WEBVTT
Kind: captions
Language: en
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So you see what we did last time was to try
to define when an analytic function is
00:01:01.220 --> 00:01:08.501
having infinity as a point of analyticity
okay and so as I told you if you recall one
00:01:08.501 --> 00:01:12.480
of the standard ways of defining analyticity
at the point is to say that the function is
00:01:12.480 --> 00:01:17.780
differentiable at that point and also in a
neighbourhood at every point in a neighbourhood
00:01:17.780 --> 00:01:22.979
of the given point okay. So this is how you
define analyticity at the point in the complex
00:01:22.979 --> 00:01:28.619
plane okay but since we are worried about
analyticity at infinity okay what we do is
00:01:28.619 --> 00:01:35.030
that we tend to look at the point at infinity
as singularity a singular point of an analytic
00:01:35.030 --> 00:01:40.250
function which is defined in a deleted neighbourhood
of infinity means that the function is analytic
00:01:40.250 --> 00:01:44.670
or mod Z greater than R or R sufficiently
large okay.
00:01:44.670 --> 00:01:50.360
Outside a circle of sufficiently large radius
the function you are given a function which
00:01:50.360 --> 00:01:57.289
is analytic and then infinity the point at
infinity becomes an isolated singular point
00:01:57.289 --> 00:02:00.960
and then you want to say that function is
analytic at infinity, so it will not help
00:02:00.960 --> 00:02:06.880
to say that the function is differentiable
at infinity because if you write you know
00:02:06.880 --> 00:02:13.240
a differential limit it does not make sense
at infinity, so what is the way out? The way
00:02:13.240 --> 00:02:19.280
out is to actually get the draw inspiration
from Riemann’s removable singularity theorem.
00:02:19.280 --> 00:02:23.520
Riemann’ removable singularity theorem tells
you that if you are looking at an isolated
00:02:23.520 --> 00:02:28.349
singularity of an analytic function at a point
in the complex plane then saying that the
00:02:28.349 --> 00:02:33.489
function is analytic at that point namely
that the which is essentially saying that
00:02:33.489 --> 00:02:37.370
the function can be extended to an analytic
function at that point including that point
00:02:37.370 --> 00:02:41.780
okay.
00:02:41.780 --> 00:02:47.319
This is one of the definitions of what the
removable singularities okay is equivalent
00:02:47.319 --> 00:02:53.110
to requiring that the function has a limit
at that point, it is which is equivalent to
00:02:53.110 --> 00:02:58.821
continuity of the function at that point,
in fact it is also equivalent to the function
00:02:58.821 --> 00:03:05.290
being bounded in a neighbourhood of that point
bounded in modulus of course, so this is so
00:03:05.290 --> 00:03:10.219
so the moral of the story is that you can
use these conditions to define the function
00:03:10.219 --> 00:03:17.040
to be analytic at infinity okay. What you
can say is that go it will not it does not
00:03:17.040 --> 00:03:22.180
make sense to will not work to say that the
function is differentiable at infinity, you
00:03:22.180 --> 00:03:27.349
can always say that the function has a removable
singularity at infinity in the sense that
00:03:27.349 --> 00:03:31.819
the function either has a limit at infinity.
00:03:31.819 --> 00:03:37.760
So the limit as Z tends to infinity f Z exist
okay that is one condition, the other condition
00:03:37.760 --> 00:03:43.480
is the function is bounded in bounded at infinity
that means there is a deleted neighbourhood
00:03:43.480 --> 00:03:48.450
of infinity where the function, the modulus
of the function can be made less than positive
00:03:48.450 --> 00:03:56.400
constant okay and in these conditions are
one and the same okay and why these conditions
00:03:56.400 --> 00:04:03.470
are one and the same is because of this other
important philosophy that studying the function
00:04:03.470 --> 00:04:12.189
at infinity, studying f of W at infinity is
the same as studying f of 1 by W at 0 okay,
00:04:12.189 --> 00:04:20.580
so and I told you the the or the justification
for that is that Z going to W equal to 1 by
00:04:20.580 --> 00:04:27.139
Z is actually a homeomorphism of the Riemann
extended complex plane onto the extended complex
00:04:27.139 --> 00:04:34.520
plane which interchange is 0 at infinity okay
and further if you throughout the point 0
00:04:34.520 --> 00:04:41.300
at infinity then you get the punctured complex
plane, the complex plane punctured at the
00:04:41.300 --> 00:04:42.300
origin.
00:04:42.300 --> 00:04:49.090
If this map Z going to 1 over Z which is equal
to W is going to be an analytic holomorphic
00:04:49.090 --> 00:04:54.990
isomorphism of the punctured complex plane
onto itself okay and under such an analytic
00:04:54.990 --> 00:05:01.639
isomorphism, the nature of the singularity
at 0 and the nature of the singularity at
00:05:01.639 --> 00:05:07.229
infinity which is an immediate 0 they should
correspond, this is a philosophy that we use.
