WEBVTT
Kind: captions
Language: en
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Okay so let me have your attention you see
we are now more or less you know comfortable
00:01:41.190 --> 00:01:46.350
with thinking of the point at infinity of
functions behaving at infinity, and analyticity
00:01:46.350 --> 00:01:53.590
at infinity, singularities at infinity, so
you know we wanted to do that because we needed
00:01:53.590 --> 00:02:02.940
to look at functions on the extended plane
okay and this viewpoint is very very important
00:02:02.940 --> 00:02:09.130
because it is not…normally when you do complex
analysis say for example in a 1st course you
00:02:09.130 --> 00:02:14.250
are worried about only questions on the plane
okay on the complex plane you are not worried
00:02:14.250 --> 00:02:20.490
about the point at infinity okay and but if
you include the point at infinity you get
00:02:20.490 --> 00:02:26.200
more information that is the point, so so
I explain this last time but let me again
00:02:26.200 --> 00:02:27.220
repeated.
00:02:27.220 --> 00:02:32.980
So I am saying or example look at let us take
an entire function okay let us take an entire
00:02:32.980 --> 00:02:41.180
function take a non-constant entire function
okay then if you look at the little Picard
00:02:41.180 --> 00:02:46.320
theorem what it will tell you is that the
image of the whole plane is going to be either
00:02:46.320 --> 00:02:52.489
the whole plane or the pole plane minus 1
point which is one value which it may not
00:02:52.489 --> 00:02:59.880
take and that is the best possible I mean
it cannot mix 2 values, what it means is that
00:02:59.880 --> 00:03:06.150
if an entire function Mrs more than one value
it has to be constant okay.
00:03:06.150 --> 00:03:16.730
So you see and of course you have but
then you know if I ask you now take this entire
00:03:16.730 --> 00:03:24.049
function and take the exterior of a circle
of sufficiently large radius okay and take
00:03:24.049 --> 00:03:31.720
its image, what will the image look like?
Okay then you know you do not have an answer,
00:03:31.720 --> 00:03:41.019
you do not have an answer, so whereas
for example if you that the if you can think
00:03:41.019 --> 00:03:50.870
of infinity as a singular point and in fact
if infinity is an essential singular point
00:03:50.870 --> 00:03:57.549
okay then you can apply the great Picard theorem
which will tell you that the image of every
00:03:57.549 --> 00:04:03.449
deleted neighbourhood of infinity will be
either the full plane or the plane minus a
00:04:03.449 --> 00:04:04.630
point okay.
00:04:04.630 --> 00:04:08.690
So you can use that which is a more powerful
theorem to say that if you have an entire
00:04:08.690 --> 00:04:15.070
function and suppose it is not algebraic namely
that infinity it is a transcendental entire
00:04:15.070 --> 00:04:22.510
function that infinity is you know essential
singularity then if you take the exterior
00:04:22.510 --> 00:04:27.530
of a circle of sufficiently large radius for
that matter if you take exterior of any circle
00:04:27.530 --> 00:04:32.290
okay because exterior of any circle is going
to be a deleted neighbourhood of the point
00:04:32.290 --> 00:04:37.820
at infinity in the extended plane and if you
apply the big Picard theorem you get the information
00:04:37.820 --> 00:04:46.850
that the image of the exterior of any circle
is always going to be the whole plane or the
00:04:46.850 --> 00:04:50.699
whole plane minus a point okay and it will
also give you the additional information that
00:04:50.699 --> 00:04:58.410
all the values it takes it takes infinitely
many times okay so you see the advantage of
00:04:58.410 --> 00:05:04.680
thinking of the point at infinity okay you
get more information.
00:05:04.680 --> 00:05:10.940
So along these lines you know that for example
you can characterise entire functions as transcendental
00:05:10.940 --> 00:05:16.000
or algebraic based on whether infinity is
an essential singularity or not if infinity
00:05:16.000 --> 00:05:20.150
is not an essential singularity then it is
a pole or a removable singularity infinity
00:05:20.150 --> 00:05:25.410
is a removable singularity then by Liouville’s
theorem the function has to be constant if
00:05:25.410 --> 00:05:31.220
infinity is a pole then the function has to
be a polynomial and otherwise it is a transcendental
00:05:31.220 --> 00:05:34.039
entire function, infinity is an essential
singularity, okay.
00:05:34.039 --> 00:05:41.509
Now what we need to do is see we are the whole
aim of this all this is you trying to prove
00:05:41.509 --> 00:05:48.030
the big Picard theorem and as a corollary
deduce the little Picard theorem okay and
00:05:48.030 --> 00:05:53.320
you know theory record or that is to do analysis
on compact spaces of Meromorphic functions
00:05:53.320 --> 00:05:58.290
okay, so you have to study the apology and
you have to understand…do some analysis
00:05:58.290 --> 00:06:04.590
using compact families of Meromorphic functions,
so we need to worry about Meromorphic functions.
00:06:04.590 --> 00:06:09.050
So let me start by telling you what a Meromorphic
function is? I have told this before but this
00:06:09.050 --> 00:06:13.930
is let us go into that this a little bit more
detail because that is something that gives
00:06:13.930 --> 00:06:16.490
you a link to algebra okay.
