WEBVTT
Kind: captions
Language: en
00:00:51.000 --> 00:00:55.020
Alright, so what we are doing now is trying
to understand what the spherical derivative
00:00:55.020 --> 00:01:04.610
of meromorphic function is, okay. So, well,
you know the reason for all this is, this
00:01:04.610 --> 00:01:15.780
idea of spherical derivative is important
to study the topology of families of meromorphic
00:01:15.780 --> 00:01:23.320
functions, okay. See the reason is that normally,
you know there are, there is a relationship
00:01:23.320 --> 00:01:32.630
between, see basically we are interested in
compactness, okay. And you know compactness
00:01:32.630 --> 00:01:43.330
is the same , is strongly related to sequential
compactness. Okay, which is, given any sequence,
00:01:43.330 --> 00:01:48.840
you have, at least you are able to find the
convergence of sequence, okay.
00:01:48.840 --> 00:01:53.400
And of course if you are worrying about euclidean
spaces, then of course compactness is the
00:01:53.400 --> 00:01:57.710
same as close and founded but then you know
for saying things like bounded you need a
00:01:57.710 --> 00:02:06.850
metric and so on and so forth. But you know
if you are working with faces, spaces of holomorphic
00:02:06.850 --> 00:02:13.459
functions or analytics functions, the point
is that you know, you will have to work only
00:02:13.459 --> 00:02:17.760
with normal convergence, you will not get
uniform convergence, okay. You will get uniform
00:02:17.760 --> 00:02:26.670
convergence only on compact subsets. And then
its, had it been only uniform convergence,
00:02:26.670 --> 00:02:33.730
then you could have taken the soup norm, okay
and you could have used to define a metric.
00:02:33.730 --> 00:02:37.280
But the point is that you do not have uniform
convergence, you have only uniform convergence
00:02:37.280 --> 00:02:42.011
only restricted to compact subsets, that is
called normal convergence. So it is not so
00:02:42.011 --> 00:02:47.690
easy to think of a metric, all right. But
then you still want to think of compactness
00:02:47.690 --> 00:02:56.090
and compactness is kind of related to sequential
compactness. And so then you know this
00:02:56.090 --> 00:03:01.160
is also connected with uniform bounded less,
it is connected with each equicontinuity,
00:03:01.160 --> 00:03:12.510
okay, and it is also connected with, for example
uhh if you want boundedness of the derivatives,
00:03:12.510 --> 00:03:17.970
okay, so these are bunch of interrelated results,
okay.
00:03:17.970 --> 00:03:23.489
In the, on the topological side this is the
so-called arzela ascoli theorem, okay. And
00:03:23.489 --> 00:03:28.709
on the holomorphic side or on the complex
analytic side, it is the so-called montel
00:03:28.709 --> 00:03:35.230
theorem, okay, which we need to prove, okay.
And, so you see that somehow we want to do
00:03:35.230 --> 00:03:39.180
it not only for analytic function, we want
to do it for meromorphic functions. Because
00:03:39.180 --> 00:03:43.640
you see if we have to worry about meromorphic
functions because that is the, these are the
00:03:43.640 --> 00:03:48.040
functions that you need to study families
of such functions to get to the proof of the
00:03:48.040 --> 00:03:50.770
picard’s theorem which is what our primary
aim is, okay.
00:03:50.770 --> 00:03:56.670
Since you are worried about meromorphic functions,
the problem is that you know they are
00:03:56.670 --> 00:04:00.980
not always different, they are not differentiable
everywhere, i mean they are not analytic everywhere.
00:04:00.980 --> 00:04:06.450
If you go to a pole, at the pole of course
the function goes to infinity, so you cannot,
00:04:06.450 --> 00:04:10.271
you cannot differentiate the function at the
pole, it is not differentiable because it
00:04:10.271 --> 00:04:16.750
is a singular point basically, okay. So, for
your usual derivative will not work, your
00:04:16.750 --> 00:04:24.880
usual derivative will not work at a pole.
So what we will do? The method is that you
00:04:24.880 --> 00:04:29.570
introduce a spherical derivative because spherical
derivative is something that will work even
00:04:29.570 --> 00:04:32.550
at own, that is the whole point, okay.
00:04:32.550 --> 00:04:39.020
So i want to tell you in general while we
are getting so worried about, why we are making
00:04:39.020 --> 00:04:44.550
so much noise about the spherical derivative,
because see that is what we need, that is
00:04:44.550 --> 00:04:50.030
a thing that will work even for meromorphic
functions, it will work even at poles, okay.
00:04:50.030 --> 00:04:54.229
Whereas ordinary derivative you cannot think
of at the pole, because it is a, the moment
00:04:54.229 --> 00:04:59.270
you say the pole is a singular point and singular
point derivative does not exist, then you
00:04:59.270 --> 00:05:03.890
know you are a lot of trouble. That is the
reason we introduced spherical derivative.
00:05:03.890 --> 00:05:07.040
So i was, so let me continue with what i was
telling you last time.