00:05:07.229 --> 00:05:14.780
Now what I want to say is that I want to go
ead with with this and so so now that we
00:05:14.780 --> 00:05:20.741
have defined function being analytic at infinity
okay and the weakest definition or a function
00:05:20.741 --> 00:05:24.949
being analytic at infinity that it is bounded
at infinity namely there is a point I mean
00:05:24.949 --> 00:05:30.580
there is a small neighbourhood of infinity
okay which should be thought of as all Z,
00:05:30.580 --> 00:05:37.740
so I said mod Z greater than R for R sufficiently
large then mod f Z should be made you should
00:05:37.740 --> 00:05:44.110
be able to make mod f Z is less than M okay
for some positive constant M okay so so that
00:05:44.110 --> 00:05:50.259
is exactly what you want to do, so what I
want to do next is I want to worry about
00:05:50.259 --> 00:05:57.711
how I want to worry about what it means to
say that a function has a removable singularity
00:05:57.711 --> 00:06:01.870
at infinity okay, so we want to analyse this,
right.
00:06:01.870 --> 00:06:10.639
So let us take the let us take the…so we
will look at the case of an entire function
00:06:10.639 --> 00:06:18.199
alright but even before that I wanted to say
that you know this there is one more aspect
00:06:18.199 --> 00:06:24.400
that we need to be need to actually look into
okay and this aspect is about this aspect
00:06:24.400 --> 00:06:37.000
is about the Laurent series okay, so you see
you know in some sense of the study the isolated
00:06:37.000 --> 00:06:41.000
singularity of a function at a point in the
complex plane at the point Z naught the complex
00:06:41.000 --> 00:06:42.180
plane.
00:06:42.180 --> 00:06:46.870
One of the ways of studying this is by looking
at the Laurent expansion of the function centred
00:06:46.870 --> 00:06:52.470
at Z naught that means you expand the function
in positive and negative powers of Z minus
00:06:52.470 --> 00:06:57.590
Z naught that is the Laurent expansion you
get the coefficients are the Laurent coefficients
00:06:57.590 --> 00:07:02.919
and the Laurent theorem says that there is
such a Laurent expansion okay which is in
00:07:02.919 --> 00:07:12.090
general valid in an annulus is alright and
then you know that the nature of the Laurent
00:07:12.090 --> 00:07:18.289
expansion to be more specific the nature of
the principal part or the singular part of
00:07:18.289 --> 00:07:23.700
the Laurent expansion tell you what kind of
singularity Z naught is, so you know that
00:07:23.700 --> 00:07:29.820
if the Laurent expansion as only negative
powers I mean it has only finitely many negative
00:07:29.820 --> 00:07:36.629
powers of Z minus Z naught know it is a pole
if it has no that is if the principal part
00:07:36.629 --> 00:07:40.289
has only finitely many terms then it is a
pole okay.
00:07:40.289 --> 00:07:45.370
If the principal part does not exists namely
if the principal part is 0 then it is a removable
00:07:45.370 --> 00:07:52.800
singularity okay and if the principal part
has infinitely many negative powers of Z minus
00:07:52.800 --> 00:07:57.430
Z naught then it is an essential singularity
okay this is this are trying to classify
00:07:57.430 --> 00:08:01.980
singularity based on the Laurent expansion
okay which is something that you know. Now
00:08:01.980 --> 00:08:08.330
the question is what is an analog of this
when Z naught is the point at infinity, not
00:08:08.330 --> 00:08:12.479
when Z naught is the point at the complex
plane but when Z naught is the point at infinity,
00:08:12.479 --> 00:08:15.960
so how do you get a handle on this? How do
you get a handle on this?
00:08:15.960 --> 00:08:26.840
So you see so let me write that down, so my
point is what is this Laurent series at infinity
00:08:26.840 --> 00:08:34.490
which is rather funny thing again you
know you must remember that the way you deal
00:08:34.490 --> 00:08:38.450
with infinity is you have to be very clever,
there are certain things you can do with infinity,
00:08:38.450 --> 00:08:42.360
there are certain things you cannot do with
infinity so for example when you want to define
00:08:42.360 --> 00:08:46.410
a function to be analytic at infinity you
do not take the road of trying to define it
00:08:46.410 --> 00:08:50.930
differentiable at infinity because derivative
at infinity is not defined is not easy
00:08:50.930 --> 00:08:52.940
you cannot define it so easily, right.
00:08:52.940 --> 00:08:58.410
Of course you can go to another level of abstraction
called what is called of what is called Riemann’s
00:08:58.410 --> 00:09:03.420
surface okay and you can make the you can
make the complex plane along with the point
00:09:03.420 --> 00:09:08.940
at infinity namely the extended complex in
into a Riemann’s surface and once you have
00:09:08.940 --> 00:09:12.710
Riemann’s surface and you have a holomorphic
function then you can talk about derivative
00:09:12.710 --> 00:09:17.290
at any point by you seeing local coordinates
okay but then we do not want to get in to
00:09:17.290 --> 00:09:24.700
that amount of generality you can look up
that point of reasoning if you look at my
00:09:24.700 --> 00:09:29.300
video course on Riemann surfaces which is
available on the web okay but then let us
00:09:29.300 --> 00:09:32.530
not go to that level of abstraction okay.
00:09:32.530 --> 00:09:39.750
At this moment at this level of our exposition
we do not try to define derivative at infinity
00:09:39.750 --> 00:09:44.810
but trying to say it is analytic at infinity
you get away by using a drawing inspiration
00:09:44.810 --> 00:09:49.120
Riemann’s removable singularity theorem
okay by simply for example just requiring
00:09:49.120 --> 00:09:54.390
a function continues at infinity okay, so
in the same way if you look at the Laurent
00:09:54.390 --> 00:10:01.960
series at infinity, how do you define this?