00:06:16.490 --> 00:06:22.100
So the reason why a lot of complex geometry
is connected to algebra is because of the
00:06:22.100 --> 00:06:27.720
fact that all the Meromorphic functions they
form a field naturally and that field is an
00:06:27.720 --> 00:06:32.600
extension field of the complex numbers because
the complex number is always can be thought
00:06:32.600 --> 00:06:37.480
of as one functions. Every complex number
corresponds to the constant function given
00:06:37.480 --> 00:06:44.880
by that number, so you know and constant functions
are of course Meromorphic okay they are analytic,
00:06:44.880 --> 00:06:50.010
so you see that the field of Meromorphic function
as the field extension of the complex field
00:06:50.010 --> 00:06:57.320
is...the properties of this field extension
in algebra for example in and things like
00:06:57.320 --> 00:07:03.160
that, that determining a lot about the geometry
of the domain on which you are studying the
00:07:03.160 --> 00:07:09.840
Meromorphic functions okay and this is extensively
used for example in classifying Riemann surfaces
00:07:09.840 --> 00:07:13.710
and more generally if you want to classify
manifolds and so on complex manifolds you
00:07:13.710 --> 00:07:15.320
can use this theory.
00:07:15.320 --> 00:07:22.400
It gives you an interface between complex
analysis and topology and algebraic geometry,
00:07:22.400 --> 00:07:29.150
complex geometry okay and algebra in that
sense okay. So what is the Meromorphic function?
00:07:29.150 --> 00:07:37.250
The definition is pretty simple, it is a function
which has which is analytic on a domain okay
00:07:37.250 --> 00:07:41.890
but the only singularity it has are poles,
okay that is the definition of a Meromorphic
00:07:41.890 --> 00:07:42.890
function, okay.
00:07:42.890 --> 00:08:01.300
So let me write that down, so the field of
Meromorphic functions
00:08:01.300 --> 00:08:16.550
so here is the definition. Let D in the extended
complex plane be a domain okay, so I am looking
00:08:16.550 --> 00:08:23.410
at a domain in the extended complex plane
by mind you it means that it is an open connected
00:08:23.410 --> 00:08:32.529
set and of course we always assume that it
is nonempty okay. A Meromorphic function on
00:08:32.529 --> 00:09:10.029
D is either an analytic function
on D or a function that is analytic on D except
00:09:10.029 --> 00:09:21.130
for poles, okay. So Meromorphic function by
definition is a function that might that could
00:09:21.130 --> 00:09:31.920
be define on D or D minus points where those
points will be of course singularities okay
00:09:31.920 --> 00:09:35.610
and the point is that they have to be isolated
in fact they have to be poles, okay.
00:09:35.610 --> 00:09:42.570
This is the this is the definition okay, so
so we say that we often use the abbreviation
00:09:42.570 --> 00:09:51.041
and I mean we use this short phrase analytic
except for poles that is the that is what
00:09:51.041 --> 00:09:58.510
Meromorphic means okay and now we need to
make a few observations, the 1st thing is
00:09:58.510 --> 00:10:04.760
that you see the number of poles okay the
set of poles of the function it may be of
00:10:04.760 --> 00:10:10.029
course empty in which case the function is
actually analytic or holomorphic but if it
00:10:10.029 --> 00:10:14.740
is not analytic then it has poles at least
one pole and the fact is that the set of poles
00:10:14.740 --> 00:10:20.279
in D will be accountable set, so the 1st thing
is that you cannot have an uncountable set
00:10:20.279 --> 00:10:25.810
of poles, so a Meromorphic function will have
only accountably many poles and what is the
00:10:25.810 --> 00:10:31.600
reason for that, the reason for that is because
of 2nd accountability okay.
00:10:31.600 --> 00:10:37.300
The complex plane is just R2 and you know
Euclidian spaces RN they are all 2nd countable
00:10:37.300 --> 00:10:45.779
because they have accountable bases, so you
can take all the open balls centered at points
00:10:45.779 --> 00:10:51.779
with rational coordinates and with radii given
by rational numbers okay and use the fact
00:10:51.779 --> 00:10:58.839
that the rational number form a countable
set okay and then you get this collection
00:10:58.839 --> 00:11:07.899
of open balls centred at points with rational
coordinates and have a rational radii as accountable
00:11:07.899 --> 00:11:14.430
collection of open sets and they will form
a basis or the topology okay in the sense
00:11:14.430 --> 00:11:22.809
that any open set is a union of such sets
okay and the intersection of any 2 sets like
00:11:22.809 --> 00:11:27.059
that of this types again is union of sets
of that form okay.
00:11:27.059 --> 00:11:32.329
So it is a basis for topology so topology
has countable bases this is 2nd countability
00:11:32.329 --> 00:11:37.309
and now in particular For example the plane
are 2 or the complex plane as topologically
00:11:37.309 --> 00:11:44.709
is 2nd countable so you know what you can
do is and of course any subspace of a 2nd
00:11:44.709 --> 00:11:49.720
countable space is also 2nd countable because
all you have to do is that you have to take
00:11:49.720 --> 00:11:54.819
the countable base, countable base for the
bigger space and intersect it with the subspace
00:11:54.819 --> 00:11:59.800
to get a countable base for the subspace okay.
