WEBVTT
Kind: captions
Language: en
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You see we have been looking at a point at
infinity and what we have seen is situation
00:01:34.000 --> 00:01:40.620
when infinity is an isolated singularity okay
and I have already told you what happens when
00:01:40.620 --> 00:01:49.020
infinity is a removable singularity or infinity
is a pole and even more generally we have
00:01:49.020 --> 00:01:56.910
also seen the so-called residue theorem for
the extended complex plane okay, so the point
00:01:56.910 --> 00:02:02.960
that one has to remember is that the residue
theorem for the extended complex plane which
00:02:02.960 --> 00:02:10.390
we discussed in the last lecture in that the
contribution of the residue at infinity will
00:02:10.390 --> 00:02:20.940
always be there okay irrespective of whether
the point at infinity is a removable singularity
00:02:20.940 --> 00:02:22.640
or a pole or an essential singularity.
00:02:22.640 --> 00:02:30.750
So the issue is that you know even if infinity
is a removable singularity okay for example
00:02:30.750 --> 00:02:41.240
for the function f of w equal to 1 by w okay
infinity is w equal to infinity is a removable
00:02:41.240 --> 00:02:48.830
singularity okay because the function tends
to 0 as w tends to infinity but still the
00:02:48.830 --> 00:02:55.040
residue at infinity is not 0 which does not
happen for a point in the usual plane, in
00:02:55.040 --> 00:03:01.210
the complex plane at a point at which the
function has a removable singularity is a
00:03:01.210 --> 00:03:04.530
point where the function can be redefined
so that it becomes analytic and therefore
00:03:04.530 --> 00:03:09.820
if you calculate the residue at that point
you will get 0 okay whereas this does not
00:03:09.820 --> 00:03:15.470
happen for the point at infinity that is 1
point that you should always keep in mind.
00:03:15.470 --> 00:03:20.380
Now we are going to talk about the situation
when infinity is an essential singularity
00:03:20.380 --> 00:03:26.440
okay, so so what does it mean to say that
infinity is an essential singularity, so basically
00:03:26.440 --> 00:03:32.270
you are looking at a function for which the
point at infinity is a singular point and
00:03:32.270 --> 00:03:36.630
isolated singular point that means it is defined
in a neighbourhood of infinity in a deleted
00:03:36.630 --> 00:03:40.030
neighbourhood of infinity and it is analytic
in a deleted neighbourhood of infinity which
00:03:40.030 --> 00:03:44.670
means that basically you are looking at an
analytic function which is analytic outside
00:03:44.670 --> 00:03:50.130
a circle of sufficiently large radius okay
and because that is what neighbourhood of
00:03:50.130 --> 00:03:54.950
infinity is okay you know that the stereographic
projection.
00:03:54.950 --> 00:04:00.560
Now what does it mean to say that the function
has infinity as an essential singularity,
00:04:00.560 --> 00:04:06.990
it is the same as sayingůthere are 2 ways
of saying it of course in fact 3 ways of saying
00:04:06.990 --> 00:04:15.910
it and all the 3 ways of saying it are correct,
so there is one way which is using the Laurent
00:04:15.910 --> 00:04:22.170
expansion at infinity, so so one way of saying
that infinity is an essential singularity
00:04:22.170 --> 00:04:32.640
is by saying that if you take Laurent expansion
at infinity then the singular part has
00:04:32.640 --> 00:04:37.410
infinitely many terms okay and there what
you must remember is that the singular part
00:04:37.410 --> 00:04:45.150
is actually a polynomial part I mean it is
theůif you expand it in powers of the variable
00:04:45.150 --> 00:04:51.730
okay if w is the variable then you are going
to expand the function in positive and negative
00:04:51.730 --> 00:04:58.210
powers of w okay and you should look at an
expansion which is valid outside a circle
00:04:58.210 --> 00:05:02.850
of a sufficiently large radius mind you that
is the Laurent expansion in a neighbourhood
00:05:02.850 --> 00:05:11.910
of infinity okay and because you are looking
at a point at infinity the positive powers,
00:05:11.910 --> 00:05:17.370
the terms involving positive powers of w that
is a singular part okay that is a singular
00:05:17.370 --> 00:05:23.450
part and the term involving the constant and
the negative powers of w is the analytic part
00:05:23.450 --> 00:05:27.250
because negative powers of w behave well at
infinity okay.