So see the idea of so you have to think
00:10:01.960 --> 00:10:07.610
like this, the idea of what is a Laurent series
see the whole point about Laurent series,
00:10:07.610 --> 00:10:13.670
a Laurent series is a generalisation of
Taylor series and what is the Taylor series?
00:10:13.670 --> 00:10:20.831
Taylor series is trying to express an analytic
function in terms of simple analytic functions.
00:10:20.831 --> 00:10:25.250
Functions which are simplest possible analytic
functions, so you know if when I say Taylor
00:10:25.250 --> 00:10:28.850
series of an analytic function at a point
Z naught in the complex plane I am simply
00:10:28.850 --> 00:10:34.490
expanding the function in powers of Z minus
Z naught and I know Z… the powers of Z minus
00:10:34.490 --> 00:10:39.750
Z naught they are they are simple function
they are simple polynomials and I am trying
00:10:39.750 --> 00:10:46.060
to expand I am trying to write the function
a given analytic function as the limit of
00:10:46.060 --> 00:10:51.480
polynomials, after all a power series for
that matter any functional series is by definition
00:10:51.480 --> 00:10:55.670
when it converges is just the limit of partial
sums okay and if you take a Taylor series
00:10:55.670 --> 00:11:01.550
or a power series then the partial sums are
all polynomials okay and it is the limit of
00:11:01.550 --> 00:11:08.190
these polynomials that gives you the given
function okay which is the limit of the series.
00:11:08.190 --> 00:11:14.890
So the purpose of a Taylor series is to expand
a function as a series in terms of simple
00:11:14.890 --> 00:11:20.040
function is that is the idea okay and this
is this is at the point where the function
00:11:20.040 --> 00:11:24.940
is analytic okay but suppose a point you are
in question is not a point of analyticity
00:11:24.940 --> 00:11:29.400
suppose it is a point where the function as
an isolated singularity then then what comes
00:11:29.400 --> 00:11:34.920
in is the Laurent expansion, the Laurent expansion
says that well you can still get an expansion
00:11:34.920 --> 00:11:39.250
of the function in the form of the series
but then now you will have to allow also negative
00:11:39.250 --> 00:11:40.640
powers okay.
00:11:40.640 --> 00:11:46.790
Now so in general we think of philosophically
we think of the Laurent series as a generalisation
00:11:46.790 --> 00:11:53.990
of the Taylor series and the guiding philosophy
is that number 1 is that it allows you to
00:11:53.990 --> 00:12:00.690
expand the function in terms of simple functions
that is point number 1, point number 2 is
00:12:00.690 --> 00:12:06.210
that the Laurent series and the broken up
into pieces, there is one part of the Laurent
00:12:06.210 --> 00:12:12.820
series which consist of positive and 0 powers
of Z minus Z naught Z not is the point where
00:12:12.820 --> 00:12:18.240
you are looking at the Centre of the series
that is called the analytic part of the Laurent
00:12:18.240 --> 00:12:24.410
series okay and then there is also the part
of the series that involves the negative powers
00:12:24.410 --> 00:12:28.590
of Z minus Z naught which you call as the
singular part or the principal part of the
00:12:28.590 --> 00:12:29.590
Laurent series.
00:12:29.590 --> 00:12:33.680
So you see the general idea of Laurent series
is that it breaks the function into 2 functions,
00:12:33.680 --> 00:12:40.580
it breaks the function into 2 pieces it expresses
the function as a sum of 2 pieces. One piece
00:12:40.580 --> 00:12:45.562
is the analytic part of the function at that
point in the neighbourhood of the point, the
00:12:45.562 --> 00:12:49.230
other piece is the principal part of the function
at that point which is not analytic at that
00:12:49.230 --> 00:12:56.320
point okay. Now using these 2 guiding philosophies
you can also define what a Laurent series
00:12:56.320 --> 00:13:07.600
at infinity means okay and well the point
is that the point is as follows, so you suppose
00:13:07.600 --> 00:13:18.871
f of Z is analytic at Z equal to infinity
or let me not even start with analytic at
00:13:18.871 --> 00:13:19.871
infinity.
00:13:19.871 --> 00:13:27.270
Let me just say let me say this analytic at
neighbourhood of infinity suppose f of Z is
00:13:27.270 --> 00:13:36.820
analytic in a neighbourhood of infinity say
in and you know for obvious reasons let me
00:13:36.820 --> 00:13:46.410
do the following thing let me not use Z let
me use W okay say in mod W greater than R
00:13:46.410 --> 00:13:53.520
okay so I am using W as a variable because
I will always you know when I want to study
00:13:53.520 --> 00:13:58.580
W at infinity I will rather study you know
there is one of the tactics that we have been
00:13:58.580 --> 00:14:03.270
using is that you study W equal to 1 by Z
at 0, so that is why I want to reserve W for
00:14:03.270 --> 00:14:15.830
1 by Z okay, so suppose f is analytic at neighbourhood
of infinity say in mod W greater than R, so
00:14:15.830 --> 00:14:19.870
of course by this I do not mean the function
is analytic at infinity mind you okay, so
00:14:19.870 --> 00:14:24.520
so you have to be a little careful when I
say function is analytic in a neighbourhood
00:14:24.520 --> 00:14:29.470
of a point in the complex plane it is understood
that the function is also analytic at that
00:14:29.470 --> 00:14:34.270
point okay but then when I am saying f of
W is analytic in the neighbourhood of infinity
00:14:34.270 --> 00:14:38.750
I am not necessarily meaning that it is also
analytic at at the point at infinity. The
00:14:38.750 --> 00:14:42.790
point at infinity could is a singularity okay
it is an isolated singularity I did not know
00:14:42.790 --> 00:14:44.350
whether it is analytic or not okay.