00:05:07.040 --> 00:05:14.570
I was trying to tell you that the spherical
derivative, if f is a, if small f is a meromorphic
00:05:14.570 --> 00:05:27.580
function, if you can see here, so let me,
so let me use a different colour for the moment.
00:05:27.580 --> 00:05:36.870
So if you have this f which is a meromorphic
function on the domain d in the plane, then
00:05:36.870 --> 00:05:44.300
there is a spherical derivative, okay. F hash
of z, and in fact this is spherical derivative
00:05:44.300 --> 00:05:49.660
in absolute value, mind you i have
put 2 times mod f dash of z by 1plus mod fz
00:05:49.660 --> 00:05:55.630
the whole square, so it is nonnegative real
valued function, okay. Normally it derivative
00:05:55.630 --> 00:05:58.930
should be, derivative of a complex valued
functions would again be a complex early functions
00:05:58.930 --> 00:06:04.250
but this is not exactly a derivative, this
is complex valued, it is actually positive
00:06:04.250 --> 00:06:06.530
real, nonnegative real value.
00:06:06.530 --> 00:06:12.280
So you should think of it as absolute value
of the derivative, okay. So when we say that
00:06:12.280 --> 00:06:17.120
spherical derivative, i mean absolute, in
absolute value, okay. And why is it that we
00:06:17.120 --> 00:06:22.620
are interested in this absolute value, because
it is a scaling factor. You see, so you see
00:06:22.620 --> 00:06:27.420
what is happening, see the point is that,
you know as i told you, as i was telling you
00:06:27.420 --> 00:06:33.810
last time, see if you look at this, if you
take this function w equal to f of z, to the
00:06:33.810 --> 00:06:41.220
transformation from the z plane to that the
blue plane, okay. Then you know, you assume
00:06:41.220 --> 00:06:46.160
it is, it is an analytic function, you can
assume it is not constant.
00:06:46.160 --> 00:06:49.260
So then what happens is that you know if it
is nonconstant analytic function, then the
00:06:49.260 --> 00:06:54.740
image of any open set is open. So if i start
with this open set d here, which is supposed
00:06:54.740 --> 00:07:01.180
to be for example in this diagram the interior
of this dotted boundary, okay, then the image
00:07:01.180 --> 00:07:10.030
of that which is f of d is an open set. And
if i take a curve gamma inside d, the image
00:07:10.030 --> 00:07:15.509
of gamma will be f of gamma and this f of
r gamma is now going to be a curve in the
00:07:15.509 --> 00:07:20.600
image which is f of d which is open. And you
know if i calculate the length of gamma, it
00:07:20.600 --> 00:07:27.460
is given by the formula, namely integrating
mod dz, okay because mod dz is the infinitesimal
00:07:27.460 --> 00:07:31.720
version of the euclidean distance which is
more z1 minus z2, okay.
00:07:31.720 --> 00:07:37.070
So you take, if you take 2 points z1 and z2
on the complex plane, then the distance between
00:07:37.070 --> 00:07:41.470
them is given by mod z1 minus z2. If these
points are very close to each other, you can
00:07:41.470 --> 00:07:47.190
call one point as z1, you can call the next
point as z1 plus delta z, then mod z1 minus
00:07:47.190 --> 00:07:52.520
z2 will become more than delta z and you replace
delta by d to get the infinitesimal version,
00:07:52.520 --> 00:08:02.610
so you get mod dz. So mod dz is just the infinitesimal
version, it is called, you may also call it
00:08:02.610 --> 00:08:09.520
the element of arc length which you have to
use to integrate. And if you integrate the
00:08:09.520 --> 00:08:13.230
curve over the arc length, you will get the
length of the curve, okay. And of course it
00:08:13.230 --> 00:08:16.919
is very important that the curves needs to
be rectifiable, okay, it should be a curve
00:08:16.919 --> 00:08:18.880
which is finite length, okay.
00:08:18.880 --> 00:08:23.680
And that is why we always put a condition
that we work only with contours and contours
00:08:23.680 --> 00:08:29.180
are you know they are continuous images of
closed bounded intervals which are in fact
00:08:29.180 --> 00:08:38.330
piecewise smooth and, in fact piecewise continuously
differentiable, okay. And for such, for such
00:08:38.330 --> 00:08:44.180
curves, such contours, the length will always
be a finite quantity and, so, so you have
00:08:44.180 --> 00:08:49.480
this. So the point is that you integrate mod
dz, you will get the length of this gamma,
00:08:49.480 --> 00:08:57.950
all right. Now on the other hand what happens
if you, if the, by the same philosophy if
00:08:57.950 --> 00:09:06.240
you integrate mod dw here over its
image which is f of gamma, you should get
00:09:06.240 --> 00:09:10.930
the length of f of gamma. Which is the, f
of gamma is the image of gamma under f, it
00:09:10.930 --> 00:09:12.640
is the image curve.
