:02.260
sequence, all right. And so you have compactness
is somehow strong related to sequential compactness,
00:30:02.260 --> 00:30:03.260
okay.
00:30:03.260 --> 00:30:08.820
So basically you want, basically given, given
a sequence you always want a convergence of
00:30:08.820 --> 00:30:14.220
sequence, all right. And now you want this
also to happen for meromorphic functions,
00:30:14.220 --> 00:30:19.380
that is the, that is the central idea, the
central idea is give me a bunch of, give me
00:30:19.380 --> 00:30:27.799
a sequence of meromorphic functions on a domain
and now you try to find conditions, topological
00:30:27.799 --> 00:30:32.980
conditions, that will tell you, topological,
of course includes analytics conditions. That
00:30:32.980 --> 00:30:38.010
will tell you that i always will be able to
find from this sequence i find a sub sequence
00:30:38.010 --> 00:30:39.190
which converges.
00:30:39.190 --> 00:30:43.779
But mind you now it is not just convergence,
it is normal convergence, because in the context
00:30:43.779 --> 00:30:51.050
of complex analysis, in the context of holomorphic
functions, uniform convergence will not work.
00:30:51.050 --> 00:30:56.399
You will get only uniform convergence on restricted
subsets, namely only on compact subsets, that
00:30:56.399 --> 00:31:02.870
is called normal convergence, okay. So you
have to worry about this and in order to do
00:31:02.870 --> 00:31:10.380
all this, see i need to, see the boundedness
of the derivatives for example if something
00:31:10.380 --> 00:31:15.649
that is strongly related to all this. So i
want to be able to work with derivatives.
00:31:15.649 --> 00:31:19.650
But the problem is that functions i am trying
to work with are all what, they are meromorphic
00:31:19.650 --> 00:31:20.650
functions.
00:31:20.650 --> 00:31:24.020
And meromorphic functions are not differentiable,
at the poles they are not differentiable.
00:31:24.020 --> 00:31:28.090
So what do i will do at the poles with meromorphic
functions? What i do is, the clever thing
00:31:28.090 --> 00:31:31.422
is i do not look at the ordinary derivative,
i look at the spherical derivative. Spherical
00:31:31.422 --> 00:31:38.059
derivative makes sense even at pole, that
is where spherical derivative comes in. Okay,
00:31:38.059 --> 00:31:42.840
that is what you offer understands all this
is required for me to do analysis on a space
00:31:42.840 --> 00:31:47.789
of meromorphic functions. And that is a kind
of, see that is the kind of you know
00:31:47.789 --> 00:31:51.140
analysis you have to do to prove the picard’s
theorem, okay.
00:31:51.140 --> 00:31:58.940
Fine, so okay so what i want to do next is
i want to tell you , well that this other
00:31:58.940 --> 00:32:05.570
fact that i was telling you last time that
you know the spherical derivative has another
00:32:05.570 --> 00:32:12.470
important advantage. The spherical derivative
allows you to, you know forget meromorphicity.
00:32:12.470 --> 00:32:20.740
It is a very clever trick, you see on the,
you introduce a spherical derivative because
00:32:20.740 --> 00:32:27.929
you want to look at derivative of a function
which is meromorphic at the pole for example,
00:32:27.929 --> 00:32:35.000
okay. That is why you introduce a spherical
derivative but i, the, it is beautiful, once
00:32:35.000 --> 00:32:41.080
you introduce this notion, you can in most
cases you can even forget the pole.
00:32:41.080 --> 00:32:46.029
Which means you can reduce everything to just
studying analytics functions. So that is the
00:32:46.029 --> 00:32:50.840
beauty, the reason is, the spherical derivative
of meromorphic functions is the same as the
00:32:50.840 --> 00:32:57.450
spherical derivative of its reciprocal, okay.
And what is the advantage of passing the reciprocal,
00:32:57.450 --> 00:33:03.010
passing to the reciprocal, the advantage is
that a pole becomes is zero, okay. And a zero
00:33:03.010 --> 00:33:07.230
is a very nice thing, the function is differentiable
there, all right, so that is the advantage.