00:14:44.350 --> 00:14:58.780
So let me let me state that we do not know
we do not know if f is analytic at infinity
00:14:58.780 --> 00:15:06.630
okay. Now what you do with this? Of course
you know let us go by the philosophy that
00:15:06.630 --> 00:15:18.330
to study f of W at infinity you study f of
1 by Z at 0 okay. The behaviour the behaviour
00:15:18.330 --> 00:15:35.850
of f of Z, f of W at W equal to infinity is
the same as the behaviour of f of 1 by Z which
00:15:35.850 --> 00:15:45.500
is g of Z mind you f of 1 by Z is the same
as W, f of W where W is equal to 1 by Z at
00:15:45.500 --> 00:16:08.430
Z equal to 0 okay. Note that g of Z is defined
in defined an analytic define an analytic
00:16:08.430 --> 00:16:13.500
or holomorphic in a deleted neighbourhood
of the origin which is just given by 0 less
00:16:13.500 --> 00:16:19.850
than mod Z is less than 1 by R so which is
actually writing this mod W with greater than
00:16:19.850 --> 00:16:28.120
R in terms of Z okay putting W equal to 1
by Z okay, so and 0 less than 0 strictly less
00:16:28.120 --> 00:16:32.320
than mod Z strictly less than 1 by R is a
deleted neighbourhood of the origin it is
00:16:32.320 --> 00:16:39.390
a circle I mean it is an interior of a circle
with the origin removed radius one by R okay.
00:16:39.390 --> 00:16:46.560
Now but then you know now you are looking
at the point 0 in the complex plane and we
00:16:46.560 --> 00:16:53.170
have Laurent’s theorem since now 0 is an
isolated singularity g of Z okay and the idea
00:16:53.170 --> 00:16:59.660
is that a nature of the singularity of g at
0 should be the same as the nature of singularity
00:16:59.660 --> 00:17:04.579
of f at infinity okay that is the idea okay
and of course you know why that is correct
00:17:04.579 --> 00:17:12.949
because Z going to W which is Z going to 1
by Z is an isomorphism okay of deleted neighbourhood,
00:17:12.949 --> 00:17:22.699
alright. So now g has a Laurent expansion
okay so g has a Laurent expansion has a Laurent
00:17:22.699 --> 00:17:26.299
expansion.
00:17:26.299 --> 00:17:34.370
So what is the Laurent expansion it is a of
Z equal to Sigma n equal to minus infinity
00:17:34.370 --> 00:17:43.929
to infinity a n Z power n, this is the Laurent
expansion of g okay and mind you I am the
00:17:43.929 --> 00:17:49.179
Centre of the expansion is the origin normally
if the centres is the point Z naught and you
00:17:49.179 --> 00:17:54.220
have to use powers of Z minus Z naught but
here Z naught is 0 so use powers of Z and
00:17:54.220 --> 00:17:59.100
the point that is a Laurent expansion at there
are negative powers of Z included as well
00:17:59.100 --> 00:18:05.170
that is why this summation is running from
minus infinity to plus infinity and well so
00:18:05.170 --> 00:18:15.529
this is the g as it is. Now you know what
you must understand is that you know if you
00:18:15.529 --> 00:18:24.899
look at if you look at this what does it mean
for F, so what this will tell you is that
00:18:24.899 --> 00:18:32.549
see this is valid for mod Z less than 1 by
R Z naught equal to 0 okay and if I plug-in
00:18:32.549 --> 00:18:38.639
see of course instead of Z I can put 1 by
W okay instead of Z can put 1 by W and g of
00:18:38.639 --> 00:18:41.559
1 by W is just f of W okay.
00:18:41.559 --> 00:18:48.399
So this is the same as writing this is equivalent
writing f of W is equal to Sigma you know
00:18:48.399 --> 00:18:57.539
n equal to minus infinity to infinity
a n W power minus n okay. So I can simply
00:18:57.539 --> 00:19:05.169
replace Z by 1 by W’s, so Z power and becomes
W power minus n and summation will run from
00:19:05.169 --> 00:19:12.070
again minus infinity to plus infinity the
only thing is that because I change my variable
00:19:12.070 --> 00:19:19.760
the powers the power of the nth power of the
variable has a negative subscript okay right,
00:19:19.760 --> 00:19:29.309
so so this is well this is the Laurent expansion
but now so you know the point is that the
00:19:29.309 --> 00:19:36.259
Laurent expansion, so this gives you a clue
as to what you should call the Laurent expansion
00:19:36.259 --> 00:19:41.190
of f at infinity you can very will call this
expression that you have written for f as
00:19:41.190 --> 00:19:46.080
a Laurent expansion at infinity okay because
it is in line with a philosophy that it is
00:19:46.080 --> 00:19:52.549
expressing it as a series okay in terms of
simple functions, the functions are just powers
00:19:52.549 --> 00:19:59.129
of Z okay, so you can very will call this
in on the right this expression of f W as
00:19:59.129 --> 00:20:03.549
Laurent expansion of f at infinity that is
fair enough but then there is a little bit
00:20:03.549 --> 00:20:04.669
more to be seen.