00:09:12.640 --> 00:09:20.180
But you know, but here the variable of integration
w is f of z, so if you integrate mod dw
00:09:20.180 --> 00:09:26.920
, i mean if you substitute for w f of z, then
mod d fz will become mod f dash of z into
00:09:26.920 --> 00:09:33.690
dz, okay. And you see the difference between
this formula here and this formula here, is
00:09:33.690 --> 00:09:41.810
that there is this x factor of mod f dash
of z, okay. So what it means is that if you
00:09:41.810 --> 00:09:47.660
simply integrate over mod dz, you will get
the length of the source curves, if integrate,
00:09:47.660 --> 00:09:53.890
if you multiply it by the modulus of the derivative,
you get the length of the target class, they
00:09:53.890 --> 00:10:00.970
may serve. So the point is that the extra
factor you have to put in the integrand to
00:10:00.970 --> 00:10:06.700
get the length of the image curve is the derivative,
the modulus of the derivative, okay.
00:10:06.700 --> 00:10:11.370
Now in the same way what has happened, going
to happen if the following thing. If you take
00:10:11.370 --> 00:10:17.970
f to be, if you take the function f to be
meromorphic function on d, okay, now what
00:10:17.970 --> 00:10:24.910
you are doing this, say you have the complex
plane, and you have this domain d inside it.
00:10:24.910 --> 00:10:31.740
And you have this function f and this is the,
i am now thinking of this function as a function
00:10:31.740 --> 00:10:36.380
into cu infinity. Mind you it is a meromorphic
function, so i am thinking of it as a continuous
00:10:36.380 --> 00:10:41.380
function into the extended complex plane because
i define the value at a pole to be infinity,
00:10:41.380 --> 00:10:44.940
okay, and with that it is continuous.
00:10:44.940 --> 00:10:49.580
And how to actually think of this? So you
see that is this, it that is this, this is
00:10:49.580 --> 00:11:05.440
identified via the stereo graphic projection
to the riemann sphere, okay. Which, i think
00:11:05.440 --> 00:11:13.110
i should put this as 2, s2, 2 sphere, okay,
centred at the origin in the 3 space, real
00:11:13.110 --> 00:11:19.170
3 space radius 1 unit, okay and with the north
pole being identified with the point at infinity,
00:11:19.170 --> 00:11:25.110
okay. Now you see what you do is you essentially
think of this, look at this function. Okay,
00:11:25.110 --> 00:11:30.200
if you look at this function, think of the
function as a map from the domain in the complex
00:11:30.200 --> 00:11:32.330
plane onto the sphere.
00:11:32.330 --> 00:11:37.050
So geometrically whenever you are thinking
of the riemann, whenever you are thinking
00:11:37.050 --> 00:11:40.230
of the extended plane, think of the riemann
sphere dramatically, that is how you should
00:11:40.230 --> 00:11:45.570
think of, okay. So this function , think of
function f as going into the riemann sphere.
00:11:45.570 --> 00:11:51.519
So if you want you know, in abuse of rotation
i will still call this as f, in fact it is
00:11:51.519 --> 00:11:55.370
f followed by the stereo graphic projection.
So in principle i should give it some other
00:11:55.370 --> 00:11:59.540
name but i will still call it as f because
i am thinking of this as an identification,
00:11:59.540 --> 00:12:02.700
i am thinking of the stereo graphic projection
as an identification, okay.
00:12:02.700 --> 00:12:07.070
I am identifying the extended plane for all
practical purposes with my riemann sphere.
00:12:07.070 --> 00:12:12.769
Now what happens, now what happens is that
on the one hand you have the complex plane,
00:12:12.769 --> 00:12:18.029
and you have the domain of the complex plane,
okay. So this is my domain in the complex
00:12:18.029 --> 00:12:24.089
plane and the variable here is n and 10 on
the other hand i have this sphere, i have
00:12:24.089 --> 00:12:34.200
the riemann sphere, okay. And what is happening
is that you know if i now take, if i now take
00:12:34.200 --> 00:12:42.290
a small, if i take the, if i take a curve
gamma here, okay in the complex plane, in
00:12:42.290 --> 00:12:46.920
my domain and take its image under f, okay.
00:12:46.920 --> 00:12:51.329
Then what will happen is that the image of
this curve will also give me a curve here,
00:12:51.329 --> 00:12:59.350
on the riemann sphere. Now, so this will become
the curve f of gamma and how will i guess
00:12:59.350 --> 00:13:05.980
the length of f of gamma? Okay, now think
about what i just told you sometime
00:13:05.980 --> 00:13:12.541
ago, to get the length of image curve, you
have to integrate over, you have to integrate
00:13:12.541 --> 00:13:24.830
over the original curve with the original,
the original infinitesimal element of arc
00:13:24.830 --> 00:13:31.790
length and then you have to scale it by the
modulus of the derivative, okay. But now you
00:13:31.790 --> 00:13:39.959
see what i am doing this i am actually trying
to get the , i am trying to get the
00:13:39.959 --> 00:13:44.640
length of the image curve it is, the length
i am getting is actually the spherical length.
