WEBVTT
Kind: captions
Language: en
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Okay so what we are looking at is you know
we are trying to understand the behaviour
00:00:51.590 --> 00:00:59.989
at infinity okay and somehow the idea is that
if you want to study f of Z at infinity then
00:00:59.989 --> 00:01:08.720
it is the same as studying f of 1 by Z at
0, so see the question was, why is this why
00:01:08.720 --> 00:01:16.170
is this justified and well I was trying to
explain that in the towards the end of the
00:01:16.170 --> 00:01:18.810
last lecture, so I will take off from there.
00:01:18.810 --> 00:01:30.430
So let me let me go back to what I was
saying, so you see so the idea is the idea
00:01:30.430 --> 00:01:36.650
is the following, the idea is the following.
The idea is that you know whenever 2 objects
00:01:36.650 --> 00:01:43.850
are isomorphic okay then they are property
should correspond alright and is not only
00:01:43.850 --> 00:01:50.330
there for example if 2 topological spaces
are isomorphic okay then you expect both of
00:01:50.330 --> 00:01:55.150
them to have the same to have the same
topological properties, so if one has a certain
00:01:55.150 --> 00:01:59.869
topological properties, you should expect
the same thing to happen of the other okay,
00:01:59.869 --> 00:02:04.260
so for example if 2 topological spaces are
isomorphic one is connected then the other
00:02:04.260 --> 00:02:09.720
should also be connected, so a disconnected
topological space cannot be isomorphic to
00:02:09.720 --> 00:02:15.190
a connected topological space because continuous
image of connected set is connected okay and
00:02:15.190 --> 00:02:16.190
so on and so forth.
00:02:16.190 --> 00:02:20.770
So this this is a very general philosophy
in mathematics for example if 2 vector space
00:02:20.770 --> 00:02:26.530
are isomorphic you then they will have
the same dimensions okay so the properties
00:02:26.530 --> 00:02:31.569
like dimensional, properties like in topology,
properties like connectedness, compactness
00:02:31.569 --> 00:02:36.050
these are all intrinsic properties and they
are not supposed to change under isomorphism.
00:02:36.050 --> 00:02:42.670
Now therefore so that is one level of thinking,
the other level of thinking is that especially
00:02:42.670 --> 00:02:48.930
when you are looking at an isomorphism of
spaces okay is not only that the properties
00:02:48.930 --> 00:02:53.909
of the spaces, geometrics properties of the
spaces should should correspond namely if
00:02:53.909 --> 00:02:57.810
one of the spaces as a geometric property
in the other should also have an by geometry
00:02:57.810 --> 00:03:04.939
here of course I mean you know geometry something
that includes both I mean it includes topology
00:03:04.939 --> 00:03:09.980
it includes algebra, it includes analysis,
it includes an interplay of all these properties.
00:03:09.980 --> 00:03:16.209
So you expect if 2 objects are isomorphic
you expect them to behave in the same way
00:03:16.209 --> 00:03:22.959
whether no matter if you look at them topologically
or algebraically or analytically okay and
00:03:22.959 --> 00:03:27.920
of course you should also assume that the
isomorphism is also going to be compatible
00:03:27.920 --> 00:03:33.640
with whatever structure you are looking at
if you are if you are looking at topological
00:03:33.640 --> 00:03:38.040
properties, the topological structure then
you expect the isomorphism to be are topological
00:03:38.040 --> 00:03:41.579
isomorphism which is a homeomorphism, if you
are looking at algebraic properties you should
00:03:41.579 --> 00:03:50.109
expect the usual thing of an isomorphism which
also you know behaves well with the algebraic
00:03:50.109 --> 00:03:55.560
structure and if you are looking at an analytic
properties for example if you are looking
00:03:55.560 --> 00:04:01.499
at manifold theoretic properties and you should
expect the isomorphism to be an isomorphism
00:04:01.499 --> 00:04:02.499
that respects their structures.
00:04:02.499 --> 00:04:10.819
So now you see the big deal is that whenever
2 spaces are isomorphic then the functions
00:04:10.819 --> 00:04:16.031
on those spaces are also isomorphic okay,
so this is a very very important idea, so
00:04:16.031 --> 00:04:21.019
and function are isomorphic the sense that
you know properties of functions carry over
00:04:21.019 --> 00:04:29.030
so so that is what was trying to explain last
time, so so look at this thing so x phi from
00:04:29.030 --> 00:04:37.300
x to y is a homeomorphism of topological spaces
and an your given this f which is a function
00:04:37.300 --> 00:04:43.220
on the topological space y with values in
another topological space Z okay and I am
00:04:43.220 --> 00:04:47.502
just saying is a function, I am not saying
that I do begin with I do not know whether
00:04:47.502 --> 00:04:52.830
it is continues or not okay but the point
is that the moment you give it is very
00:04:52.830 --> 00:04:58.460
natural theory that whenever you give me a
function on the target set you can composite
00:04:58.460 --> 00:05:02.610
with the given map from the source set to
the target set to get function on the source
00:05:02.610 --> 00:05:03.610
set.