00:11:59.800 --> 00:12:05.709
So you take any domain in the complex plane
or extended complex plane, it will be 2nd
00:12:05.709 --> 00:12:09.540
countable okay and by the way if you are looking
at the extended complex plane of course it
00:12:09.540 --> 00:12:18.970
is homeomorphic to the Riemann sphere and
the Riemann’s sphere is a subset of R3 real
00:12:18.970 --> 00:12:24.959
three-dimensional space and since real three-dimensional
space is 2nd countable the Riemann sphere
00:12:24.959 --> 00:12:29.110
being a subset is also 2nd countable okay
so the extended plane is also 2nd countable
00:12:29.110 --> 00:12:37.269
okay after all you are just adding one more
point at infinity and if you want you can
00:12:37.269 --> 00:12:45.610
take all the disk centred at infinity but
of course you should not say finite radius
00:12:45.610 --> 00:12:51.800
okay unless you are looking at the images
in the Riemann sphere with the point at infinity
00:12:51.800 --> 00:12:57.170
corresponding to the North pole okay but in
any case if you are taking a domain in the
00:12:57.170 --> 00:13:04.829
extended plane is going to be 2nd countable
and now if you take a function which has poles
00:13:04.829 --> 00:13:11.519
in that domain then the set of poles poles
by definition are isolated, poles are by definition
00:13:11.519 --> 00:13:12.920
isolated singularities.
00:13:12.920 --> 00:13:18.790
So you have an isolated subset of a 2nd countable
set of a 2nd countable space and isolated
00:13:18.790 --> 00:13:24.370
subset will always be countable okay. The
reason is because since the points of the
00:13:24.370 --> 00:13:31.439
subsets are isolated you can cover each of
those points by a member of the countable
00:13:31.439 --> 00:13:38.239
bases okay and you can choose different members
from the countable basis because the points
00:13:38.239 --> 00:13:43.689
are separated from each other because they
are isolated and in this way you get a mapping
00:13:43.689 --> 00:13:49.360
an injective mapping from the set of isolated
points in this case these are the poles to
00:13:49.360 --> 00:13:57.290
the this countable base okay and the moment
you get an injective map what it tells you
00:13:57.290 --> 00:14:05.029
is that whenever you get a injective map from
a set to another set which is countable then
00:14:05.029 --> 00:14:10.470
your original set also countable because it
is a subset of a countable set is countable
00:14:10.470 --> 00:14:11.470
okay.
00:14:11.470 --> 00:14:15.439
Therefore you see the 1st observation is that
a Meromorphic function will have only countably
00:14:15.439 --> 00:14:21.449
many poles and then if you are looking at
the Meromorphic function on the whole Riemann
00:14:21.449 --> 00:14:28.329
sphere okay which means you are taking the
domain in the extended complex plane often
00:14:28.329 --> 00:14:32.899
I keep saying Riemann sphere because I think
of the extended complex plane as Riemann sphere
00:14:32.899 --> 00:14:39.250
they are one and the same, so your domain
is the whole extended complex plane and you
00:14:39.250 --> 00:14:43.259
are looking at a Meromorphic function of the
whole extended complex plane you get more,
00:14:43.259 --> 00:14:49.149
what you get is not only is a set of poles
countable it is actually finite because an
00:14:49.149 --> 00:14:57.949
isolated set of points in a compact set 2nd
countable set is going to be only finite okay
00:14:57.949 --> 00:15:01.689
because you know one property of compactness
is that if you had in finite subset it should
00:15:01.689 --> 00:15:08.860
have an accumulation point okay therefore
the moral of the story is that if you are
00:15:08.860 --> 00:15:16.739
looking at a Meromorphic function on the whole
extended plane then it should have only finitely
00:15:16.739 --> 00:15:18.379
many poles, okay.
00:15:18.379 --> 00:15:38.709
So let me so let me write that down so here
are remarks, so remark number 1 the set of
00:15:38.709 --> 00:16:04.180
poles in D is countable this by 2nd countability
okay, so I am just saying that am just using
00:16:04.180 --> 00:16:08.739
the fact that the poles are of course isolated
and I am using 2nd accountability okay and
00:16:08.739 --> 00:16:24.939
the 2nd remark is that if D is the whole extended
complex plane then the set of poles is finite
00:16:24.939 --> 00:16:33.100
and that is because of compactness of the
extended plane okay and then there is yet
00:16:33.100 --> 00:16:47.439
another fact if you see the set of poles they
form they are set of isolated points because
00:16:47.439 --> 00:16:54.930
poles are by definition isolated but on the
whole some of them might converge. See they
00:16:54.930 --> 00:17:00.629
may be isolated points but they make it closer
and closer and closer okay.