00:13:44.640 --> 00:13:49.300
See it is the length of the riemann sphere
and the length of the riemann sphere corresponds
00:13:49.300 --> 00:13:56.519
to the length on the extended complex plane
even by the spherical metric. See the only
00:13:56.519 --> 00:14:00.880
difference, the only point for you to remember
is, in my target the metric is not euclidean
00:14:00.880 --> 00:14:07.160
metric, it is a spherical metric. I have to
use a spherical metric, i have to use the
00:14:07.160 --> 00:14:11.210
element of the spherical metric and what i
have to integrate. So what is the length of
00:14:11.210 --> 00:14:18.540
uhh f of gamma under the medical metrci?
So i put l sub s just to indicate that this
00:14:18.540 --> 00:14:20.110
is length of spherical metric.
00:14:20.110 --> 00:14:24.840
What is this, this is just i have to integrate
over f of gamma, okay, i have to integrate
00:14:24.840 --> 00:14:32.589
over f of gamma, have to integrate over modulus
of dz sub s, i will keep putting this sub
00:14:32.589 --> 00:14:41.130
s just emphasise that i have to integrate
over an element of the spherical arc length.
00:14:41.130 --> 00:14:50.180
But what is, what is the element of spherical
arc length? You see, in fact i think i should
00:14:50.180 --> 00:14:57.350
not put, so it is important that my variable,
let me call this variable as w. Okay and i
00:14:57.350 --> 00:15:01.510
should be careful inadvertently to make mistakes
like this, see i am integrating over f of
00:15:01.510 --> 00:15:08.510
gamma, so my variable should be in f of gamma,
variable of integration and that has to be
00:15:08.510 --> 00:15:10.470
not z, z is in the source.
00:15:10.470 --> 00:15:16.449
Z is on gamma, that is w is what is on f of
gamma, so you know this not have been z, i
00:15:16.449 --> 00:15:27.771
should correct this, this should be dw sub
s. It is an element of arc length, spherical
00:15:27.771 --> 00:15:35.470
arc length with respect to the variable w
on the sphere all right, where w is f of z.
00:15:35.470 --> 00:15:40.420
So this transformation is given by w equal
to f of z, okay. And the only funny thing
00:15:40.420 --> 00:15:45.410
is that this w is now on the sphere, mind
you w can take the value of the north pole,
00:15:45.410 --> 00:15:49.420
patches corresponding to w equal to infinity,
that is also allowed now, okay, because we
00:15:49.420 --> 00:15:53.440
have allowed values in c union infinity extended
plane, all right.
00:15:53.440 --> 00:16:00.170
Now you see but what is this, what is this,
what is this element of spherical arc length?
00:16:00.170 --> 00:16:07.370
I told you that last time that the element
of spherical arc length is actually 2 times
00:16:07.370 --> 00:16:17.740
mod dw, the usual euclidean element of arc
length divided by 1plus mod w the whole square.
00:16:17.740 --> 00:16:25.060
This is what the element of spherical arc
length is. And the reason why you got this,
00:16:25.060 --> 00:16:30.740
is if you want as an aside, let me write that
down. I will use a different colour, so you
00:16:30.740 --> 00:16:37.300
see, so what happens was that if you take
the spherical distance between 2 points w1
00:16:37.300 --> 00:16:46.370
and w2, okay, then that turned out to be 2
times more than w1 minus w2 by square root
00:16:46.370 --> 00:16:54.350
of mod 1plus w1 the whole square into square
root of 1plus mod w2 the whole square.
00:16:54.350 --> 00:17:01.690
This is the spherical distance between 2 points
w1 and w2 on the on the extended plane or
00:17:01.690 --> 00:17:08.409
the riemann sphere, okay. This is a spherical
distance, it is actually the, in fact i should
00:17:08.409 --> 00:17:12.600
not even say, this is a spherical distance,
sorry, this is actually they, this is the
00:17:12.600 --> 00:17:17.300
cordal distance as such. Yah, this is still
not the infinitesimal version, sorry. So this
00:17:17.300 --> 00:17:26.390
is the d sub c, this is the cordal distance.
So you take 2 points w1 here and w2 and you
00:17:26.390 --> 00:17:32.330
join them by this cord, okay. It is a line
segment, it is a line segment in 3 space,
00:17:32.330 --> 00:17:34.190
joining those 2 points on the riemann sphere.
00:17:34.190 --> 00:17:36.990
But you see, minute those 2 points are actually
points from the extended plane, i am simply
00:17:36.990 --> 00:17:41.450
identifying the extended plane with the riemann
sphere. So i am still writing w1, w-2, where
00:17:41.450 --> 00:17:46.679
actually i mean the stereo graphic projection
of, i mean the stereo graphic projection of
00:17:46.679 --> 00:17:53.200
w1 and the stereo graphic projection of w2.
So if see this is the, this is the cord, this
00:17:53.200 --> 00:17:57.890
is the cordal distance, this is the cordal
metric, and what is this metric? This metric
00:17:57.890 --> 00:18:04.340
is a metric in r3, minute this, this here,
this riemann sphere is sitting inside r 3,
00:18:04.340 --> 00:18:07.540
this is inside r3, okay.