00:05:03.610 --> 00:05:09.060
So that is called the pullback of the function,
so you know f is a function on y you can composite
00:05:09.060 --> 00:05:14.540
it with phi to get this functionality g which
is now a function on the source which is x
00:05:14.540 --> 00:05:20.210
okay and then the question is that what is
the connection between f and g in fact f and
00:05:20.210 --> 00:05:27.670
g should correspond to one another in this
isomorphism of space which induces an isomorphism
00:05:27.670 --> 00:05:35.570
of functions okay. So in fact you know what
is happening is that there is a map from functions
00:05:35.570 --> 00:05:41.500
from y of Z to function from x to Z that is
a pullback map and what is happening is f
00:05:41.500 --> 00:05:49.430
is going to g okay and this is an isomorphism
because because phi has an inverse okay this
00:05:49.430 --> 00:05:55.720
an isomorphism and the fact is that under
this isomorphism properties functions of properties,
00:05:55.720 --> 00:06:00.930
particular properties coincides, so they they
correspond, so for example if f is continuous
00:06:00.930 --> 00:06:07.530
then g is continuous and conversely if g is
continuous f is continuous because you can
00:06:07.530 --> 00:06:10.940
get f from g and g from f because phi is invertible.
00:06:10.940 --> 00:06:17.410
Now this is at the this is at the topological
level okay and then I am saying at letters
00:06:17.410 --> 00:06:22.650
look at it at a at the level of complex analysis
for example, so suppose D 1 and D 2 are domains
00:06:22.650 --> 00:06:29.100
in the complex plane and phi from D 1 to D
2 is an analytic isomorphism namely it is
00:06:29.100 --> 00:06:33.380
holomorphic map, it is an analytic map which
is injected okay and you know an injective
00:06:33.380 --> 00:06:41.230
holomorphic map is an isomorphism and then
our philosophy should tell us that the functions
00:06:41.230 --> 00:06:47.480
on D 2 correspond they are in one-to-one correspondents
with functions of D 1 okay. In particular
00:06:47.480 --> 00:06:52.021
if you are looking at complex value functions
on D 2 they should be in Bijective correspondence
00:06:52.021 --> 00:06:58.220
with complex valued function on D 1 okay and
that is again by the pullback, so if you give
00:06:58.220 --> 00:07:05.050
me a function f on D 2 with complex values
then you know I get the pullback function
00:07:05.050 --> 00:07:12.870
g okay and f is analytic if and only if g
is analytic and the reason for that is again
00:07:12.870 --> 00:07:17.820
the fact that phi is invertible, phi is analytic
and you use the fact that the composition
00:07:17.820 --> 00:07:25.910
of analytic function is analytic okay, so
so now what I want to do is that you can go
00:07:25.910 --> 00:07:28.730
one step further and you can discuss singularity.
00:07:28.730 --> 00:07:35.360
So you know so let me write this down, so
let us assume that you again have this
00:07:35.360 --> 00:07:43.360
you have a map from D 1 to D 2, phi is
a map from D 1 to D 2, phi is a homeomorphism
00:07:43.360 --> 00:07:58.970
okay and I am assuming that D 1 and D 2 are
they are domains in the complex plane
00:07:58.970 --> 00:08:05.270
okay and of course mind you I am being a little
careful, I am saying that phi is a homeomorphism
00:08:05.270 --> 00:08:11.070
am not saying that it is an an isomorphism,
it is holomorphic isomorphism, analytic isomorphism
00:08:11.070 --> 00:08:19.220
but I say the following thing suppose phi
takes a point Z naught in D 1 to point W naught
00:08:19.220 --> 00:08:27.340
in D 2 okay, so I am writing the function
phi as W is equal to phi of Z okay. Z is the
00:08:27.340 --> 00:08:33.300
is the independent variable is supposed to
vary in D 1 okay and W is phi of Z it is a
00:08:33.300 --> 00:08:37.070
dependent variable it is supposed to take
values in D 2 okay and I am assuming that
00:08:37.070 --> 00:08:50.381
phi takes Z naught to W naught okay and suppose
suppose that you know phi from well suppose
00:08:50.381 --> 00:09:01.300
phi from D 1 to D 2 is actually an analytic
isomorphism.
00:09:01.300 --> 00:09:06.860
So now I am you know I am assuming something
more am assuming that phi is not just a homeomorphism
00:09:06.860 --> 00:09:10.990
I am assuming it is an analytic isomorphism
which means in particular it is a homeomorphism
00:09:10.990 --> 00:09:17.399
okay and mind you an analytic map is continuous
okay and therefore analytic isomorphism is
00:09:17.399 --> 00:09:27.199
stronger than being a homeomorphism okay.
Well now you see now you can do the following
00:09:27.199 --> 00:09:34.260
thing, see suppose you already know that if
you have a function on D 2 I can put it back
00:09:34.260 --> 00:09:40.899
to give you a function on D 1 but suppose
I had a function on D 2 with a single point,
00:09:40.899 --> 00:09:48.749
isolated singular point a singularity at W
naught okay. Then by composing with phi I
00:09:48.749 --> 00:09:54.589
will get a function on D 1 which will have
an isolated singular point at Z naught okay.
00:09:54.589 --> 00:10:00.560
So you see suppose so so the situation is
like this you have D 2 minus the point W naught
00:10:00.560 --> 00:10:07.779
and that is being carried over by phi to D
1 minus Z naught because you know phi is after
00:10:07.779 --> 00:10:13.449
all Bijective it will take the complement
of Z naught to the complement of W naught
00:10:13.449 --> 00:10:17.050
and Z naught will go to W naught, so basically
what it is doing is that it is taking the
00:10:17.050 --> 00:10:23.019
punctured domain, punctured at Z naught and
mapping it at holomorphically, isomorphically,
00:10:23.019 --> 00:10:29.360
analytically isomorphically onto the punctured
domain at D 2 punctured at W naught okay and
00:10:29.360 --> 00:10:34.459
suppose you have a function, suppose you have
a function f here, complex valued function
00:10:34.459 --> 00:10:42.000
okay which is analytic okay then you know
if I compose this composite with phi I will
00:10:42.000 --> 00:10:48.370
get this function g which is first apply phi
and then apply f okay and of course g will
00:10:48.370 --> 00:10:53.790
also be analytic okay and mind you this inside
this is for an open subset.