00:20:04.669 --> 00:20:10.179
You see if you write if you look at g of Z
the Laurent expansion split into 2 pieces
00:20:10.179 --> 00:20:15.549
as I told you, the Laurent expansion split
into principal part is a single apart plus
00:20:15.549 --> 00:20:20.090
an analytic part okay and the principal part
or the single apart consist of negative powers
00:20:20.090 --> 00:20:26.429
of set, so so you know let me write this like
this, so this is Sigma n equal to minus infinity
00:20:26.429 --> 00:20:37.929
to minus 1 a n Z power n plus Sigma n equal
to 0 to infinity okay a n Z power n okay
00:20:37.929 --> 00:20:47.470
there are 2 pieces and this fellow here is
the so what I have written here is the
00:20:47.470 --> 00:20:57.919
principal part is the principle or the singular
part at the at the origin okay and this part
00:20:57.919 --> 00:21:15.840
is the analytic part at the origin okay, so
you know if you want let me give symbols to
00:21:15.840 --> 00:21:23.860
these things, so this is so if you want this
is g S of Z, g S of Z is the single apart
00:21:23.860 --> 00:21:32.529
okay and this is g a Z is the analytic part
alright.
00:21:32.529 --> 00:21:41.659
Now let us make this change of variable which
is Z going to W which is equal to 1 by Z okay
00:21:41.659 --> 00:21:50.740
and you know now watch carefully what is our
philosophy? Our philosophy is that if you
00:21:50.740 --> 00:21:59.669
change available from Z to 1 over Z okay then
the behaviour at 0 at Z equal to 0 should
00:21:59.669 --> 00:22:04.720
correspond to the behaviour of W equal to
1 by Z at infinity therefore if you go by
00:22:04.720 --> 00:22:15.879
this if I transform g S of Z to W which is
1 by Z, what I should get should be the singular
00:22:15.879 --> 00:22:23.200
part at infinity okay because because g S
of Z is a similar part of singular apart at
00:22:23.200 --> 00:22:29.700
Z equal to 0 if I transform g S of Z I putting
Z equal to 1 by W what I should get is a singular
00:22:29.700 --> 00:22:35.082
apart at infinity okay and similarly if I
transform g a of Z by putting Z equal to 1
00:22:35.082 --> 00:22:38.889
by W I should get the analytic part at infinity.
00:22:38.889 --> 00:22:46.769
So what do I get…see basically what I get
is I get f of W you see is so I will get Sigma
00:22:46.769 --> 00:22:55.840
n equal to minus infinity to minus 1 a n so
I will get a W to the minus n plus Sigma here
00:22:55.840 --> 00:23:05.460
I will get n equal to 0 to infinity a n W
to the minus n and if you watch carefully
00:23:05.460 --> 00:23:11.090
so so let me use a different color at this
point, if you watch carefully now you see
00:23:11.090 --> 00:23:20.460
this guy here this corresponds to…what is
this? This is just g S of 1 by this is
00:23:20.460 --> 00:23:30.120
g S of 1 by W okay because I have put Z equal
to 1 by W and but g s was a singular part
00:23:30.120 --> 00:23:35.249
and therefore g s of 1 by W should also be
the singular part so this should be in principle
00:23:35.249 --> 00:23:44.999
this must be equal to the single apart of
f at f of W okay and watch carefully this
00:23:44.999 --> 00:23:52.480
singular part of f at W has what powers of
W? It has positive powers of W okay it has
00:23:52.480 --> 00:23:55.260
positive powers of W because n is negative.
00:23:55.260 --> 00:24:04.230
So W to minus n as positive therefore the
moral of the story is that if you write if
00:24:04.230 --> 00:24:12.230
you write the Taylor series if you write a
positive power series in a variable at infinity
00:24:12.230 --> 00:24:20.139
corresponds to a singular part okay and that
is very believable because as you go to infinity
00:24:20.139 --> 00:24:23.800
okay the partial sums which a polynomial are
going to go to infinity, so infinity is a
00:24:23.800 --> 00:24:29.220
pole actually therefore at least for the partial
okay it is not bounded at infinity, so it
00:24:29.220 --> 00:24:36.110
is correct okay so the whole point is that
when you look at the…so the Taylor series
00:24:36.110 --> 00:24:44.240
at infinity should be thought of I mean
when you look at the Laurent series at infinity
00:24:44.240 --> 00:24:51.869
the principal part should will look like a
Taylor series at the origin it is because
00:24:51.869 --> 00:24:56.149
it is exactly that by the transformation Z
going to 1 over Z okay.