00:17:00.629 --> 00:17:07.850
So like the sequence of real numbers 1 by
n okay the sequence of real numbers1 by n
00:17:07.850 --> 00:17:14.540
consist of you know it is a set of isolated
points but it converges, it converges to 0
00:17:14.540 --> 00:17:22.319
okay so you could have set of poles okay you
could have a convergent sequence of poles
00:17:22.319 --> 00:17:27.939
and then the fact is that this convergence
sequence of poles as to converge only on the
00:17:27.939 --> 00:17:33.360
boundary it cannot converge in the interior
stop the reason is that if it converge in
00:17:33.360 --> 00:17:40.460
the interior you are going to get a point
which has to be in the domain of F.
00:17:40.460 --> 00:17:47.510
So it has to be either a singularity or it
has to be a point of non-singularity but then
00:17:47.510 --> 00:17:51.650
you know it cannot be kind of non-singularity
it cannot be a point of analyticity because
00:17:51.650 --> 00:17:59.190
it is approached very as close as you want
I poles and if a point is non-singular point
00:17:59.190 --> 00:18:03.830
there should be a disk around that point where
there are no singularities that since this
00:18:03.830 --> 00:18:09.650
limit is being approached by poles this cannot
be a non-singular point it cannot be a holomorphic
00:18:09.650 --> 00:18:16.830
point or analytic point it has to be therefore
necessarily a singular point but then you
00:18:16.830 --> 00:18:22.460
assume that the singular points are all going
to be poles and here you have gotten a non-isolated
00:18:22.460 --> 00:18:29.679
singular point that is a contradiction therefore
if the set of poles of the Meromorphic functions
00:18:29.679 --> 00:18:37.700
has a convergent subset then it will converge
only on the boundary okay and of course this
00:18:37.700 --> 00:18:43.440
can happen if in the case when the domain
is the whole extended plane because when the
00:18:43.440 --> 00:18:49.220
domain is the whole extended plane okay the
boundary is empty alright and there are only
00:18:49.220 --> 00:18:52.620
finitely many points okay.
00:18:52.620 --> 00:19:16.130
So well so let me write this down if the set
of poles has convergence sequence
00:19:16.130 --> 00:19:43.150
of distinct points then it has a limiting
point in the boundary of D okay that point
00:19:43.150 --> 00:19:52.360
cannot it cannot converge inside D okay, so
you know so the moral of the story is following,
00:19:52.360 --> 00:19:55.990
the moral of the story is when you are looking
at Meromorphic functions of course if you
00:19:55.990 --> 00:20:01.760
are looking at Meromorphic functions on the
whole extended plane then there is not anything
00:20:01.760 --> 00:20:05.039
we are going to see that these are exactly
the rational functions okay.
00:20:05.039 --> 00:20:11.240
We are going to prove that rational functions
which are given by you know quotient of polynomials
00:20:11.240 --> 00:20:15.179
they are exactly the same as a Meromorphic
functions on the extended plane is nothing
00:20:15.179 --> 00:20:23.289
more okay but if your domain is not the extended
plane okay then the Meromorphic functions
00:20:23.289 --> 00:20:29.580
if you look at for example if you are
looking at Meromorphic functions on the unit
00:20:29.580 --> 00:20:36.650
disk okay it is possible that you may have
a sequence of poles is converging and it is
00:20:36.650 --> 00:20:41.299
converge to a point in the boundary of the
unit disk okay and behaviour at that point
00:20:41.299 --> 00:20:46.210
of this of this Meromorphic functions or this
family of Meromorphic functions is very very
00:20:46.210 --> 00:20:55.159
important topologically okay, so so this is
very very important that you need to study
00:20:55.159 --> 00:21:02.100
the boundary points also okay especially when
you have sequence of poles tending to the
00:21:02.100 --> 00:21:07.070
boundary okay and of course what the remark
says is that if there is a sequence of poles
00:21:07.070 --> 00:21:12.179
which is converging it has to go to only to
the boundary it cannot come interior point
00:21:12.179 --> 00:21:13.840
okay.
00:21:13.840 --> 00:21:26.320
Well now so let me go ahead with this fact
that I told you that you know a rational function
00:21:26.320 --> 00:21:50.690
of Meromorphic and converse holds a rational
function is given by a quotient of polynomials
00:21:50.690 --> 00:22:01.470
f of Z is equal to P of Z by Q of Z okay,
so you know now I want to tell you something
00:22:01.470 --> 00:22:10.210
I am now all this time you know I was using
the variable w I was writing f of W. The reason
00:22:10.210 --> 00:22:15.880
why I was using the variable w is because
I wanted a study the point at w equal to infinity
00:22:15.880 --> 00:22:21.509
and the way I would do that is by studying
g of Z equal to f of 1 by Z which is f of
00:22:21.509 --> 00:22:27.630
w by making the transformation w equal to
1 by Z okay but now we have by now we understand
00:22:27.630 --> 00:22:33.490
how to deal with the point at infinity, so
I am switching back to the variable set okay.