00:10:53.790 --> 00:10:58.520
This inside this is an open subset, the complement
of a point is always open and this diagram
00:10:58.520 --> 00:11:11.430
also commutes okay and well the fact is that
what this tells you is that f has an isolated
00:11:11.430 --> 00:11:17.120
the point W naught I do not know whether the
point W naught f is analytic or not okay but
00:11:17.120 --> 00:11:24.079
in the deleted neighbourhood of W naught for
example the domain D 2 minus W naught is deleted
00:11:24.079 --> 00:11:29.400
neighbourhood of W naught and there I know
f is analytic, so W naught is an isolated
00:11:29.400 --> 00:11:36.350
singularity of f and what this diagram tells
you is that a function g that you got by pulling
00:11:36.350 --> 00:11:44.089
back f via phi is also having an isolated
singularity at Z naught okay and now you can
00:11:44.089 --> 00:11:50.629
you can believe that if you believe in the
philosophy that under a pullback functions
00:11:50.629 --> 00:11:57.800
by an isomorphism okay properties of function
should coincide, you can believe that nature
00:11:57.800 --> 00:12:02.950
of the singularity of f at W naught should
correspond should be exactly the same as the
00:12:02.950 --> 00:12:06.600
nature of singularity of g at Z naught okay.
00:12:06.600 --> 00:12:14.079
So it is natural to expect that if that f
as say a removable singularity at W naught
00:12:14.079 --> 00:12:19.050
then g should have a removable singularity
at Z naught if f has a pole at W naught of
00:12:19.050 --> 00:12:26.389
a certain order then g will also have a pole
at Z naught of that order and if f has an
00:12:26.389 --> 00:12:31.980
essential singularity at W naught then g will
have an essential singularity at Z naught
00:12:31.980 --> 00:12:37.889
and the converse will also okay so properties
of f this the nature of singularity of f at
00:12:37.889 --> 00:12:41.459
W naught should correspond to the nature should
be exactly the same as the nature of singularity
00:12:41.459 --> 00:12:49.079
of g at Z naught okay, so this is something
that that is very easy to understand and why
00:12:49.079 --> 00:12:57.329
is this why is that why is that true? That
is just true because D 1 to D 2 is an analytic
00:12:57.329 --> 00:13:03.379
isomorphism okay it is because it is an analytic
isomorphism that this is happening.
00:13:03.379 --> 00:13:11.499
So so let me write this in a let me use
a different color okay, so well so you see
00:13:11.499 --> 00:13:20.720
well let me write this here nature of
the singularity I am abbreviating it as sing
00:13:20.720 --> 00:13:37.209
of f at W naught is equal to nature of singularity
of g at W naught okay and see this is this
00:13:37.209 --> 00:13:46.459
happens basically because you can you can
see this in a moment you see, so so why is
00:13:46.459 --> 00:13:54.889
this true? So you know let us look at 3 cases,
suppose f is suppose f has a removable singularity
00:13:54.889 --> 00:14:00.060
at W naught okay then what it means is that
you know by Riemann’s removable singularity
00:14:00.060 --> 00:14:04.959
theorem you know that the limit of f as W
tends to W naught exists this is also the
00:14:04.959 --> 00:14:12.579
same as saying that the f extends to an
analytic function at W naught okay and is
00:14:12.579 --> 00:14:16.390
equivalent to saying that f is continuous
at W naught okay and if f is continuous at
00:14:16.390 --> 00:14:21.910
W naught by composition with phi it is clear
that g is also continues as Z naught okay.
00:14:21.910 --> 00:14:28.790
So so it is very clear that if f has a removable
singularity at W naught then g has a removable
00:14:28.790 --> 00:14:33.329
singularity at W naught and the converse is
also true, if g has a removable singularity
00:14:33.329 --> 00:14:38.490
at Z naught then f will have a removable singularity
at W naught because you can if you compose
00:14:38.490 --> 00:14:46.450
g with phi inverse you get f okay okay so
so this tells you that f as a removable singularity
00:14:46.450 --> 00:14:51.699
at W naught if and only if g as removable
singularity at Z naught okay. Now what about
00:14:51.699 --> 00:14:57.980
the case of W naught being of pole if f has
a pole at W naught then a limit as W tends
00:14:57.980 --> 00:15:07.119
to W naught of f is infinity okay and and
you know you see by continuity it should happen
00:15:07.119 --> 00:15:13.139
that g will also have a pole at at Z not.
00:15:13.139 --> 00:15:24.999
It is continuous at Z naught okay and g being
continuous at Z naught is the same as saying
00:15:24.999 --> 00:15:36.940
that g has a removable singularity at Z naught
okay you have this. Now what is the situation
00:15:36.940 --> 00:15:51.809
when f has a pole at W naught f has a pole
at W naught if and only if you know say
00:15:51.809 --> 00:16:02.920
of order N greater than equal to 1, what is
the condition for a pole? It is well limit
00:16:02.920 --> 00:16:14.670
one condition is of course limit Z tends
to I mean limits W tends to W naught of
00:16:14.670 --> 00:16:20.569
of f of W should be infinity this is this
is one of the conditions and that is also
00:16:20.569 --> 00:16:28.550
the same as saying that limit W tends to W
naught of W minus W naught with the power
00:16:28.550 --> 00:16:37.259
of N times f of W is non-zero okay this is
exactly the condition that f has a pole of
00:16:37.259 --> 00:16:43.629
okay and you know if you well if you translate
if you translate this limit when realizing
00:16:43.629 --> 00:16:52.069
that as W tends to W naught if and only if
Z tends to Z naught because phi is a homeomorphism
00:16:52.069 --> 00:16:59.759
okay and you know under a continuous map the
image of convergence sequence is again a convergence
00:16:59.759 --> 00:17:03.259
sequence okay the image of a limit is again
a limit alright.