00:05:27.250 --> 00:05:35.570
So the condition for the function to have
infinity as an essential singular point is
00:05:35.570 --> 00:05:43.090
that if you write out its Laurent expansion
in a which is valid outside at all points
00:05:43.090 --> 00:05:50.240
outside a sufficiently circle of sufficiently
large radius centred at the origin okay then
00:05:50.240 --> 00:05:57.260
you must get you must get the positive powers
of the variable that occur, they must be there
00:05:57.260 --> 00:06:02.740
must be infinitely many terms which is the
same as saying that the singular part at infinity
00:06:02.740 --> 00:06:08.930
has infinitely many terms, so what should
not happen is that the singular part has only
00:06:08.930 --> 00:06:12.680
finitely many terms at infinity which means
the singular part is actually a polynomial
00:06:12.680 --> 00:06:18.800
okay. So in other words what we are saying
is that you know how do you recognize that
00:06:18.800 --> 00:06:24.930
action has infinity as an essential singular
point, you write out the Laurent expansion
00:06:24.930 --> 00:06:31.780
at infinity valid in a neighbourhood of infinity
then what happens is that the singular part
00:06:31.780 --> 00:06:33.610
is not just a polynomial okay.
00:06:33.610 --> 00:06:38.810
In particular what this means is that if you
take a polynomial functions they are certainly
00:06:38.810 --> 00:06:43.340
not going to have infinity as an essential
singularity and you know that you have seen
00:06:43.340 --> 00:06:51.800
it last time a polynomial of degree polynomial
is in fact have infinity as a pole and the
00:06:51.800 --> 00:06:56.790
order of the pole is actually the degree of
the polynomial okay, so this is one way of
00:06:56.790 --> 00:07:03.010
defining infinity as an essential similar
point of course the other way is that the
00:07:03.010 --> 00:07:12.870
limit of the function as you approach infinity
does not exist okay that is also correct okay.
00:07:12.870 --> 00:07:17.620
So usually there are 3 characterisations one
of any singular point there are 3 characterisations
00:07:17.620 --> 00:07:25.240
one is you look at the Laurent series and
look at the condition of the Laurent series
00:07:25.240 --> 00:07:32.169
whether it has no terms at all in the singular
part or whether it has finitely many terms
00:07:32.169 --> 00:07:35.840
in the singular part or whether it has infinitely
many terms in the singular part and the 2nd
00:07:35.840 --> 00:07:41.270
condition is on the limit of the function
as you approach that point okay. Of course
00:07:41.270 --> 00:07:45.250
the 3rd condition is about the bounded mass
of the function okay or the unboundedness
00:07:45.250 --> 00:07:50.410
of the function but of course the boundedness
is equivalent to the point being a removable
00:07:50.410 --> 00:07:54.790
singularity okay and the unboundedness means
that it is either it could either be a pole
00:07:54.790 --> 00:07:56.530
or an essential singularity okay.
00:07:56.530 --> 00:08:14.780
So so let me write these these points down
infinity as an essential singularity of f
00:08:14.780 --> 00:08:26.520
of W, so you see so f of w is assumed to be
analytic in a deleted neighbourhood of infinity
00:08:26.520 --> 00:08:37.390
okay that is a standard assumption blanket
assumption and well so, so what are the conditions
00:08:37.390 --> 00:09:01.130
well you write a Laurent expansion expansion
of f at infinity that is f of w equal to Sigma
00:09:01.130 --> 00:09:18.500
and equal is to minus infinity to infinity
a n w power n valid in a deleted neighbourhood
00:09:18.500 --> 00:09:40.000
of infinity has in finitely many terms
in its singular part and what is the singular
00:09:40.000 --> 00:09:41.000
part?