00:22:33.490 --> 00:22:39.929
So Z will be our complex variable from now
on, so rational function is just quotient
00:22:39.929 --> 00:22:53.799
of polynomials, so P Q polynomial in Z with
coefficients in complex numbers of course
00:22:53.799 --> 00:22:59.320
you know coefficient of polynomial of course
include even constants okay you can take Q
00:22:59.320 --> 00:23:07.590
to be 1 okay constants are also treated as
polynomials of the be 0 okay. In fact in particular
00:23:07.590 --> 00:23:14.830
0 is also considered as a polynomial, constant
polynomial and therefore you must understand
00:23:14.830 --> 00:23:26.539
that constants are also Meromorphic functions
okay and of course you know of force a function
00:23:26.539 --> 00:23:33.909
rational function like this is certainly a
Meromorphic function on the extended plane
00:23:33.909 --> 00:23:37.680
that is very clear because if you take a function
of this form if you take a function which
00:23:37.680 --> 00:23:43.960
is coefficient of polynomial is okay it is
going to…where is it going to have problems
00:23:43.960 --> 00:23:49.149
with analyticity is going to have problem
with analyticity where the denominator vanishes.
00:23:49.149 --> 00:23:55.019
So you take those…and where the denominator
vanishes is just the set of zeros over polynomial
00:23:55.019 --> 00:24:02.679
that is only finitely many points okay and
maybe some of these may be the numerator polynomial
00:24:02.679 --> 00:24:08.770
so has zeros at some of these points, so some
of the zeros they cancel out okay but then
00:24:08.770 --> 00:24:16.679
the fact is that this function cannot have
poles at winds more than the set of zeros
00:24:16.679 --> 00:24:23.009
of the denominator polynomial okay and at
all other points it is analytic, so it is
00:24:23.009 --> 00:24:29.049
an analytic function defined on the whole
you know in the extended plane with having
00:24:29.049 --> 00:24:33.679
only you know finitely many points which are
going to be poles.
00:24:33.679 --> 00:24:37.630
So it satisfies the definition of Meromorphic
functions, so you see there for a rational
00:24:37.630 --> 00:24:42.120
function is certainly a Meromorphic function
okay and the fact the theorem is that the
00:24:42.120 --> 00:24:56.039
converse is also true okay, so that is what
I am going to about. A rational function is
00:24:56.039 --> 00:25:29.919
analytic on C union infinity except possibly
at the zeros of denominator polynomial, hence
00:25:29.919 --> 00:25:50.341
it is Meromorphic on the extended plane, so
the theorem is that the converse holds, the
00:25:50.341 --> 00:26:13.490
converse holds so here is the so essentially
here is the theorem. If f of Z is Meromorphic
00:26:13.490 --> 00:26:33.950
on the extended plane then f of Z is a rational
function okay, so there is no difference between
00:26:33.950 --> 00:26:41.370
Meromorphic functions and quotient of polynomials
okay there is absolutely no difference and
00:26:41.370 --> 00:26:48.799
you know now this should you should now think
of it like this.
00:26:48.799 --> 00:26:56.490
If you look at the polynomials in one variable
say in the variable Z with the complex coefficients
00:26:56.490 --> 00:27:01.120
that gives you the polynomial ring in one
variable over the complex numbers C square
00:27:01.120 --> 00:27:09.330
bracket Z okay and you know that is an integral
domain if we have studied that in algebra
00:27:09.330 --> 00:27:14.850
and then this integral domain has what is
known as a field of fractions, a quotient
00:27:14.850 --> 00:27:20.940
yield this is just like you get the rational
number as a field of fractions when you look
00:27:20.940 --> 00:27:27.019
at the integral domain consisting of the integers,
so if you look at the field of fractions of
00:27:27.019 --> 00:27:34.450
the polynomial ring in one variable over complex
numbers you are going to get just quotient
00:27:34.450 --> 00:27:39.450
of polynomial is and these are precisely the
Meromorphic functions on the extended plane.
00:27:39.450 --> 00:27:46.000
So the moral of the story is that if you look
at the extended plane okay the set of Meromorphic
00:27:46.000 --> 00:27:52.590
functions automatically is a field it is none
other than the field of fractions of the polynomial
00:27:52.590 --> 00:27:58.769
ring in one variable over C okay, so you can
see that.
00:27:58.769 --> 00:28:09.909
So well so let us try to prove that theorem
let me use another color
00:28:09.909 --> 00:28:19.120
so proof, so you see so I am given a function
f Z the function is supposed to be considered
00:28:19.120 --> 00:28:23.419
on the extended plane okay which means you
are also considered the point at infinity
00:28:23.419 --> 00:28:29.960
okay and then the only information is given
to you is that a function is Meromorphic it
00:28:29.960 --> 00:28:36.470
means you know that it has only poles and
since the set in consideration with the extended
00:28:36.470 --> 00:28:39.830
plane which is compact you know I have already
told you there are going to be only finitely
00:28:39.830 --> 00:28:45.000
many poles okay and it is possible that infinity
may be a pole or not okay.
00:28:45.000 --> 00:28:52.559
So 1st let deal with infinity first okay and
so you look at the Laurent expansion of the
00:28:52.559 --> 00:28:57.970
function at infinity alright and you know
that the Laurent expansion of the function
00:28:57.970 --> 00:29:03.049
at infinity will consist of both the positive
and negative powers of Z okay and of course
00:29:03.049 --> 00:29:08.419
the constant term and you know the singular
part at infinity is the part that consist
00:29:08.419 --> 00:29:19.039
of positive powers of Z okay and you know
since, so now you see infinity there are 3
00:29:19.039 --> 00:29:26.899
choices for infinity, see infinity can be
either the removable singularity okay or it
00:29:26.899 --> 00:29:35.330
could be a pole okay and of course it cannot
be an essential singularity because we assume
00:29:35.330 --> 00:29:40.740
the function is Meromorphic so it cannot have
any singularities other than poles okay.