00:17:03.259 --> 00:17:08.521
So you know so if you translate that you did
you will actually if you if you change the
00:17:08.521 --> 00:17:20.850
variable it will tell you that Z minus Z naught
to the power of N g of Z is nonzero if you
00:17:20.850 --> 00:17:26.270
lead Z tends to Z not, okay. So the easiest
thing to do is the following thing, what you
00:17:26.270 --> 00:17:31.200
do is you take this this is the easiest probably,
is the easiest thing to do what you do is
00:17:31.200 --> 00:17:35.910
you put W is equal to phi Z okay if you put
W is equal to phi Z you will get limit phi
00:17:35.910 --> 00:17:43.010
Z tends to phi Z not, f of phi of Z but f
of phi of Z is g Z okay and the limit phi
00:17:43.010 --> 00:17:47.770
of Z tends to phi of Z naught is the same
as limit Z tends to Z not, so you get limit,
00:17:47.770 --> 00:17:54.299
so this is the same as saying that limit Z
tends to Z not, g of Z is infinity okay. See
00:17:54.299 --> 00:17:59.320
this is this is plainly equivalent okay so
this is what I want, so this is what we want,
00:17:59.320 --> 00:18:09.309
so this is this is plainly equivalent to limit
Z tends to Z naught g of Z is equal to infinity.
00:18:09.309 --> 00:18:17.150
It is just you just make a change of variable
from W to Z so you will have to so so you
00:18:17.150 --> 00:18:23.320
put W is equal to phi Z okay then you will
get f of phi Z, f of phi Z is just g Z by
00:18:23.320 --> 00:18:32.340
definition okay and W tends to W naught will
read phi Z tends to phi Z naught but phi Z
00:18:32.340 --> 00:18:36.899
tends to phi Z naught is equivalent to Z to
Z naught because phi is a homeomorphism so
00:18:36.899 --> 00:18:42.409
this is the same as this and add this condition
limit Z tends to Z naught g Z is infinity
00:18:42.409 --> 00:18:47.892
is will tell you that it is a pole Z naught
is a pole of g okay and you will have to do
00:18:47.892 --> 00:18:53.790
a little bit more work to for example
compare Laurent series say that the poll is
00:18:53.790 --> 00:19:01.780
exactly at the same order on both sides okay,
so so you see therefore essentially I am just
00:19:01.780 --> 00:19:09.230
using the fact that phi is a homeomorphism
I am not using anything more than that alright,
00:19:09.230 --> 00:19:20.929
so well so so let me write this here g
has a pole at Z naught and a little bit more
00:19:20.929 --> 00:19:26.070
work will tell you that the pole will also
have order m okay fine.
00:19:26.070 --> 00:19:36.090
So you know you can if you try to if you try
to prove this thing below okay that may not
00:19:36.090 --> 00:19:46.060
be so easy to do at the face of it okay, fine.
So anyway what this tells you is that if f
00:19:46.060 --> 00:19:51.440
has a pole at W naught then g has a pole at
Z naught and conversely okay and of course
00:19:51.440 --> 00:19:55.000
you can get the converse because instead of
phi you can use phi inverse which is also
00:19:55.000 --> 00:20:03.880
a homeomorphism okay and well and then
the last case is the left out case if you
00:20:03.880 --> 00:20:08.490
will have an essential singularity at W naught
if and only if g has an essential singularity
00:20:08.490 --> 00:20:14.860
at Z naught this is just by tautology it is
just biologic logic because essential singularities
00:20:14.860 --> 00:20:21.210
are singularities which are neither removable
nor poles okay and you already shown that
00:20:21.210 --> 00:20:24.970
removable singularity is correspond you showing
that polls correspond therefore essential
00:20:24.970 --> 00:20:28.460
singularities which is complement of these
2 should also correspond okay.
00:20:28.460 --> 00:20:43.240
So let me write that down it follows that
f has an essential singularity at at Z naught
00:20:43.240 --> 00:20:57.809
at W naught if and only if g has an essential
singularity at Z naught okay this is this
00:20:57.809 --> 00:21:03.740
is very clear and in fact if you want we can
also say it in another way, what is the condition
00:21:03.740 --> 00:21:10.659
that what is the condition that a function
has an essential singularity that point one
00:21:10.659 --> 00:21:18.190
condition for example is at the limit as pro
set point does not exist. If limit Z if limit
00:21:18.190 --> 00:21:24.289
W tends to W naught f W does not exist then
f must have an essential singularity at W
00:21:24.289 --> 00:21:30.440
naught and again you know if you if you use
a substitution W equal to phi Z and remember
00:21:30.440 --> 00:21:37.559
that phi is a homeomorphism it is very clear
that the limit as W tends to W naught f W
00:21:37.559 --> 00:21:44.450
will not exist if and only if limit as Z tends
to Z naught g of Z does not exist and this
00:21:44.450 --> 00:21:49.640
will tell you that W naught being an essential
singularity or f is the same as Z naught being
00:21:49.640 --> 00:21:51.970
an essential singularity for g okay.