00:09:41.000 --> 00:09:48.750
The singular part is Sigma, the singular part
is the positive powers of so it is Sigma n
00:09:48.750 --> 00:09:59.230
equal to 1 to infinity a n w power n and that
is the same as saying that a n not equal to
00:09:59.230 --> 00:10:16.740
0 for many in finitely many n for infinitely
many n, so and you see this definition is
00:10:16.740 --> 00:10:24.720
just the same as usual definition that in
order to say that a certain point for example
00:10:24.720 --> 00:10:28.780
in the complex plane and isolated singularity
is an essential singularity all you have to
00:10:28.780 --> 00:10:32.839
do is write out the Laurent expansion at that
point then you find that the Laurent expansion
00:10:32.839 --> 00:10:37.790
has in finite singular point okay and there
it will be in finitely many negative powers
00:10:37.790 --> 00:10:39.600
okay.
00:10:39.600 --> 00:10:52.280
So so of course I should tell you that this
is equivalent to g of Z is equal to f of 1
00:10:52.280 --> 00:11:09.090
by Z has an essential singularity at 0 okay,
so g of Z f of 1 by Z has an essential singularity
00:11:09.090 --> 00:11:22.930
singularity at 0 at Z equal to 0 okay and
this is of course you know this is of course
00:11:22.930 --> 00:11:28.590
based on this philosophy that the behaviour
of f of w at infinity is the same as the behaviour
00:11:28.590 --> 00:11:37.360
of f of 1 by Z at 0 okay you put w equal to
1 by Z we already know that this is a homeomorphism
00:11:37.360 --> 00:11:44.100
okay and it is a holomorphic isomorphism of
the punctured plane, the plane minus the origin
00:11:44.100 --> 00:11:51.780
onto itself okay and so this is one thing.
Then the other thing is so this is one
00:11:51.780 --> 00:12:02.930
condition the other condition is limit w tends
to infinity f of w does not exist this is
00:12:02.930 --> 00:12:08.810
this is the other condition and of course
this is the same as saying that limit Z tends
00:12:08.810 --> 00:12:13.700
to infinity g of Z does not exist.
00:12:13.700 --> 00:12:30.260
So this is also something that you know and
so well of course the usually we will
00:12:30.260 --> 00:12:33.990
have another conditions which will be on the
behaviour of the function in the neighbourhood
00:12:33.990 --> 00:12:46.970
of that point and well certainly you cannot
expect the function to be bounded in a neighbourhood
00:12:46.970 --> 00:12:55.740
of that point okay because that is equivalent
to the function being actually analytic at
00:12:55.740 --> 00:13:04.089
that point and that is Riemannĺs removable
singularity is theorem right. Anyway so so
00:13:04.089 --> 00:13:13.310
let me go ahead and say some other things,
so maybe I will put some I will change color
00:13:13.310 --> 00:13:30.480
and put a few boxes here, so so this is equivalent
to this and this part is the same as this
00:13:30.480 --> 00:13:31.890
okay.
00:13:31.890 --> 00:13:42.770
Alright so now what I want to say next is
that you know let us analyse this condition
00:13:42.770 --> 00:13:49.210
that infinity is an essential singularity
okay, now theůso for example what are the
00:13:49.210 --> 00:13:54.760
functions? What are the entire functions which
have infinity as an essential singularity,
00:13:54.760 --> 00:14:00.160
you can ask this question? Okay so we have
already answered a similar question for poles
00:14:00.160 --> 00:14:06.430
okay and for removable singularities. See
you take an entire function mind you an entire
00:14:06.430 --> 00:14:11.260
function means a function which is analytic
on the whole plane okay and the moment it
00:14:11.260 --> 00:14:14.830
is analytic on the whole plane the whole plane
is also mind you a deleted neighbourhood of
00:14:14.830 --> 00:14:18.630
the point at infinity you must not forget
that there for a function is analytic on the
00:14:18.630 --> 00:14:23.310
whole plane is also having infinity as an
isolated singular point automatically okay.