00:33:07.230 --> 00:33:12.740
So that is a, that is an added advantage you
get free, okay. And why is this true, this
00:33:12.740 --> 00:33:20.299
is true because the spherical length, that
is invariant under inversion, okay.
00:33:20.299 --> 00:33:35.190
So that is what i want to tell you about next.
So you see, so recall that the spherical distance
00:33:35.190 --> 00:33:49.980
d sub s is invariant under inversion, so you
have the spherical distance between z1 and
00:33:49.980 --> 00:33:58.250
z2 is the same as the spherical distance between
1 by z1 and 1 by z2 where you know z1 and
00:33:58.250 --> 00:34:03.830
z2 are now taken to be in the external complex
plane. And in the external complex plane mind
00:34:03.830 --> 00:34:10.710
you 1 by 0 is defined to be infinity and 1
by infinity is defined to be 0, this is the
00:34:10.710 --> 00:34:15.610
condition. So you know the spherical distance
is the, is actually the spherical distance
00:34:15.610 --> 00:34:20.490
on the riemann sphere, okay which means the
distance between 2 points on the riemann sphere
00:34:20.490 --> 00:34:25.310
is given by the length of the minor arc of
the biggest, bigger circle that passes through
00:34:25.310 --> 00:34:28.899
those 2 points and which lies on the riemann
sphere, okay.
00:34:28.899 --> 00:34:36.909
And that length, how do you get it, and that
length you get it by, for example, using basic
00:34:36.909 --> 00:34:42.510
analytic geometry, it is after all length
of the arc of a circle and you can always
00:34:42.510 --> 00:34:46.640
derive’s formula, okay. And that length
you transport it via the stereo graphic projection
00:34:46.640 --> 00:34:50.909
to the extended complex plane. So the advantage
of that is that i can measure for example
00:34:50.909 --> 00:34:55.409
the distance between the point in the complex
plane and the point that infinity. Okay, i
00:34:55.409 --> 00:35:01.089
do, which i cannot do with the usual euclidean
distance because you, euclidean distance becomes
00:35:01.089 --> 00:35:05.610
unbounded as the point, as one of the points
is fixed and the other goes to infinity, all
00:35:05.610 --> 00:35:06.610
right.
00:35:06.610 --> 00:35:13.110
So and i told you in an earlier lecture that
you the mapping z going to 1 over z which
00:35:13.110 --> 00:35:19.570
is the inversion mapping, that is a map of
c union infinity onto itself, it is a, it
00:35:19.570 --> 00:35:27.410
is a, in fact it is a homeomorphism and the
fact is that under that homeomorphism
00:35:27.410 --> 00:35:31.670
, see what it does is that she has just maps
the extended complex plane back to the extended
00:35:31.670 --> 00:35:36.280
complex plane, if exchanges 0 and infinity.
Infinity goes to 0 and 0 goes to infinity,
00:35:36.280 --> 00:35:42.680
okay. But on the other hand since it is, it
is itself homeomorphism of the extended complex
00:35:42.680 --> 00:35:47.730
plane, it will also induce itself homeomorphism
of the riemann sphere because after all the
00:35:47.730 --> 00:35:50.500
riemann sphere is homeomorphism to the extended
complex plane.
00:35:50.500 --> 00:35:55.540
See whenever in mathematics whenever one object
has an isomorphism and you take another isomorphic
00:35:55.540 --> 00:36:00.680
object, then an isomorphism on the 1st object
will automatically induce an isomorphism on
00:36:00.680 --> 00:36:06.250
the 2nd object which is transported by this
isomorphism between them. So the inversion
00:36:06.250 --> 00:36:12.270
will also induce an, a self isomorphism, a
self isomorphism, a self homeomorphism of
00:36:12.270 --> 00:36:17.030
the, of the riemann sphere and what is it?
It is nothing, i have asked you to check this,
00:36:17.030 --> 00:36:22.470
you should do it, i hope you have done it.