00:21:51.970 --> 00:21:55.690
So essential singularity corresponds but of
course here I am using the fact that the limit
00:21:55.690 --> 00:22:02.480
does not exist and where did that come from
that basically came from application of actually
00:22:02.480 --> 00:22:06.169
if you go back it is an application of Riemann’s
removable singularity theorem which says that
00:22:06.169 --> 00:22:10.990
the limit exists then it is removable okay
and if the limit exists and is infinite then
00:22:10.990 --> 00:22:16.169
it is of pole okay and if the limit does not
exists then it is an essential singularity
00:22:16.169 --> 00:22:23.600
and all these ifs are actually if and only
ifs okay fine. Now now having said all of
00:22:23.600 --> 00:22:30.799
this, now how do we deal…I want to get back
to try to you know tell you the story that
00:22:30.799 --> 00:22:35.179
saying that f of Z studying f of Z at infinity
at Z equal to infinity is the same as studying
00:22:35.179 --> 00:22:43.590
f of 1 by Z at 0 okay and what is the, why
is that justified in the light of this argument,
00:22:43.590 --> 00:23:01.880
so it is justified in the following way, so
so let me write that down you know justify
00:23:01.880 --> 00:23:12.870
behaviour of f of Z… Let me use f of W because
so that I am consistent with my notation,
00:23:12.870 --> 00:23:29.980
behaviour of f of W at W equal to infinity
is the same as that of f of 1 by W at W equal
00:23:29.980 --> 00:23:32.970
0 okay.
00:23:32.970 --> 00:23:39.850
So so this is if you if you look if you have
gone through the 1st course in complex analysis
00:23:39.850 --> 00:23:47.269
and behaviour at infinity was covered then
you would see people saying that f has a has
00:23:47.269 --> 00:23:51.330
a… The nature of singularity of f at infinity
is the same as the nature of singularity of
00:23:51.330 --> 00:23:58.720
f of 1 by Z at 0 okay and why is that true?
Is true in the light of following argument
00:23:58.720 --> 00:24:03.570
which is based on what we have been saying,
you take this map from C star let me not
00:24:03.570 --> 00:24:12.120
use C star, take this map from C union infinity
which is extended complex plane to C union
00:24:12.120 --> 00:24:16.830
infinity. This is extended complex plane to
the extended complex plane okay, so now we
00:24:16.830 --> 00:24:20.210
are making use of the point at infinity and
we are also making use of the topology of
00:24:20.210 --> 00:24:21.340
the extended complex plane.
00:24:21.340 --> 00:24:26.630
So and you know and what is the map, the map
is just Z going to 1 by Z you take this map,
00:24:26.630 --> 00:24:35.600
so this is my map fee, so here my phi of Z
is 1 by Z, so W is phi of Z which is 1 by
00:24:35.600 --> 00:24:41.940
Z so W is 1 by Z this is my map and this is
the well-defined map you see that the point
00:24:41.940 --> 00:24:48.500
is that you have to the you have to just send
infinity to 0 and 0 to infinity this is the
00:24:48.500 --> 00:24:54.730
obvious thing that you will do and so you
know so let me write that down, you send 0
00:24:54.730 --> 00:24:59.830
to the point at infinity and you said infinity
the point at 0 okay and mind you when you
00:24:59.830 --> 00:25:07.950
send, when you make these definitions it feel
continuous to be a homeomorphism okay. See
00:25:07.950 --> 00:25:12.120
limit Z tends to infinity phi of Z is what?
00:25:12.120 --> 00:25:21.340
Limit Z tends to infinity of phi of Z is just
limit Z tends to infinity of 1 by Z which
00:25:21.340 --> 00:25:31.070
is 0 okay and what this will tell you?
so limit Z tends to infinity phi of Z is 0
00:25:31.070 --> 00:25:36.130
that is exactly phi of infinity as per our
definition because we have sent infinity to
00:25:36.130 --> 00:25:42.120
0 and what does this tells you? This tells
you phi is continuous at 0 at infinity
00:25:42.120 --> 00:25:46.480
and the same kind of argument will tell you
that phi is also continuous at 0, so if you
00:25:46.480 --> 00:25:55.929
take limit Z tends to 0 phi of Z is limit
Z tends to 0 of 1 by Z and this is infinity
00:25:55.929 --> 00:26:03.240
which is phi of 0 okay and mind you we have
defined what limit Z tends to infinity means
00:26:03.240 --> 00:26:10.059
we have define what when limit is in finite,
we are using all those definitions, we are
00:26:10.059 --> 00:26:17.120
using the fact that of infinity actually is
a point okay and we are using and we are also
00:26:17.120 --> 00:26:21.200
thinking of infinity as a value okay do not
a finite value.
00:26:21.200 --> 00:26:26.610
So the fact is that if you look at this map
now from extended complex plane to extended
00:26:26.610 --> 00:26:33.130
complex plane Z going to 1 over Z this is
actually homeomorphism this is the important
00:26:33.130 --> 00:26:42.090
point is a homeomorphism and if you throw
away both the origin and the point at infinity
00:26:42.090 --> 00:26:49.279
okay you get C star which is punctured complex
plane C minus 0 okay and C minus 0 goes to
00:26:49.279 --> 00:26:57.520
C minus 0 and if you take if you restrict
this map to C minus 0 it is an analytic, it
00:26:57.520 --> 00:27:03.270
is a holomorphic isomorphism that is the point.
This map is a holomorphic isomorphism which
00:27:03.270 --> 00:27:11.159
extends to infinity okay, so…in a continuous
way okay so so let me write that down.