00:14:23.310 --> 00:14:28.670
If you think of the Riemannĺs stereographic
projection you see that the infinity corresponds
00:14:28.670 --> 00:14:36.300
to the North pole and the whole plane corresponds
to the remaining part of the Riemann sphere
00:14:36.300 --> 00:14:40.530
which is the Riemann sphere minus the North
pole okay by this geographic projection, so
00:14:40.530 --> 00:14:47.049
the plane itself is a neighbourhood of the
point at infinity and therefore an entire
00:14:47.049 --> 00:14:52.270
function will always have the point at infinity
as an isolated singularity. Now what happens
00:14:52.270 --> 00:14:58.910
if that isolated singularity is a removable
singularity? Well then you are saying that
00:14:58.910 --> 00:15:04.270
the entire function as infinity as a removable
singularity okay which is the same as saying
00:15:04.270 --> 00:15:10.309
that at infinity it is bounded or it has a
limit okay and then by Liouvilleĺs theorem
00:15:10.309 --> 00:15:12.140
it will reduce to a constant okay.
00:15:12.140 --> 00:15:17.120
So the moral of the story is that a nonconstant
entire function cannot have infinity as a
00:15:17.120 --> 00:15:20.420
removable singularity we have already seen
this okay and then you can ask the question
00:15:20.420 --> 00:15:28.209
is when will an entire function have infinity
as a pole okay and have seen that that will
00:15:28.209 --> 00:15:38.411
happen if and only if the entire function
is well is a polynomial okay and the degree
00:15:38.411 --> 00:15:43.050
of the polynomial will be the order of the
pole okay. So now we ask the question, when
00:15:43.050 --> 00:15:47.610
will an entire function have infinity as an
essential singularity? And the answer to that
00:15:47.610 --> 00:15:55.579
will be that it should not be a polynomial
okay basically if you write out the if you
00:15:55.579 --> 00:16:02.730
write out its Taylor expansion at any point,
the Taylor expansion of course you will get
00:16:02.730 --> 00:16:05.850
you have to ride only a Taylor expansion because
it is an entire function okay.
00:16:05.850 --> 00:16:12.170
There is no question of Laurent expansion
okay, so at any point you choose any point
00:16:12.170 --> 00:16:16.710
in the plane it is analytic everywhere, the
plane so you choose any point and write the
00:16:16.710 --> 00:16:20.330
Taylor expansion at that point. That Taylor
series will have in finite radius of convergence
00:16:20.330 --> 00:16:27.930
because this function is entire okay and the
point is that the Taylor series should be
00:16:27.930 --> 00:16:36.710
a power series, it should not be a polynomial
okay, so the moral of the story is an entire
00:16:36.710 --> 00:16:43.000
function as infinity as an essential singularity
if and only if its Taylor series is not a
00:16:43.000 --> 00:16:46.730
polynomial okay.
00:16:46.730 --> 00:16:52.929
So and the Taylor series of an entire function
being a polynomial is the same as you entire
00:16:52.929 --> 00:16:58.610
function itself being polynomial okay therefore
all you are saying is that you know an entire
00:16:58.610 --> 00:17:08.680
function if you wanted to have an essential
singularity at infinity okay then it should
00:17:08.680 --> 00:17:16.049
not be a polynomial alright and for this reason
we call such entire function as transcendental
00:17:16.049 --> 00:17:22.309
okay so usually a polynomial function and
the Meromorphic functions which are given
00:17:22.309 --> 00:17:27.380
by quotients of polynomials they are called
algebraic and everything that is not algebraic
00:17:27.380 --> 00:17:32.760
is called transcendental, so such entire functions
which have infinity as an essential singular
00:17:32.760 --> 00:17:35.250
point are called transcendental entire function
okay.
00:17:35.250 --> 00:17:45.049
So so let me write that down an entire
function
00:17:45.049 --> 00:18:11.799
that has infinity as an essential singularity
is one that is not a polynomial okay, so and
00:18:11.799 --> 00:18:17.860
you know what will happen if it is a polynomial,
if it is a polynomial infinity is a pole okay,
00:18:17.860 --> 00:18:22.280
if it is a polynomial of positive degree of
course if youůwe also consider conscience
00:18:22.280 --> 00:18:26.940
as polynomials, polynomial is of degree 0
and that is the case of a constant function,
00:18:26.940 --> 00:18:32.640
so if you are looking at a non-constant entire
function okay then it isůthe only way infinity
00:18:32.640 --> 00:18:37.330
is an essential singularity is that it is
not a polynomial of positive degree okay.