00:29:40.740 --> 00:29:50.090
Now if infinity is a removable singularity
okay it means that if you take the Laurent
00:29:50.090 --> 00:29:55.580
expansion at infinity okay the principal part
which consist of positive powers of Z
00:29:55.580 --> 00:30:05.899
there is a singular part has no terms okay
and if you assume, so it consist of only the
00:30:05.899 --> 00:30:14.340
constant part and the negative powers of Z
okay that is if you assume infinity is a removable
00:30:14.340 --> 00:30:19.590
singularity. If you assume infinity is a pole
which is the only other possibility and you
00:30:19.590 --> 00:30:24.019
know that the principal part of the singular
part consisting of positive powers of Z is
00:30:24.019 --> 00:30:30.879
to be finite, so it has to be polynomial of
positive degree okay without a constant term
00:30:30.879 --> 00:30:36.309
and of course the degree will be the order
of the pole at infinity okay.
00:30:36.309 --> 00:30:45.031
So in any case you are going to get the similar
part as either 0 or a polynomial that is what
00:30:45.031 --> 00:30:52.159
I am trying to say at infinity and if you
take the function and remove that singular
00:30:52.159 --> 00:30:57.990
part whatever is left is going to be analytic
at infinity that is what you must understand.
00:30:57.990 --> 00:31:04.070
So this is something that we use repeatedly,
what is the point about the Laurent expansion
00:31:04.070 --> 00:31:10.740
at a point of a function. The Laurent expansion
consist of a singular part or principal part
00:31:10.740 --> 00:31:20.191
and an analytic part and if you take the function
and subtract the singular part what you will
00:31:20.191 --> 00:31:24.080
get is only a Taylor series which will be
actually the Taylor series of the analytic
00:31:24.080 --> 00:31:27.610
function which is given by the difference
of the function and its singular point.
00:31:27.610 --> 00:31:32.639
The moment you take away the singular part,
the principal part the function becomes analytic
00:31:32.639 --> 00:31:41.529
okay, so this is a trek that you always use
if you want to extract the analytic part what
00:31:41.529 --> 00:31:46.090
do you do? You take the function and subtract
the singular part or the principal part okay,
00:31:46.090 --> 00:31:59.330
so let me write that down so let us 1st deal
with the point at infinity the point Z equal
00:31:59.330 --> 00:32:20.009
to infinity if the Laurent expansion
of f of Z at infinity is f of Z is equal to
00:32:20.009 --> 00:32:35.730
Sigma n equal is to minus infinity to infinity
a n Z power n that is Sigma so let me write
00:32:35.730 --> 00:32:49.389
it like this Sigma n equal to 0 to minus infinity
a n Z power n plus Sigma n equal to 1 to infinity
00:32:49.389 --> 00:33:00.379
a n Z power n. If Z is this and now let me
call this fellow as p infinity of Z okay,
00:33:00.379 --> 00:33:05.799
this p infinity of Z is what this is the singular
or principal part at infinity okay.
00:33:05.799 --> 00:33:22.559
So this is the singular or sensible part at
Z equal to infinity okay and of course whatever
00:33:22.559 --> 00:33:37.279
is left out here this is the analytic part
okay, so you see of course you should take
00:33:37.279 --> 00:33:43.529
this Laurent expansion to be valid for mod
Z greater than R for R sufficiently large,
00:33:43.529 --> 00:33:57.909
so valid for mod Z greater than R, R sufficiently
large, so this is very important okay and
00:33:57.909 --> 00:34:07.159
so if you take, so the point is that so you
should take f of Z minus P infinity of Z okay.
00:34:07.159 --> 00:34:16.590
This is going to be this is analytic at infinity
okay this is going to be analytic at infinity
00:34:16.590 --> 00:34:22.440
okay because f of Z minus p infinity of Z
will only consist of a constant term a naught
00:34:22.440 --> 00:34:27.540
and negative powers of terms involving negative
powers of Z and negative powers of Z behave
00:34:27.540 --> 00:34:32.480
well at infinity okay and of course should
tell you something if you look at the
00:34:32.480 --> 00:34:40.040
point at infinity since we have assumed that
f is Meromorphic this p infinity of Z is not
00:34:40.040 --> 00:34:46.830
actually a power series is only a polynomial
okay p infinity of Z will be 0 if infinity
00:34:46.830 --> 00:34:53.110
is a removable singularity at is a point of
analyticity or f and it will be a polynomial
00:34:53.110 --> 00:34:58.670
of positive degree equal to the order of the
pole of f at infinity if infinity is a pole
00:34:58.670 --> 00:34:59.670
okay.