00:27:11.159 --> 00:27:16.539
So let me draw the diagram again, so I have
the C union infinity here and I have this
00:27:16.539 --> 00:27:26.840
homeomorphism to C union infinity and this
is the map phi which is sending Z to 1 over
00:27:26.840 --> 00:27:36.899
Z okay which is W right and what is sitting
inside this is C star which is C minus 0 with
00:27:36.899 --> 00:27:43.700
the punctured plane and on this side also
I have C star that is the image of C star
00:27:43.700 --> 00:27:48.399
because if Z is a nonzero complex number and
1 by Z is also a nonzero complex number and
00:27:48.399 --> 00:27:54.790
this is correspondences is an Z going to 1
by Z is analytic if Z is not 0 because it
00:27:54.790 --> 00:27:59.519
has the derivative minus 1 over Z square you
know that pretty well Z equal to 0 is a is
00:27:59.519 --> 00:28:07.759
a pole of a is a simple pole of 1 by Z okay
so to the point is that if you restrict phi
00:28:07.759 --> 00:28:12.740
to C star what you get here is not just a
homeomorphism it is a holomorphic analytic
00:28:12.740 --> 00:28:13.740
isomorphism.
00:28:13.740 --> 00:28:20.879
So this is the this is so let me write that
here it is a holomorphic analytic isomorphism,
00:28:20.879 --> 00:28:30.120
this is what you get and now watch suppose
I have a function which is defined in a neighbourhood
00:28:30.120 --> 00:28:35.140
of infinity okay suppose f is a function which
is defined in the neighbourhood of infinity,
00:28:35.140 --> 00:28:42.210
what is the neighbourhood of infinity? A neighbourhood
of infinity is the exterior of a circle
00:28:42.210 --> 00:28:48.080
of sufficiently large radius, so you know
if I have a function defined on mod Z say
00:28:48.080 --> 00:28:53.850
greater than R, R sufficiently large this
water neighbourhood of infinity is. This is
00:28:53.850 --> 00:28:59.019
this is a neighbourhood so let me so let me
write that neighbourhood of infinity in where?
00:28:59.019 --> 00:29:04.730
This is the neighbourhood of infinity in the
extended complex plane mind you that is how
00:29:04.730 --> 00:29:10.820
the topology on the extended complex plane
has been given okay and in in fact it is if
00:29:10.820 --> 00:29:16.009
you look at it in the extended complex plane
then you are including the point at infinity
00:29:16.009 --> 00:29:19.889
and if you do not include the point at infinity
is a deleted neighbourhood okay.
00:29:19.889 --> 00:29:24.760
So since I am looking at it in C star minus
0 is a deleted neighbourhood of the point
00:29:24.760 --> 00:29:31.029
at infinity okay so this is a deleted neighbourhood,
a deleted neighbourhood of infinity where
00:29:31.029 --> 00:29:37.120
infinity being deleted okay and and you know
under this map Z going to 1 over Z this should
00:29:37.120 --> 00:29:44.200
correspond to a deleted neighbourhood of the
origin which is mod Z less than 1 by R in
00:29:44.200 --> 00:29:51.049
no R sufficiently large so one by are sufficiently
small, mod Z is less than 1 by R is the…so
00:29:51.049 --> 00:29:55.980
here probably since my target variable is
W I should not have use Z let me correct that
00:29:55.980 --> 00:30:03.220
this should have been W, so I have reserved
W of the target variable and W is…so if
00:30:03.220 --> 00:30:08.120
you plug in their W equal to 1 over Z I will
get 1 by mod Z is greater than R which is
00:30:08.120 --> 00:30:11.409
same as saying mod Z is less than 1 by R that
is the…
00:30:11.409 --> 00:30:19.659
So this corresponds to mod Z less than 1 by
R which is and you know of course this
00:30:19.659 --> 00:30:24.630
is the this is the neighbourhood of the origin
but if I but since I have not included the
00:30:24.630 --> 00:30:30.750
point at infinity on the right-hand side
I am not the thing that I got get on the
00:30:30.750 --> 00:30:34.850
left-side is going to not include the origin
because I am already you know say I am considering
00:30:34.850 --> 00:30:39.070
this as a subset here, so infinity is not
included and I am considering this as a subset
00:30:39.070 --> 00:30:43.470
here 0 is not included, so this is a deleted
neighbourhood of the origin, so this is so
00:30:43.470 --> 00:30:52.240
let me write this deleted neighbourhood of
this okays origin and mind you Z going to
00:30:52.240 --> 00:30:57.641
1 over Z is still a holomorphic isomorphism
of this small punctured disk centred at the
00:30:57.641 --> 00:31:06.830
origin radius one by our open disk with
the exterior of the disk with the radius
00:31:06.830 --> 00:31:15.500
R alright, so now suppose I have a function
defined in a neighbourhood of infinity okay
00:31:15.500 --> 00:31:21.639
that means I have a function f here, I have
my function f here okay and it is taking complex
00:31:21.639 --> 00:31:24.539
values alright then if I use.
00:31:24.539 --> 00:31:30.360
So you know this so you know this diagram
commutes basically this is just this diagram
00:31:30.360 --> 00:31:34.980
commutes means that I am just restricting
the map phi that is all okay, so this map
00:31:34.980 --> 00:31:42.899
from this punctured disk surrounding the origin
to exterior of the disk with radius R is just
00:31:42.899 --> 00:31:48.470
the holomorphic isomorphism Z going to 1 over
Z which is as phi and you compose that with
00:31:48.470 --> 00:31:56.279
f you get as before you get g, so g is just
first apply phi then apply f as before but
00:31:56.279 --> 00:32:08.029
what is that, so you know f is f of W okay
and W and and so you see that if if I calculate
00:32:08.029 --> 00:32:14.820
g of Z, g of Z will then be f of phi of Z
okay but what is phi of Z? Phi of Z is one
00:32:14.820 --> 00:32:21.769
over Z, so g of Z is nothing but f over 1
over Z okay, so so what you see that if you
00:32:21.769 --> 00:32:31.400
look at the map Z going to 1 over Z, the pullback
of f of W becomes f of 1 over Z okay and now
00:32:31.400 --> 00:32:39.340
you go back to this philosophy that whenever
you have an isomorphism of punctured domain
00:32:39.340 --> 00:32:45.279
and you have an analytic function on the target
domain and you pull it back to an analytic
00:32:45.279 --> 00:32:50.950
function on the source domain than the nature
of the singularity of the function at
00:32:50.950 --> 00:32:56.340
the target on the target spaces is the
same as the nature of the singularity in the
00:32:56.340 --> 00:32:57.340
source space.