00:18:37.330 --> 00:19:01.080
So or a polynomial, if constant
has infinity has a removable singularity
00:19:01.080 --> 00:19:21.810
and if nonconstant has infinity as a pole
of order equal to its degree okay, so a polynomial
00:19:21.810 --> 00:19:27.370
is certainly not an entire function that has
an essential singularity at infinity and conversely
00:19:27.370 --> 00:19:32.100
if an entire function as an essential singularity
at infinity because you take any point and
00:19:32.100 --> 00:19:40.700
you write out the retailers expansion at that
point you will see that if infinity is
00:19:40.700 --> 00:19:50.080
a is a pole then the Taylor expansion should
terminate and it has to be a polynomial, so
00:19:50.080 --> 00:19:55.610
if infinity is not a pole then your Taylor
series will have infinitely many terms okay
00:19:55.610 --> 00:20:01.490
which is the same as saying that there are
in finitely many terms in the Laurent expansion
00:20:01.490 --> 00:20:11.760
at infinity okay, mind you if you take a polynomial
the polymer normal itself is the Taylor expansion
00:20:11.760 --> 00:20:20.520
of the function that it represents at the
origin okay and sense that expansion is valid
00:20:20.520 --> 00:20:25.610
on the whole plane it is also a Lauren expansion
at infinity.
00:20:25.610 --> 00:20:31.250
The expression for the polynomial itself is
a Lauren expansion at infinity and it is except
00:20:31.250 --> 00:20:37.430
for the constant part the positive part terms
involving positive part of the variable, that
00:20:37.430 --> 00:20:42.120
is automatically the singular part at infinity
and that it has only finitely many terms tells
00:20:42.120 --> 00:20:54.930
you that infinity is a pole okay. So let me
write that down, conversely if f of w has
00:20:54.930 --> 00:21:12.850
infinity as an essential singularity then
its Taylor expansion
00:21:12.850 --> 00:21:38.770
at any point of C has to have, at any point
of C has to have
00:21:38.770 --> 00:22:01.940
in finitely many terms, so does not a polynomial
okay, so say things in short a transcendental
00:22:01.940 --> 00:22:06.000
entire function is something that is different
from a polynomial.
00:22:06.000 --> 00:22:15.800
Now I want to make the following statement
that take any entire function okay exponentiated
00:22:15.800 --> 00:22:24.330
a, so you take f of w to be an entire function
okay, f of w may have infinity as you know
00:22:24.330 --> 00:22:31.940
either a pole or may be an essential but if
you take E power f of w okay if you take E
00:22:31.940 --> 00:22:39.930
power f of W, I claimed that it will always
be transcendental okay, so that means I am
00:22:39.930 --> 00:22:46.780
saying that or the power f of W, w equal to
infinity will always be an essential singularity
00:22:46.780 --> 00:22:53.720
okay that is also very easy to see and how
do you see it let me tell you the argument
00:22:53.720 --> 00:23:02.630
and words you see so I know that f of w is
entire and am looking at E power f of w okay.
00:23:02.630 --> 00:23:06.830
Of course E power f of w is also entire because
it is a composition of entire functions, exponential
00:23:06.830 --> 00:23:12.481
function is of course entire okay and E power
f of w is f of w allowed by the exponential
00:23:12.481 --> 00:23:18.650
function it is composition of entire function,
so it is entire okay and what other possibilities
00:23:18.650 --> 00:23:24.270
of E power f of w at infinity? Infinity can
either be removable singularity or it can
00:23:24.270 --> 00:23:30.440
be a pole or it can be an essential singularity.