So it is just rotation of the riemann sphere
00:36:22.470 --> 00:36:27.210
about the x axis about 180 degrees, that is
all, that is what it is.
00:36:27.210 --> 00:36:34.700
And you know if you if you take 2 points on
a sphere, and you take the spherical distance
00:36:34.700 --> 00:36:41.100
between them, that arc distance, now if you
rotate the sphere, that is not going to change,
00:36:41.100 --> 00:36:46.829
okay. So it is invariant under that rotation,
all right. And therefore the net effect is
00:36:46.829 --> 00:36:53.760
that the spherical distance is invariant under
inversion, okay. Now this implies that the
00:36:53.760 --> 00:37:02.369
spherical derivative of f is the same as the
spherical derivative of 1 by f, okay. And
00:37:02.369 --> 00:37:13.470
uhh so so why is this, this is for f in
meromorphic, f is a meromorphic function on
00:37:13.470 --> 00:37:14.530
d.
00:37:14.530 --> 00:37:22.950
And why is this true? Because you see, i will
tell you if you want you can try to
00:37:22.950 --> 00:37:28.640
do direct calculations, you can do a direct
calculation. So you know the what
00:37:28.640 --> 00:37:39.280
is f hash f hash is just 2 mod f dash z by
1plus mod fz the whole square, this is what
00:37:39.280 --> 00:37:46.900
it is, all right. Now this is okay where f
dash exists, this formula is correct where
00:37:46.900 --> 00:37:53.839
f dash exists. Let us assume that, for, for
simplicity let us assume that f dash is not
00:37:53.839 --> 00:38:00.250
0 at the point. Suppose derivative does not
vanishes at the point, 1 by f also makes sense
00:38:00.250 --> 00:38:04.430
at that point, the usual derivative of 1 by
f also makes sense at that point.
00:38:04.430 --> 00:38:12.380
So if you calculate, if you calculate, it
will, what it will be, it will be 2 times,
00:38:12.380 --> 00:38:24.710
you know i will get modulus of 1 by f dash
of z divided by 1plus mod 1 by f of z the
00:38:24.710 --> 00:38:35.620
whole squared. Now if you calculate this what
will i get, i will get is well you
00:38:35.620 --> 00:38:44.880
know 2 times, if you take the derivative of
1 by f, i am going to get minus1 by f
00:38:44.880 --> 00:38:55.750
squared into f dash, this is what i will get,
okay, if i use a chain rule. And then i have
00:38:55.750 --> 00:39:05.030
to divide by 1plus 1 by f of z the whole
square mod, okay. And now if my simplified,
00:39:05.030 --> 00:39:13.380
you see, i will again end up with 2 times
mod f dash of z divided by 1plus mod z
00:39:13.380 --> 00:39:15.970
the whole square.
00:39:15.970 --> 00:39:21.720
If i simply multiply numerator and denominator
by mod fz the whole square, i will end up
00:39:21.720 --> 00:39:26.810
with this which is the same as f hash, okay.
So this is a very heuristic, i mean it is
00:39:26.810 --> 00:39:30.700
a very simple calculation, the only thing,
the only problem with this calculation is
00:39:30.700 --> 00:39:40.300
that you know, so i am, i am cancelling out
mod f dash, i am cancelling out more f. And
00:39:40.300 --> 00:39:47.190
to cancel out mod f, f should not vanish,
okay, otherwise i cannot cancel mod f in the
00:39:47.190 --> 00:39:53.050
numerator and denominator. So this is okay
if f is, f is, f of z is, it is okay at the
00:39:53.050 --> 00:40:04.510
point where f is not 0, okay, so let me write
that, valid if f is nonzero, that is one thing
00:40:04.510 --> 00:40:07.490
in the 2nd thing is that f dash should exist,
okay.