00:32:57.340 --> 00:33:02.580
So you know if you apply that philosophy you
can see that f is at the nature of singularity
00:33:02.580 --> 00:33:09.960
of f of W at W equal to infinity is must correspond
must be the same as the nature of singularity
00:33:09.960 --> 00:33:16.470
of g of Z at Z equal to 0 but what is g of
Z? g of Z is f of 1 by Z okay so you know
00:33:16.470 --> 00:33:23.470
this justifies the statement that nature of
singularity of f of W at W equal to infinity
00:33:23.470 --> 00:33:30.059
is the same as nature of the singularity of
f of 1 by Z at Z equals to 0 okay, so so you
00:33:30.059 --> 00:33:34.669
must so you should understand what is going
on, why this is a very natural thing to do?
00:33:34.669 --> 00:33:42.919
Okay fine so so the moral of the story is
that we have a justification as to why
00:33:42.919 --> 00:33:48.700
studying f at infinity f of W at infinity
is the same as studying f of 1 by Z at Z equals
00:33:48.700 --> 00:33:57.200
to 0 okay, now let us go and try to look at
what we are going to get okay and so we will
00:33:57.200 --> 00:34:05.799
we will get 3 cases as usual because we
are trying to classify the singularity of
00:34:05.799 --> 00:34:12.070
f at infinity mind you in all these things
will talk about to be able to talk about these
00:34:12.070 --> 00:34:16.430
kinds of things the function f should be defined
in neighbourhood of infinity which means the
00:34:16.430 --> 00:34:23.760
function should be defined the exterior of
the circle okay for all values of the variable
00:34:23.760 --> 00:34:28.490
in the exterior of a circle of sufficiently
large radius okay which is what a neighbourhood
00:34:28.490 --> 00:34:29.840
of infinity is.
00:34:29.840 --> 00:34:35.330
So so what is the 1st what is the 1st case,
the 1st case is and will you say that
00:34:35.330 --> 00:34:43.980
f has infinity as a removable singularity
okay or more you know removable singularity
00:34:43.980 --> 00:34:50.300
is the same as a point where the function
is analytic okay that is exactly what the
00:34:50.300 --> 00:34:54.520
Riemann’s removable singularity theorem
says. It says that if you take a function
00:34:54.520 --> 00:34:59.610
which has an isolated singularity at a point
in the complex plane then it has a removable
00:34:59.610 --> 00:35:02.950
singularity at that point if and only if it
can be extended to an analytic function at
00:35:02.950 --> 00:35:07.510
that point and of course the weakest condition
is that it is even bounded in the neighbourhood
00:35:07.510 --> 00:35:11.960
of that point that is the strongest that is
the strongest part of the Riemann’s removable
00:35:11.960 --> 00:35:18.710
singularity theorem, so I would like to ask
when will f be analytic at infinity okay.
00:35:18.710 --> 00:35:22.579
Now you know you must be careful when you
think about the point at infinity because
00:35:22.579 --> 00:35:28.099
there are issues, so for example you should
not be attempted to say that f normally what
00:35:28.099 --> 00:35:31.920
is the definition of at a point in the complex
plane the definition the simplest definition
00:35:31.920 --> 00:35:35.640
of analyticity is that the function is differentiable
at that point and in every neighbourhood in
00:35:35.640 --> 00:35:39.950
in some neighbourhood of that point, at every
point in some neighbourhood of that point.
00:35:39.950 --> 00:35:44.100
Now you cannot adapt this definition at infinity
you cannot say a function f is analytic at
00:35:44.100 --> 00:35:48.510
infinity if it is differentiable at infinity
and it is also differentiable in neighbourhood
00:35:48.510 --> 00:35:52.710
of infinity okay that the function is already
differentiable in neighbourhood of infinity
00:35:52.710 --> 00:35:57.270
is given because it is already given to me
because infinity is an isolated singularity
00:35:57.270 --> 00:36:02.640
what trying to say that the function is differentiable
at infinity will not make sense because the
00:36:02.640 --> 00:36:04.070
derivative at infinity does not make sense.
00:36:04.070 --> 00:36:09.109
So what is a derivative at infinity if you
really try to define f dash of infinity if
00:36:09.109 --> 00:36:15.280
you want to define it like this you will write
limit Z tends to infinity f of Z minus f of
00:36:15.280 --> 00:36:20.920
infinity divided by Z minus infinity which
really does not make any sense. See f of infinity
00:36:20.920 --> 00:36:27.400
might make sense because if if for example
f extends to something continuous at infinity
00:36:27.400 --> 00:36:33.380
f of infinity could be defined as limit Z
tends to infinity f Z okay that is fine but
00:36:33.380 --> 00:36:40.680
this Z minus infinity is absolutely absurd
and trying to let Z tends to infinity so
00:36:40.680 --> 00:36:45.849
you know this is not trying to make f
differentiable at infinity is not going to
00:36:45.849 --> 00:36:51.859
help because there is no way to do it okay,
so how will you do it so the trick is you
00:36:51.859 --> 00:37:00.510
do it in rather you know indirect way you
recall the Riemann’s removable singularity
00:37:00.510 --> 00:37:07.790
theorem which says that if you look at a point
in a finite complex plane then the function
00:37:07.790 --> 00:37:14.400
is analytic there at that point which is an
isolated singularity if and only if it is
00:37:14.400 --> 00:37:18.860
if you want to bounded at that point in a
neighbourhood of that point or if it has a
00:37:18.860 --> 00:37:23.480
limit at that point which means it extends
to a continuous function at that point okay.