00:18:07.540 --> 00:18:11.190
And in r3 i am simply measuring the distance,
and i asked you to check that this is the,
00:18:11.190 --> 00:18:17.460
this is an exercise for you to check that,
it is an exercise for you to check that this
00:18:17.460 --> 00:18:22.330
is the distance formula, okay. I asked you
to do that, you should do it if you have not
00:18:22.330 --> 00:18:27.840
done it so far. Now you see, this is the,
this is a cordal arc length. If i want the
00:18:27.840 --> 00:18:31.289
spherical, what is the spherical arc length?
Spherical arc length is this, this arc length
00:18:31.289 --> 00:18:36.280
and what is that arc length? I take the great
circle, that is only one big circle on the
00:18:36.280 --> 00:18:40.420
rim, on the sphere which passes through these
2 points.
00:18:40.420 --> 00:18:46.690
And that circle, with these 2 points, 2 points
on a circle determine a minor arc and a major
00:18:46.690 --> 00:18:52.490
arc, okay. And you take the length of the
minor arc, that is the definition of spherical
00:18:52.490 --> 00:19:00.630
distance. So, so if i want the, if i want
the spherical length, okay, and i want the
00:19:00.630 --> 00:19:07.190
infinitesimal spherical length, that is, that
is what this quantity is. This dw sub s if
00:19:07.190 --> 00:19:12.100
the infinitesimal element of spherical length
and for that what i will have to do is i will
00:19:12.100 --> 00:19:19.840
have to bring w1 and w-2 very very close,
okay and as i bring w1 and w-2 very very close,
00:19:19.840 --> 00:19:26.860
the cordal distance will approximate the,
it will come close to the spherical distance,
00:19:26.860 --> 00:19:27.860
okay.
00:19:27.860 --> 00:19:33.570
So what i do is you know in this calculation,
and this formula what you do is you put w2
00:19:33.570 --> 00:19:40.840
is equal to w1 plus delta w, okay and then,
and then you write it in such a way that you
00:19:40.840 --> 00:19:47.120
only allow, you only worry about dw and do
not worry about delta w whole square, delta
00:19:47.120 --> 00:19:53.450
w whole cube, higher-order terms because you
think of them as being very small and negligible.
00:19:53.450 --> 00:19:59.210
And then you change the delta w to dw, so
what will happen is that will, this thing,
00:19:59.210 --> 00:20:06.080
this formula as w1 tends to w-2, okay, this
formula will give you this formula.
00:20:06.080 --> 00:20:10.090
So you see in the in the in the numerator
instead of w-2 if you put w1 plus delta w,
00:20:10.090 --> 00:20:18.330
the numerator will become 2 delta w and this,
both of these quantities will become
00:20:18.330 --> 00:20:26.570
equal to root of 1plus w square. So you will
get the square of that which is 1plus mod
00:20:26.570 --> 00:20:32.310
w the whole square, okay. So each of these
quantities will become square root of, so
00:20:32.310 --> 00:20:40.150
the 1st, the 2nd term will become square root
of 1plus w1 plus delta w mod the whole square,
00:20:40.150 --> 00:20:45.230
okay. And as w tends to 0, you will get this
quantity. So this is the, this is how you
00:20:45.230 --> 00:20:50.040
get this infinitesimal element of spherical
arc length.
00:20:50.040 --> 00:20:57.830
And integrating over that, integrating that
over the curve f of omega should give you
00:20:57.830 --> 00:21:00.970
the length of f of omega, spherical length
of f of omega, which is what we are interested
00:21:00.970 --> 00:21:10.320
in, okay. But then in this you plug-in what
w is. Your w is fz, so plug fz inside that.
00:21:10.320 --> 00:21:15.790
If you plug fz inside that, what will get,
you will end up, well, let me go back to the
00:21:15.790 --> 00:21:25.970
other colour that i was using. So what will
i get it, i will get, well, i will get integral,
00:21:25.970 --> 00:21:30.940
since i have changed, i made a change of variable
w equal to f of z, now the variable becomes
00:21:30.940 --> 00:21:36.960
z and z now varies over gamma, so i will put
gamma here, okay.
00:21:36.960 --> 00:21:43.900
And if i now calculate this mod dfz will become
mod f dash of z into dz, so what i will get
00:21:43.900 --> 00:21:54.990
is able to mod f dash of z mod dz by 1plus
mod f of z the whole square. And what is this,
00:21:54.990 --> 00:22:00.130
if you look at it carefully, this is just
2 times, sorry this is just integral over
00:22:00.130 --> 00:22:10.840
gamma, this is just integral over gamma 2
mod f dash of z divided by 1plus mod fz the
00:22:10.840 --> 00:22:17.960
whole square, this whole thing multiplied
by mod dz. So what are you getting, you are
00:22:17.960 --> 00:22:27.460
getting the length of the image curve in the
spherical metric of gamma under f is this
00:22:27.460 --> 00:22:33.370
formula. And go, you go back to what i was
telling you some time ago, if you want the
00:22:33.370 --> 00:22:41.130
length of the image curve, you have to multiply
by the modulus of the derivative, okay.