00:40:07.490 --> 00:40:16.930
Derivative should exist, otherwise i cannot
write, so f dash exists. So what i am saying
00:40:16.930 --> 00:40:21.170
is that the spherical derivative of f and
the spherical derivative of 1 by f, they are
00:40:21.170 --> 00:40:26.090
the same, you can verify it at all points
of f which is different from the zeros and
00:40:26.090 --> 00:40:32.510
poles, okay. Now what you do is, you check
it at a pole using the same calculation that
00:40:32.510 --> 00:40:39.820
we did last time, all right and you will see
the mobile last time i mean just some time
00:40:39.820 --> 00:40:50.080
ago, we did this calculation to calculate
the f hash z0 at a pole z0, that is by basically
00:40:50.080 --> 00:40:57.619
writing out f locally at the point z0 which
is a pole in this form, f is equal to g by
00:40:57.619 --> 00:41:00.630
z minus z0 to the n, all right.
00:41:00.630 --> 00:41:03.530
And you use the same calculation, if you use
the same calculation, what will happen is
00:41:03.530 --> 00:41:11.430
that you can see that when f is 0 at a point,
that is at the point where f has a zero or
00:41:11.430 --> 00:41:14.990
at a point where f has a pole, the same calculation
is correct. What you do is you calculate in
00:41:14.990 --> 00:41:19.900
the deleted neighbourhood and then you let
limit z tends to z0. You do it both at the
00:41:19.900 --> 00:41:24.760
pole and at t0 and you will see that the limit
will exist and whether you calculate the limit
00:41:24.760 --> 00:41:32.490
for 1 by f or whether you calculate the limit
for f, you will get the same thing. Okay,
00:41:32.490 --> 00:41:35.839
so that, that i leave it to you as an exercise,
okay.
00:41:35.839 --> 00:41:41.580
So one of the, one of the important things
is that, see one of the important things is
00:41:41.580 --> 00:41:51.260
that you know if you take the spherical length
of f of gamma, okay, this is going to be by
00:41:51.260 --> 00:41:59.800
definition integral over gamma f hash of z
mod dz. And since f hash is 1 by f hash, this
00:41:59.800 --> 00:42:10.210
is also integral over gamma, 1 by f hash of
z mod dz and this is by definition spherical
00:42:10.210 --> 00:42:20.470
length of 1 by f of gamma. And f of gamma
and 1 by f of gamma, they only differ by an
00:42:20.470 --> 00:42:27.120
inversion and under the inversion, the spherical
length should not change, so it is correct,
00:42:27.120 --> 00:42:28.120
okay.
00:42:28.120 --> 00:42:35.589
So this way also you see that the, you
know, you can, when you calculate the spherical
00:42:35.589 --> 00:42:39.460
derivative, whether you calculate for f or
whether you calculate for 1 by f, there is
00:42:39.460 --> 00:42:44.250
no difference, okay. So what is the advantage,
suppose you are proving something, involving
00:42:44.250 --> 00:42:49.690
spherical derivative and suppose you have
to deal with a point which is a pole, okay.
00:42:49.690 --> 00:42:56.280
Suppose i have to deal with a function f at
a point which is a pole and suppose i am working
00:42:56.280 --> 00:43:02.270
with the spherical derivative, without loss
of generality i can replace f by 1 by f. Because
00:43:02.270 --> 00:43:07.180
by replacing f by 1 by f, my spherical derivative
does not change but my pole for f becomes
00:43:07.180 --> 00:43:11.160
is 0 for 1 by f and 1 by f becomes analytic.
00:43:11.160 --> 00:43:16.089
So i am dealing with a nice analytic function,
okay. So that is the advantage of having the
00:43:16.089 --> 00:43:20.780
spherical derivative, okay. So now what we
will do is in the forthcoming classes, we
00:43:20.780 --> 00:43:28.780
will use all this, all this background that
we have developed so far do you know in a
00:43:28.780 --> 00:43:33.510
series of lemmas and propositions and finally
theorems, we will prove the Picard’s theorems,
00:43:33.510 --> 00:43:37.740
okay. And on the way we get the very important
Montel’s theorem, okay. So i will stop here.