00:24:56.149 --> 00:25:02.580
So this is the this is the similar part of
f and then and then this guy here this
00:25:02.580 --> 00:25:18.289
is this is f a or W which is g a of 1
by W and this is the analytic part at infinity
00:25:18.289 --> 00:25:25.750
okay so you see f of W is also split into
a singular part f S W plus f a of W which
00:25:25.750 --> 00:25:31.409
is an analytic part at infinity okay and mind
you the analytic part at infinity contains
00:25:31.409 --> 00:25:36.629
all the negative powers including the constant
because I have put n equal to 0 is also here,
00:25:36.629 --> 00:25:44.410
so a not is here in the analytic part of infinity
and so you see when you look at the variable
00:25:44.410 --> 00:25:54.320
at 0 okay then the analytic part consist of
the non-negative terms and the single apart
00:25:54.320 --> 00:26:00.259
consist of the negative terms but when you
look at the variable at infinity the analytic
00:26:00.259 --> 00:26:07.429
part consist of the negative terms and including
the constant and the principal part consist
00:26:07.429 --> 00:26:13.179
of the positive terms this is the this is
exactly what happens and it is correct okay.
00:26:13.179 --> 00:26:18.809
Now why do you thing that this analytic why
do you think that the negative powers are
00:26:18.809 --> 00:26:23.900
the portion of the expansion which involves
the negative power is analytic at infinity?
00:26:23.900 --> 00:26:28.789
That is correct because you see as variable
approaches infinity the negative power approach
00:26:28.789 --> 00:26:35.149
0, so you see that it is bounded essentially
okay so it is analytic and because our definition
00:26:35.149 --> 00:26:39.679
of analytic at infinity is either that it
should be bounded or it should tend to a limit
00:26:39.679 --> 00:26:47.210
okay and no positive power of a variable will
ever tend to will ever be bounded or will
00:26:47.210 --> 00:26:53.379
ever tend to 0 if you let the variable go
to infinity okay.
00:26:53.379 --> 00:26:58.049
So so everything is fine so you know so here
is the so here is our definition our definition
00:26:58.049 --> 00:27:04.549
is you take a function is an analytic in a
deleted neighbourhood of infinity okay right
00:27:04.549 --> 00:27:11.559
out its Laurent expansion okay basically
the Laurent expansion is the Laurent expansion
00:27:11.559 --> 00:27:20.490
of gotten by changing the variable to its
reciprocal okay and then what you do is you
00:27:20.490 --> 00:27:28.570
take the positive part of the Laurent expansion
okay in the original variable and call
00:27:28.570 --> 00:27:34.500
that as a singular part okay and the negative
part including the constant term is what is
00:27:34.500 --> 00:27:39.470
called the analytic okay so now we have this
clear definition of what a Laurent series
00:27:39.470 --> 00:27:43.519
at infinity should mean okay, fine.
00:27:43.519 --> 00:27:48.190
Now once you have made this definition of
Laurent series at infinity what you need to
00:27:48.190 --> 00:27:55.830
know is that whether this fits in well with
with the theory that you do on the finite
00:27:55.830 --> 00:28:01.440
complex plane, so for example you can ask
this question suppose you know for a point
00:28:01.440 --> 00:28:09.120
Z naught in the finite complex plane a function
is analytic at that point if you assume to
00:28:09.120 --> 00:28:13.450
begin with that that point is an an isolated
singularity the function is analytic at that
00:28:13.450 --> 00:28:17.299
point if and only if you write the Laurent
expansion about that point it has no singular
00:28:17.299 --> 00:28:23.809
part its principal part is 0 okay so if you
go by that philosophy for the function f which
00:28:23.809 --> 00:28:28.309
is defined in a neighbourhood of infinity
I mean for which infinity is an isolated singular
00:28:28.309 --> 00:28:34.350
point the function will be analytic at infinity
if the Laurent expansion at infinity as no
00:28:34.350 --> 00:28:37.210
singular part that is what should happen.
00:28:37.210 --> 00:28:42.510
Now does that happen? It does you see
here is my function f W defined in the neighbourhood
00:28:42.510 --> 00:28:49.639
of infinity its singular part is this is this
part which consist of positive powers of W
00:28:49.639 --> 00:28:56.940
okay, if that singular part is not there okay
that means that I can express f only in terms
00:28:56.940 --> 00:29:03.460
of negative powers of W then that is of course
analytic at infinity cause all these go to
00:29:03.460 --> 00:29:10.929
0 as W goes to infinity okay. So the moral
of the story is that our definition is correct,
00:29:10.929 --> 00:29:16.330
so you know the point is that sometimes you
may have to make definitions based on certain
00:29:16.330 --> 00:29:20.919
philosophy and then you have to check whether
it matches with what what happens, what you
00:29:20.919 --> 00:29:29.539
expect to happen as morally correct okay,
so so from this it is very clear that a function
00:29:29.539 --> 00:29:34.200
is analytic at infinity if and only if its
singular part at infinity vanishes okay.
00:29:34.200 --> 00:29:40.919
If you take the Laurent expansion at infinity
okay then its singular part vanishes so so
00:29:40.919 --> 00:29:53.789
let me write few things so this is the this
is called the singular part of f W at infinity
00:29:53.789 --> 00:30:11.129
and this guy here is called the analytic part
of f W at infinity and now you know if you
00:30:11.129 --> 00:30:17.489
if you look at it in a very simple way you
know when you are looking at infinity? The
00:30:17.489 --> 00:30:24.179
good functions are negative powers of W because
they go to 0 and the bad functions of positive
00:30:24.179 --> 00:30:28.889
power is W because we go to infinity so if
you expect a function to be good at infinity
00:30:28.889 --> 00:30:33.000
it should be expressible only in terms of
negative powers of W and that is why we negative
00:30:33.000 --> 00:30:38.429
powers of W along with constant that part
is analytic part at infinity okay.