00:34:59.670 --> 00:35:17.500
So so let me write that down, note that p
infinity of Z is 0 if infinity is a removable
00:35:17.500 --> 00:35:47.880
singularity of f and is a polynomial of positive
degree equal to the order of f the order of
00:35:47.880 --> 00:35:59.900
the pole of f at infinity. It is no other
possibility you do not have the situation
00:35:59.900 --> 00:36:05.280
when infinity is an essential singular point
because we have assumed f is Meromorphic the
00:36:05.280 --> 00:36:09.870
only singularity that are allowed are poles
okay, fine.
00:36:09.870 --> 00:36:17.741
So now what you do is now let us look at the
other singular point, see I have assumed f
00:36:17.741 --> 00:36:23.160
is Meromorphic so it has only finitely many
poles because it is Meromorphic on the extended
00:36:23.160 --> 00:36:27.400
plane which is compact that is very very important
okay there are only finitely many poles in
00:36:27.400 --> 00:36:31.970
the usual complex plane and forget the point
at infinity because I have already dealt with
00:36:31.970 --> 00:36:37.020
it. I now want to keep track of the points
on the plane where f has poles there are only
00:36:37.020 --> 00:36:44.340
finitely many let me call those points Z 1
through let us say Z n okay, so let us write
00:36:44.340 --> 00:36:45.340
that down.
00:36:45.340 --> 00:37:00.570
Let Z 1, Z 2 and so on Z n be the poles of
f of Z in the complex plane this okay and
00:37:00.570 --> 00:37:04.960
again let me stress this is very important
that you getting finitely only finitely many
00:37:04.960 --> 00:37:12.080
poles because you have assume that f is Meromorphic
on the extended plane. The compactness of
00:37:12.080 --> 00:37:16.960
the extended plane is doing a big job here
otherwise you need not get finitely many poles
00:37:16.960 --> 00:37:29.650
okay. So you take these holes now whatever
you did at infinity you do at each of those
00:37:29.650 --> 00:37:37.690
poles okay, so take any of those poles then
in a deleted neighbourhood of those poles
00:37:37.690 --> 00:37:43.090
each of those poles f that means a Laurent
expansion and you know if you take the Laurent
00:37:43.090 --> 00:37:49.130
expansion you take the singular part it is
going to have only finitely many terms okay
00:37:49.130 --> 00:37:56.140
and of course the highest negative power occurring
will be equal to the order of the pole okay.
00:37:56.140 --> 00:38:19.080
So let a Laurent expansion
of f of Z around Z k be f of Z is equal to
00:38:19.080 --> 00:38:25.860
Sigma m, m equal to minus infinity to
infinity and I will call the coefficients
00:38:25.860 --> 00:38:36.430
as a m k and mind you it will be Z minus Z
k to the power of m okay this will be the
00:38:36.430 --> 00:38:43.500
Laurent expansion, this is the Laurent expansion
centred at Z k okay and valid in a deleted
00:38:43.500 --> 00:38:49.160
neighbourhood of Z k and of course you know
that it will be a actually disk centred at
00:38:49.160 --> 00:39:00.220
Z k and the radius will be equal to the distance
from Z k to the nearest of the other Z j,
00:39:00.220 --> 00:39:19.760
so valid in 0 less than mod Z minus k
there of course r k I am not writing it r
00:39:19.760 --> 00:39:27.050
k is a distance of Z k to the nearest of the
other Z j, J naught equal to k okay and what
00:39:27.050 --> 00:39:33.480
is the principal part at Z k it is going to
have only finitely many terms okay and the
00:39:33.480 --> 00:39:39.140
highest negative power of Z minus Z k you
are going to get that is going to be the pole
00:39:39.140 --> 00:39:40.340
of f at Z k.
00:39:40.340 --> 00:40:02.310
So the principal part
of f at Z k is therefore p Z k of f which
00:40:02.310 --> 00:40:13.370
is going to be let me write this as p Z k
of Z that is going to be Sigma it is going
00:40:13.370 --> 00:40:34.140
to be m equal to minus 1 to minus let
me put t k, a m k into Z minus Z k to the
00:40:34.140 --> 00:40:51.300
power of M okay and mind you where t k is
equal to order of the pole Z k okay of f mind
00:40:51.300 --> 00:40:55.811
you these are all the powers of Z minus Z
k r in the denominator starting with m equals
00:40:55.811 --> 00:41:04.870
to minus 1 and going all the way up to minus
t k and of course a minus t k, k is not 0,
00:41:04.870 --> 00:41:17.000
a minus sub t k, k is not 0 I mean this is
the coefficient of the highest negative power
00:41:17.000 --> 00:41:28.440
of Z minus Z k, right. Well so this is the
principal part at Z okay and now what you
00:41:28.440 --> 00:41:34.360
do is that you do the same trick as before
you use the fact that if you take the function
00:41:34.360 --> 00:41:39.790
and subtract away, take away the principal
part whatever you going to be left with is
00:41:39.790 --> 00:41:43.030
going to be analytic because it is going to
be a power series.