00:37:23.480 --> 00:37:29.750
So you do that so what you do is instead of
trying to define the function to be analytic
00:37:29.750 --> 00:37:33.910
at infinity if it is differentiable at infinity
which is wrong because you cannot define the
00:37:33.910 --> 00:37:37.900
derivative at infinity, what you do is? You
say you define the function to be analytic
00:37:37.900 --> 00:37:47.050
at infinity if either it is bounded in a neighbourhood
of infinity or it is it has a limit at
00:37:47.050 --> 00:37:53.010
infinity namely that is continuous at infinity
okay you make this definition and then you
00:37:53.010 --> 00:37:57.550
are in a very good shape and it will also
agree well with… both these definition will
00:37:57.550 --> 00:38:03.780
agree well with the earlier philosophy that
the nature of the singularity at infinity
00:38:03.780 --> 00:38:08.880
of f of W at infinity is the same as the nature
of singularity of f of 1 by Z at 0, so you
00:38:08.880 --> 00:38:21.800
can see that so so here is so defined f is
analytic at infinity okay.
00:38:21.800 --> 00:38:27.230
So mind you I am so this definition of
analytic is very very funny okay it is not
00:38:27.230 --> 00:38:31.910
the definition that the function is differentiable
at that point and differentiable in a neighbourhood
00:38:31.910 --> 00:38:35.810
of that point, it is not the definition okay
but it is a definition that, that point is
00:38:35.810 --> 00:38:41.369
the removable singularity okay it is an indirect
way of defining it, so f is analytic at infinity,
00:38:41.369 --> 00:38:58.880
so let me write f of W if well limit W
tends to infinity f of W exist okay or limit
00:38:58.880 --> 00:39:13.070
let me let me write this here f is bounded
at infinity or f is continuous at infinity
00:39:13.070 --> 00:39:21.500
okay, so and and you know all these things
all these things are equivalent is the
00:39:21.500 --> 00:39:27.099
3 different ways of trying to define the function
is analytic at infinity but the philosophy
00:39:27.099 --> 00:39:34.540
is that you are using you know you are using
Riemann’s removable singularity theorem
00:39:34.540 --> 00:39:35.660
and why are they equivalent?
00:39:35.660 --> 00:39:41.190
They are equivalent because of the following
thing because you see you see this is equivalent
00:39:41.190 --> 00:39:51.400
to saying that limit Z tends to 0 f of 1 by
Z exists which is which we have called as
00:39:51.400 --> 00:39:59.220
g Z because the map Z going to 1 over Z is
a homeomorphism these 2 are equivalent okay
00:39:59.220 --> 00:40:08.500
and well f is bounded at infinity is the same
as saying that f of 1 by Z is bounded
00:40:08.500 --> 00:40:15.160
at 0 g of Z is bounded which is defined by
f of 1 by Z mind you g of Z is by our original
00:40:15.160 --> 00:40:26.119
notation g of Z is f phi of Z and phi of Z
is 1 by Z which is W okay, so g is bounded
00:40:26.119 --> 00:40:38.089
at 0 okay and the 3rd thing as well g of Z
is equal to f of 1 by Z is continuous at 0
00:40:38.089 --> 00:40:45.980
and mind you all these 3 are in fact equivalent,
all these 3 conditions are equivalent or the
00:40:45.980 --> 00:40:54.740
function g of Z, why because now I am looking
at a function g of Z with 0 as an isolated
00:40:54.740 --> 00:41:00.010
singularity and all the 3 conditions are equivalent
by Riemann’s removable singularity theorem
00:41:00.010 --> 00:41:08.570
to saying that g is actually having a removable
singularity at 0 okay, so these 3 are
00:41:08.570 --> 00:41:16.440
so all these 3 are equivalent any here this
is Riemann’s theorem.
00:41:16.440 --> 00:41:25.140
So I need to make some more space to write
down, so let me write it here so this
00:41:25.140 --> 00:41:38.869
is by Riemann’s theorem on removable singularity
and because these 3 are equivalent therefore
00:41:38.869 --> 00:41:41.390
the conditions that I have written on the
left side are equivalent that is the point
00:41:41.390 --> 00:41:49.410
I want you to notice. See this is equivalent
to this is equivalent to this, this is therefore
00:41:49.410 --> 00:41:56.960
this therefore comes from the right side okay,
so so this will tell you that so you know
00:41:56.960 --> 00:42:01.049
now we have reconciled all the definitions
f has a removable singularity at infinity
00:42:01.049 --> 00:42:06.130
if f of 1 by Z as removable singularity at
0 is same as saying f of 1 by Z is analytic
00:42:06.130 --> 00:42:10.400
at 0, f of 1 by Z is continues at 0, f of
1 by Z is bounded in the neighbourhood of
00:42:10.400 --> 00:42:16.290
0 and you see we have used to things we have
use the fact that Z going to 1 by Z is a homeomorphism
00:42:16.290 --> 00:42:20.400
and we have also use the fact that we are
using the Riemann’s removable singularity
00:42:20.400 --> 00:42:26.430
theorem for a point singularity in the finite
plane okay, so so we will continue in the
00:42:26.430 --> 00:42:26.789
next talk.