00:22:41.130 --> 00:22:46.539
If you simply integrate, you will get the
length of the source curve. But if you multiply
00:22:46.539 --> 00:22:51.890
by the modulus of derivative, you will get
the length of the image curve. So you see
00:22:51.890 --> 00:22:55.440
what this tells you, this tells you that if
you want the length, spherical length of the
00:22:55.440 --> 00:23:03.270
image curve f gamma under f, you will have
to multiply by this, this quantity here. And
00:23:03.270 --> 00:23:07.971
that quantity therefore has to be the absolute
value of the spherical derivative. Therefore
00:23:07.971 --> 00:23:13.340
the spherical derivative, so this is what
we call as, so this is what we call as f hash
00:23:13.340 --> 00:23:25.039
z, this is called the spherical derivative
of f, okay.
00:23:25.039 --> 00:23:35.320
And mind you this is, this is i should write
it in bracket, it is an absolute value, this
00:23:35.320 --> 00:23:41.030
is an absolute value, all right. Because it
is, so you must remember, go back to your,
00:23:41.030 --> 00:23:46.150
your 1st course in complex analysis, when
you take f dash of z, okay. If you take a
00:23:46.150 --> 00:23:54.060
point z0, if you take f dash of z0 where suppose
z0 is a point where function is analytic,
00:23:54.060 --> 00:23:58.320
so the derivative exists, you take f dash
of z. What is the, what is the geometric significance
00:23:58.320 --> 00:24:04.600
of f dash of z0? See the modulus of f dash
of z0 is the scaling fraction, it is locally
00:24:04.600 --> 00:24:07.590
the factor by which an image is scaled.
00:24:07.590 --> 00:24:15.520
If you take some, if you take a small square
containing z0, a very small square containing
00:24:15.520 --> 00:24:21.820
z0 and integrate its image under f, you will
get something that looks like a square, okay,
00:24:21.820 --> 00:24:32.400
okay and this area will be, you know its length
will be scaled by mod f dash. So the modulus
00:24:32.400 --> 00:24:36.779
with derivative is a scaling factor, the argument
of derivative is a rotating factor, is a factor
00:24:36.779 --> 00:24:44.529
of rotation, okay. The argument of z0 is that
the angle by which the tangent rotates, okay.
00:24:44.529 --> 00:24:48.520
If you have a source point and you have a
curve passing through the source point z0
00:24:48.520 --> 00:24:51.000
and you have tangent at that point.
00:24:51.000 --> 00:24:55.950
Now you take the image curve which will pass
through f of z0, the image point and you take
00:24:55.950 --> 00:25:01.120
the tangent there, the difference in the angles
that the tangent makes with the x axis is
00:25:01.120 --> 00:25:07.799
precisely the argument of f dash of z0, okay.
So the geometric meaning of f dash of z0 is
00:25:07.799 --> 00:25:15.659
that the modulus of f dash of z0 gives locally
at z0 the magnifications factor and the argument
00:25:15.659 --> 00:25:20.650
of f dash of z0 gives locally the friction
factor. This is house geometrically the map
00:25:20.650 --> 00:25:27.590
f behaves locally. And you know if the derivative
f, f dash is nonzero, then you know it is
00:25:27.590 --> 00:25:31.669
conformal, you would have studied this in
a 1st course, conformal means you know it
00:25:31.669 --> 00:25:35.840
will preserve angle between the curves.
00:25:35.840 --> 00:25:40.799
So for example if you take something like,
something like a square, its image will be
00:25:40.799 --> 00:25:45.669
something like a distorted square, all right.
If you take something like a circle, its image
00:25:45.669 --> 00:25:50.120
will be something like a distorted circle,
you can expect it to be like for example something
00:25:50.120 --> 00:25:55.040
like an ellipse or something like that. And
this is of course, if you consider it sufficiently
00:25:55.040 --> 00:26:00.290
small and at a point where the derivative
does not vanish, okay. And this is why it
00:26:00.290 --> 00:26:03.320
has got so many applications to engineering
because of conformality.
00:26:03.320 --> 00:26:07.020
So you see why i am trying to tell you all
this is that the modulus of the derivative
00:26:07.020 --> 00:26:12.160
is the magnification factor. And therefore
multiplying by, multiplying the infinitesimal
00:26:12.160 --> 00:26:17.550
arc length by the modulus of the derivative
always give you the length of the arc length
00:26:17.550 --> 00:26:23.200
of the image curve and that is what is happening
here, you see this is a multiplication factor,
00:26:23.200 --> 00:26:26.960
this multiplication factor is therefore called
the spherical derivative. Now i want to tell
00:26:26.960 --> 00:26:31.950
you a few things, few very very important
things in this integral.
00:26:31.950 --> 00:26:39.330
See the 1st and foremost, the amazing thing
about this is that, you know i told you f
00:26:39.330 --> 00:26:49.710
is a meromorphic function, okay. F is a meromorphic
function, so you see, f could have poles,
00:26:49.710 --> 00:26:57.419
okay, f could have poles, of course they are
isolated but f can have poles. And the beautiful
00:26:57.419 --> 00:27:04.360
thing is your curve gamma, see your curve
gamma can pass through those poles, okay.