00:41:43.030 --> 00:41:47.200
So it is going to represent an analytic action
whose Taylor expansion at Z k is exactly the
00:41:47.200 --> 00:41:52.190
power series which is the analytic part of
the Laurent expansion. See you must always
00:41:52.190 --> 00:41:55.860
remember this analytic part of the Laurent
expansion is actually the Taylor series of
00:41:55.860 --> 00:42:01.140
what analytic function? It is the Taylor series
of the analytic function which is given by
00:42:01.140 --> 00:42:09.350
a result function minus the principal part
okay. So so let me write that down we need
00:42:09.350 --> 00:42:26.170
to use it f of Z minus p Z k of Z is analytic
at Z k okay so you see so now look at the
00:42:26.170 --> 00:42:33.120
scenario, the scenario is have there is Meromorphic
function f there are these poles, finitely
00:42:33.120 --> 00:42:39.170
many poles that one through Z n okay and at
each of these poles there is a principal part
00:42:39.170 --> 00:42:44.390
okay and if you take the principal part away,
what you get is something that is analytic
00:42:44.390 --> 00:42:49.130
at that point and of course there is also
in a principal part at infinity okay, now
00:42:49.130 --> 00:42:54.660
what I do is I take the function and remove
all principal part is okay.
00:42:54.660 --> 00:42:59.330
You take the function subtract the principal
part at Z 1, subtract the principal part at
00:42:59.330 --> 00:43:04.560
Z 2 and so on and you subtract the principal
part at infinity you subtract all the principal
00:43:04.560 --> 00:43:10.200
part and what are you going to get? You are
going to get an entire function on the whole
00:43:10.200 --> 00:43:16.320
Riemann sphere and what is going to be? It
is going to be a constant because of Liouville’s
00:43:16.320 --> 00:43:21.850
theorem. An entire function which is… If
you are going to get an analytic function
00:43:21.850 --> 00:43:25.840
on the extended plane which means you are
getting an entire function which is analytic
00:43:25.840 --> 00:43:31.930
at infinity, an entire function which is analytic
at infinity by Liouville’s has to be a constant
00:43:31.930 --> 00:43:36.220
it is a bounded entire function, analytic
at infinity means bounded at infinity, bounded
00:43:36.220 --> 00:43:39.950
at infinity means is a bounded entire function
and it is constant.
00:43:39.950 --> 00:43:45.440
So you moral of the story is you take this
Meromorphic function and subtract all the
00:43:45.440 --> 00:43:49.480
principle arts you are going to get a constant
and now you push all the principal part to
00:43:49.480 --> 00:43:55.160
the other side you will get that the Meromorphic
functions is a constant plus all these principal
00:43:55.160 --> 00:44:00.450
parts but each of these principal parts are
rational functions and a constant is also
00:44:00.450 --> 00:44:04.280
a rational function, so you have express the
Meromorphic function as sum of finitely many
00:44:04.280 --> 00:44:08.410
rational functions therefore it is rational
and that proofs the theorem okay, so that
00:44:08.410 --> 00:44:12.160
is all, so let me write that down.
00:44:12.160 --> 00:44:30.380
Now consider g of Z is equal to f of Z minus
Sigma k equal to 1 to n, p Z k of Z so these
00:44:30.380 --> 00:44:38.000
are the principal parts at those end poles
and then also take away p infinity of Z okay.
00:44:38.000 --> 00:44:50.120
Now what you get is g of Z is analytic on
the whole extended plane and by Liouville
00:44:50.120 --> 00:45:15.620
therefore it is a constant. g is analytic
on C union infinity hence a constant by Liouville,
00:45:15.620 --> 00:45:36.080
so f of Z is equal to constant plus Sigma
k equal to 1 to n, p Z k of said plus p infinity
00:45:36.080 --> 00:45:51.310
of Z which is Meromorphic
on the extended plane and that is the proof
00:45:51.310 --> 00:45:59.230
of that theorem that a function which is Meromorphic
and the extended plane is just in fact what
00:45:59.230 --> 00:46:11.780
I want to say is not… of course it is Meromorphic
and the point is its rational okay that is
00:46:11.780 --> 00:46:19.760
the point, we want to show that Meromorphic
function on the extended plane is a rational
00:46:19.760 --> 00:46:21.260
function okay.
00:46:21.260 --> 00:46:29.780
So what we have got as you prove that Meromorphic
function f is constant plus these principal
00:46:29.780 --> 00:46:36.900
parts, finitely many principal parts, so this
part p infinity of Z is a polynomial okay
00:46:36.900 --> 00:46:45.010
it could be 0 and these are all involving
negative powers of Z minus k finitely many
00:46:45.010 --> 00:46:51.870
for each k okay and this is of course a rational
function. If you take LCM you will see that
00:46:51.870 --> 00:46:56.110
you will get a quotient of polynomials therefore
it is a rational function and the beauty of
00:46:56.110 --> 00:47:01.290
this proof is that this proof also tells you
that you get for every Meromorphic functions
00:47:01.290 --> 00:47:06.760
you get the partial fraction decomposition,
the each p Z k they are all the various terms
00:47:06.760 --> 00:47:10.990
of the partial fraction decomposition. So
this proof in one stroke tells you that the
00:47:10.990 --> 00:47:15.850
Meromorphic function as partial fraction decomposition
and is actually a rational function okay that
00:47:15.850 --> 00:47:19.430
is the advantage of this proof, okay. So I
will stop with that.