00:30:38.429 --> 00:30:45.590
So so the definition is very clear and with
this definition you see that Riemann’s removable
00:30:45.590 --> 00:30:51.989
singularity theorem is also valid in its various
forms at the point at infinity a function
00:30:51.989 --> 00:30:57.490
namely a function which has infinity as an
isolated singularity has that singularity
00:30:57.490 --> 00:31:01.179
as a removable singularity if and only if
it is bounded in a neighbourhood of infinity,
00:31:01.179 --> 00:31:06.749
if and only if it tends to a limit at infinity
and that is also equivalent to saying that
00:31:06.749 --> 00:31:12.279
the Laurent series at infinity as no single
part they are all equivalent okay, so you
00:31:12.279 --> 00:31:17.869
get the same version of the theorem as you
would get in the case of a finite point, a
00:31:17.869 --> 00:31:22.080
point in the usual complex plane okay, so
everything fits well. The only thing that
00:31:22.080 --> 00:31:27.629
does not work is trying to define a derivative
at infinity that does not work okay, fine.
00:31:27.629 --> 00:31:34.289
So now what I am going to do is I am going
to ask this question as to what it means to
00:31:34.289 --> 00:31:42.239
having the removable singularity at infinity
for example for a good function For example
00:31:42.239 --> 00:31:46.879
a function like an entire function okay so
what does it mean and you will see that travel
00:31:46.879 --> 00:31:54.570
throughout connections with Liouville’s
theorem and so on so see so let us analyse
00:31:54.570 --> 00:31:55.720
this.
00:31:55.720 --> 00:32:22.909
Suppose that f of W as W equal to infinity
as removable singularity okay so so f W is
00:32:22.909 --> 00:32:31.480
f analytic so it has no singular part
it has only the analytic part of it expansion
00:32:31.480 --> 00:32:36.419
and that is what this is writable as the you
know the analytic part of the expansion at
00:32:36.419 --> 00:32:41.059
infinity will involve a negative powers of
the variable and also the constant term so
00:32:41.059 --> 00:32:48.899
it will be n equal to 0 to infinity if you
want I can call it as b n W power okay
00:32:48.899 --> 00:32:53.679
so this is the analytic part at infinity,
now let us analyse what it means say that
00:32:53.679 --> 00:33:02.259
the function is For example you know entire,
suppose…so I am looking the following case
00:33:02.259 --> 00:33:07.059
suppose I have an entire function and suppose
it is analytic at infinity, what happens?
00:33:07.059 --> 00:33:13.020
We will see that it will reduce to a constant
okay and that is just another avatar of the
00:33:13.020 --> 00:33:18.010
Liouville’s theorem okay so how do you see
that see suppose f has a removable singularity
00:33:18.010 --> 00:33:23.259
at W equal to infinity then f of W have this
expansion which is analytic at infinity then
00:33:23.259 --> 00:33:31.140
you see g of Z which is f of 1 by Z where
I put W equal to 1 by Z what I will get is
00:33:31.140 --> 00:33:38.010
I will get Sigma n equal to 0 we infinity
I will get B n Z power n which is you can
00:33:38.010 --> 00:33:42.259
see that that is clearly Taylor series
at the origin it is a power series at the
00:33:42.259 --> 00:33:47.230
origin okay centred at the origin, so it has
to represent an analytic function and that
00:33:47.230 --> 00:33:51.480
is correct because f is analytic f W is
analytic at W equal to infinity if and only
00:33:51.480 --> 00:33:59.450
if g Z which is f of 1 by Z that is analytic
at Z equals to 0. Let us go back to Riemann’s
00:33:59.450 --> 00:34:06.989
removable singularity theorem okay saying
that f is saying that f is analytic at infinity
00:34:06.989 --> 00:34:11.780
is the same as saying that f is bounded at
infinity okay so it means f is bounded in
00:34:11.780 --> 00:34:14.020
a deleted neighbourhood of infinity alright.
00:34:14.020 --> 00:34:38.770
So by Riemann’s removable singularity theorem
f is bounded so I am using bdd as an abbreviation
00:34:38.770 --> 00:34:47.990
for bounded at W equal to infinity, so f of
W if you want so that exist an positive constant
00:34:47.990 --> 00:35:02.870
M greater than 0 such that you know mod f
of W is less than M for if well if mod W if
00:35:02.870 --> 00:35:10.390
mod W is greater than R okay so this is this
is what bounded at infinity means. In a neighbourhood
00:35:10.390 --> 00:35:14.550
of infinity the function in modulus can be
bounded by positive constant okay and this
00:35:14.550 --> 00:35:20.140
is this is equivalent to infinity being good
point namely it is equivalent to infinity
00:35:20.140 --> 00:35:34.020
being a removable singularity okay. Now watch,
see for mod W less than or equal to R,
00:35:34.020 --> 00:35:39.500
so mod W greater than R the modulus of the
function is bounded by M okay and look at
00:35:39.500 --> 00:35:44.050
mod W less than or equal to R. I am using
the assumption that so I am putting this extra
00:35:44.050 --> 00:35:52.560
condition that f is entire and mind you I
am saying f is entire as a function of W okay.