00:27:04.360 --> 00:27:13.000
Now that is the amazing thing, we normally
when you do integration, you never try, the
00:27:13.000 --> 00:27:16.190
integrand is always supposed to be continuous,
okay.
00:27:16.190 --> 00:27:22.419
When you integrate analytic function or for
example whenever you do cauchy’s theorem
00:27:22.419 --> 00:27:30.270
or you know you you do argument principle,
in all these, in all these things when you
00:27:30.270 --> 00:27:34.440
want to apply, you always make sure that the
contour does not pass through any singular
00:27:34.440 --> 00:27:39.460
points. It cannot pass through poles, it cannot,
there are cases when you are doing the logarithmic
00:27:39.460 --> 00:27:44.581
integral in the, in the argument principle,
you assume that the contour does not pass
00:27:44.581 --> 00:27:49.430
through any poles and also through any zeros,
okay. You do not allow the contrary which
00:27:49.430 --> 00:27:53.430
passes through 0 because you are integrating
the logarithmic derivative exists f dash by
00:27:53.430 --> 00:27:54.549
f.
00:27:54.549 --> 00:28:00.860
And if there is a zero, then denominator f
will have a zero, then you cannot integrate.
00:28:00.860 --> 00:28:06.240
So in all these things that you have been,
that we have been doing so far, we always
00:28:06.240 --> 00:28:11.270
make sure that the contour on which we are
integrating does not pass through any zeros
00:28:11.270 --> 00:28:16.240
or poles. It should never pass through poles
of course but also not true zeros if you are
00:28:16.240 --> 00:28:21.450
trying to apply the argument principle. But
now mind you we are not, we are dealing with
00:28:21.450 --> 00:28:27.090
meromorphic functions, they have poles. And
my point is, now the contour gamma can pass
00:28:27.090 --> 00:28:31.000
through as many poles as you want, it will
not create any problem.
00:28:31.000 --> 00:28:38.090
It will not create any problem for this integral
because you see that is a matter of calculations
00:28:38.090 --> 00:28:48.820
that you have to understand. I will you roughly,
suppose your your contour gamma passes through
00:28:48.820 --> 00:28:52.919
some pole, my new there can be only finitely
many such poles on the contour. Because you
00:28:52.919 --> 00:28:57.950
see the set of poles is anyways an isolated
said, by definition of poles it is isolated,
00:28:57.950 --> 00:29:03.720
it is an isolated scenario. So set of poles
is an isolated set and if you take the set
00:29:03.720 --> 00:29:08.770
of poles lying on gamma, it is an isolated
said, it is an isolated subset of a compact
00:29:08.770 --> 00:29:09.800
set.
00:29:09.800 --> 00:29:17.760
Mind you gamma is a , gamma is compact,
any contour is compact, it is closed unbounded
00:29:17.760 --> 00:29:23.520
because it is actually continuous this image
of an interval, closed unbounded interval,
00:29:23.520 --> 00:29:28.230
so it is closed unbounded, it is compact.
And you know any isolated subset of a compact
00:29:28.230 --> 00:29:36.790
set is finite because if it were infinite,
it will have a limit point, okay. And that
00:29:36.790 --> 00:29:47.460
limit point will not be isolated, okay. Therefore
what will happen is that you have, okay if
00:29:47.460 --> 00:29:52.830
gamma passes through poles of f, mind you
f is a meromorphic, it could have poles, if
00:29:52.830 --> 00:29:56.549
gamma passes through poles of f, it can pass
through only finitely many poles because gamma
00:29:56.549 --> 00:29:58.100
is compact.
00:29:58.100 --> 00:30:02.470
And what happens at the poles? See nothing
happens to the integrand at the poles, it
00:30:02.470 --> 00:30:08.010
is bounded, that is a beautiful thing, that
is why this integral is valid even if gamma
00:30:08.010 --> 00:30:14.100
passes through a pole or several poles. That
is because you just imagine, suppose f has
00:30:14.100 --> 00:30:24.309
a pole at z0, okay, then in fact
you know we can write this down. Suppose,
00:30:24.309 --> 00:30:25.780
so let me write this down.
00:30:25.780 --> 00:30:37.100
Suppose f of z, so, let me rub this and probably
go down a little bit so that i have more space.
00:30:37.100 --> 00:30:51.660
Suppose, suppose f has a pole at z0 which
is lying on gamma, this will create, this
00:30:51.660 --> 00:30:55.549
will not create any problems, this will not
create any problems, why? Because you see
00:30:55.549 --> 00:31:09.549
in a small disc centred at gamma, centred
at z0, you see f of z will look like g of
00:31:09.549 --> 00:31:26.770
z by z minus z0 to the power n where n is
the order of the pole, okay. Now you calculate,
00:31:26.770 --> 00:31:38.500
you calculate this quantity, okay, notice
that and of course g of z0 is not 0, okay.
00:31:38.500 --> 00:31:47.540
G of z0 is not 0 and of course this
is how pole, locally a function looks locally
00:31:47.540 --> 00:31:49.659
at the pole of order n, all right.