00:35:52.560 --> 00:35:58.470
So I am trying to look at an entire function
which is having a removable singularity at
00:35:58.470 --> 00:36:08.380
infinity, so f of W itself is an entire function
even for W finite okay, so so you see if if
00:36:08.380 --> 00:36:25.850
f of W is entire okay then you know f is analytic
at 0, so is analytic at 0 okay f is analytic
00:36:25.850 --> 00:36:30.760
at 0 because the entire function supposed
to be analytic at every point, at every finite
00:36:30.760 --> 00:36:36.130
point, so if when I say f of W is entire as
a function of W it should be analytic at for
00:36:36.130 --> 00:36:40.570
all values of W in the complex plane in particular
it should be analytic at 0 and if it is an
00:36:40.570 --> 00:36:49.360
analytic at 0 and you know if you write out
it should tend to limit as W tends to 0 okay
00:36:49.360 --> 00:36:57.780
but then look at this expression, look at
this expression these expression mind you
00:36:57.780 --> 00:37:03.340
normally this expression will be will be valid
it is supposed to be a Laurent expansion at
00:37:03.340 --> 00:37:04.340
infinity.
00:37:04.340 --> 00:37:08.680
So it should be valid only in a neighbourhood
of infinity but since the function is entire
00:37:08.680 --> 00:37:13.550
it is valid everywhere, it is valid on the
whole on the whole complex plane, so it
00:37:13.550 --> 00:37:22.230
is valid at 0 also okay and if it is valid
at 0 you can see all the b n for n nonzero
00:37:22.230 --> 00:37:30.510
they all should be 0 okay because the moment
I get negative power of W at W equal to 0
00:37:30.510 --> 00:37:33.290
it is not going to give you a finite limit
is going to go to infinity because is going
00:37:33.290 --> 00:37:38.110
to become like a pole okay so the moral of
the story is that if you assume f is entire
00:37:38.110 --> 00:37:48.010
then f is analytic at 0 and this will imply
that all the b n is 0 for n not equals 0 and
00:37:48.010 --> 00:37:58.060
this this implies that f is a constant okay,
so so that is very obvious so all am trying
00:37:58.060 --> 00:38:03.270
to say is that if you have an entire function
which has a removable singularity at infinity
00:38:03.270 --> 00:38:05.750
then it is a constant.
00:38:05.750 --> 00:38:09.690
What is the Contra positive of that? The Contra
positive of that is supposed you have a non-constant
00:38:09.690 --> 00:38:15.230
entire function then infinity is certainly
not a removable singularity, for a non-constant
00:38:15.230 --> 00:38:23.910
entire function infinity cannot be a removable
singularity because the only entire function
00:38:23.910 --> 00:38:31.250
entire functions which are analytic at infinity
are constants okay and you can the reason
00:38:31.250 --> 00:38:40.530
why I got into this this stuff about modulus
is because I want to say that this is actually
00:38:40.530 --> 00:38:47.220
another avatar of Liouville’s theorem see
because you see, look at this look at this
00:38:47.220 --> 00:38:54.620
stuff that I have written in between see f
is analytic at infinity so outside a circle
00:38:54.620 --> 00:38:59.660
of sufficiently large radius mod W greater
than R, mod f W is bounded okay but if you
00:38:59.660 --> 00:39:03.720
look at the if you look at the interior of
that circle and the boundary of that circle
00:39:03.720 --> 00:39:08.670
I will get mod W less than or equal to R and
you see mod W less than or equal to R is a
00:39:08.670 --> 00:39:15.570
compact set it is both closed end bounded
and I have assumed f is entire so it is continuous,
00:39:15.570 --> 00:39:20.580
you know in a continuous function on a compact
set is bounded because the image of a compact
00:39:20.580 --> 00:39:26.500
set under continuous map is again compact
and compact is will imply bounded.
00:39:26.500 --> 00:39:32.300
So what this will tell you is that, there
is a bound for f even in mod W less than or
00:39:32.300 --> 00:39:38.980
equal to R that combined with the bound for
mod W greater than R will tell you that f
00:39:38.980 --> 00:39:43.460
is an entire function which is bounded on
the whole plane and then Liouville’s theorem
00:39:43.460 --> 00:39:47.590
will tell you that f is a constant, so that
is the point and I want to tell you, I want
00:39:47.590 --> 00:39:53.810
to tell you that see this argument that an
entire function which is analytic at infinity
00:39:53.810 --> 00:40:00.230
is constant is actually another avatar of
Liouville’s theorem okay it is actually
00:40:00.230 --> 00:40:05.220
another avatar of Levin’s theorem that is
what you have to understand, so the moral
00:40:05.220 --> 00:40:10.980
of the story is that whenever you are looking
at a non-constant entire function infinity
00:40:10.980 --> 00:40:15.790
is certainly a singularity it is a it is not
a removable singularity, so it can either
00:40:15.790 --> 00:40:21.820
be a pole or it can be an essential singularity
it cannot be removable singularity okay and
00:40:21.820 --> 00:40:27.890
the only exemptions are constants which which
are very uninteresting okay, so I will stop
00:40:27.890 --> 00:40:28.670
here.