00:31:49.659 --> 00:32:02.770
And g is of course analytic, okay, g analytic
at z0, okay. Now you see, just look at this
00:32:02.770 --> 00:32:07.100
expression that i have written about, if i
take f dash, if i take the derivative, the
00:32:07.100 --> 00:32:12.870
derivative will continue to have a pole of
one more order, if i differentiate this gz
00:32:12.870 --> 00:32:17.950
by z minus z0 power n, if you want using quotient
rule, then i will get z minus z0 to the n
00:32:17.950 --> 00:32:22.230
plus1 in the denominator. So what i will get,
i will get, i will get a pole of higher-order,
00:32:22.230 --> 00:32:33.519
okay, of greater order. And if you go to the
denominator, the denominator will have 1 by,
00:32:33.519 --> 00:32:38.519
it will have mod gz the whole square by z
minus z0 to the power 2n.
00:32:38.519 --> 00:32:47.679
And as z tends to z0, you will see that the
numerator tends to , i mean this
00:32:47.679 --> 00:32:57.260
whole quantity will go to either 0 or 2 a
finite value, okay. Because what, because
00:32:57.260 --> 00:33:03.179
what is actually happening is as z is tending
to z0, z minus z0 is going to go to 0, so
00:33:03.179 --> 00:33:08.730
f is going to infinity, all right. But the
fact is that the denominator will go to infinity
00:33:08.730 --> 00:33:15.309
faster than the numerator because the denominator
contains f square. F square has a pole of
00:33:15.309 --> 00:33:22.990
order 2n, where as the numerator has a pole
of order only n plus1, okay. Therefore the
00:33:22.990 --> 00:33:29.520
denominator goes to infinity faster than the
numerator as a result the integrand is bounded.
00:33:29.520 --> 00:33:34.220
So the point is that this integral is valid
even at a pole, that is a big deal here. So
00:33:34.220 --> 00:33:39.480
this integral is, you gamma can pass through
poles of f, there is no problem. And geometrically
00:33:39.480 --> 00:33:45.559
also you should believe this because if gamma
passes through a pole of f, its image will
00:33:45.559 --> 00:33:49.929
pass through the north pole on the riemann
sphere. After all at a pole of f, f is taking
00:33:49.929 --> 00:33:53.140
the value infinity and that corresponds to
the north pole on the riemann sphere, so after
00:33:53.140 --> 00:33:56.559
all what you are saying is that the image
curve is passing through the north pole of
00:33:56.559 --> 00:34:02.789
the sphere, how does it matter. It is not
going to affect, it is, north pole of the
00:34:02.789 --> 00:34:06.970
sphere is in no way different from any other
point on the sphere, okay.
00:34:06.970 --> 00:34:23.090
So the moral of the story is that, it shows
that l s of f gamma is well-defined, even
00:34:23.090 --> 00:34:34.560
if gamma passes through poles, okay. So this
formula always works, it works even for meromorphic
00:34:34.560 --> 00:34:39.890
function, it even works is gamma passes through
poles, that is a very very important thing.
00:34:39.890 --> 00:34:47.130
And then there is, so this something you need
to know. And there is another fact that it
00:34:47.130 --> 00:34:52.190
will expand into the next lecture, the factors
that if you take the reciprocal of f, you
00:34:52.190 --> 00:34:56.350
see f is a meromorphic function, then 1f is
also a meromorphic function. And the beautiful
00:34:56.350 --> 00:35:00.751
thing is that if you take it spherical derivative,
we will get exactly the same as spherical
00:35:00.751 --> 00:35:03.079
derivative of f.
00:35:03.079 --> 00:35:10.580
The reason is because of the fact that the
spherical distance is invariant under inversion.
00:35:10.580 --> 00:35:14.560
I told you the inversion on the complex plane
translates to a rotation of the riemann sphere
00:35:14.560 --> 00:35:19.420
about the x axis. And it leaves spherical
distances invariant. Therefore the spherical
00:35:19.420 --> 00:35:26.140
derivative is also invariant if you invert
the function, okay. So this is another important
00:35:26.140 --> 00:35:31.200
fact that we use. And the advantage that you
can replace f by 1 by f is that whenever f
00:35:31.200 --> 00:35:36.510
has a pole, 1 by f has a zero. So you can
reduce from the meromorphic case to the analytic
00:35:36.510 --> 00:35:39.210
case, so you can feel works with the analytic
functions.
00:35:39.210 --> 00:35:46.650
You see this is the advantage of having spherical
derivative, okay. So spherical derivative
00:35:46.650 --> 00:35:50.800
does not distinguish between f and 1 by f,
the advantage of moving from f to 1 by f is
00:35:50.800 --> 00:35:56.349
that you can, what is a pole for f becomes
a zero for 1 by f and zeros are very friendly,
00:35:56.349 --> 00:36:02.970
more friendlier than poles. In the neighbourhood
of zero you can usual, you apply usual analytic
00:36:02.970 --> 00:36:07.600
function theory because the function is after
all analytics. So that is the advantage, so
00:36:07.600 --> 00:36:11.150
that is another motivation for having the
spherical derivative. So i will stop here.