If infinity is a removable singularity you
00:23:30.440 --> 00:23:35.150
know then the power f of w must be a constant
because of the Liouvilleĺs theorem and if
00:23:35.150 --> 00:23:40.800
E power f of w is a constant then f has to
be a constant because f is if E power
00:23:40.800 --> 00:23:46.360
f of w is a constant that constant is a complex
number which are not be 0 because exponential
00:23:46.360 --> 00:23:51.410
function never takes the value 0 and f has
to be log of that okay.
00:23:51.410 --> 00:24:03.700
F can be one of the logarithms of that constant,
nonzero constant okay and so so in that case
00:24:03.700 --> 00:24:09.450
you power f so what am trying to say is that
if you are looking at an entire function a
00:24:09.450 --> 00:24:14.151
nonconstant entire function okay the power
have has to be transcendental okay job the
00:24:14.151 --> 00:24:21.190
only case you will have to worry about is
the constant function. When of course E power
00:24:21.190 --> 00:24:28.080
a constant is also a constant, so if you have
a non-constant entire function okay the power
00:24:28.080 --> 00:24:36.270
f also will be nonconstant right, so infinity
or the power f if infinity is a removable
00:24:36.270 --> 00:24:40.910
singularity then f is a constant, so if you
are looking at a nonconstant f then he power
00:24:40.910 --> 00:24:44.550
f will not have infinity as a removable singularity.
00:24:44.550 --> 00:24:51.630
So it can be a pole now if E power f is a
pole has a pole at infinity, infinity is a
00:24:51.630 --> 00:24:56.770
pole for E power f then E power f has to be
a polynomial because we have seen that the
00:24:56.770 --> 00:25:02.310
only entire functions which have pole at infinity
are the polynomials so E power f has to be
00:25:02.310 --> 00:25:08.800
a polynomial but that is not possible because
you see if you take a polynomial of constant
00:25:08.800 --> 00:25:14.730
a nonconstant polynomial of positive degree
the fundamental theorem of algebra says that
00:25:14.730 --> 00:25:19.930
it will have zeros, so there areůit will
assume the value 0 but that is equal to E
00:25:19.930 --> 00:25:27.850
power f and E power anything can never be
0 okay therefore E power f cannot be a polynomial
00:25:27.850 --> 00:25:32.970
alright at means that infinity cannot be a
pole for E power f so therefore the only thing
00:25:32.970 --> 00:25:37.810
that is left is E power f should have infinity
as an essential singularity, so the moral
00:25:37.810 --> 00:25:42.930
of the story is that you take any nonconstant
entire function and you take E power that,
00:25:42.930 --> 00:25:48.430
the resulting function will certainly have
infinity as an essential singularity in other
00:25:48.430 --> 00:25:53.330
words I am saying that the resulting function
it will always be transcendental okay.
00:25:53.330 --> 00:26:12.650
So so let me write that down okay so let
me use a different color an entire function
00:26:12.650 --> 00:26:42.080
that has infinity as an essential singularity
is called okay, an entire function that has
00:26:42.080 --> 00:26:48.740
infinity as an essential singularity is called
transcendental and so all entire functions
00:26:48.740 --> 00:27:01.710
which are not polynomials are transcendental
okay. Thus the entire functions, the transcendental
00:27:01.710 --> 00:27:31.990
entire functions are precisely the non-polynomials,
precisely those that are not polynomials okay
00:27:31.990 --> 00:28:00.690
if f of w is entire and nonconstant then
he power f of w is always transcendental.
00:28:00.690 --> 00:28:19.900
For if E power f w has w equal to infinity
as a removable singularity
00:28:19.900 --> 00:28:55.800
then E power f w reduces to a constant by
Liouville and contradicts f being nonconstant
00:28:55.800 --> 00:29:29.750
also E power f also infinity cannot be a pole
as then he power f would be a polynomial which
00:29:29.750 --> 00:29:36.520
must have zeros okay.
00:29:36.520 --> 00:29:43.210
So that also cannot happen so as a result
you take any entire function and you which
00:29:43.210 --> 00:29:50.120
is not constant and the exponentiated what
you get is a transcendental entire function
00:29:50.120 --> 00:29:57.200
okay. So in this context let me also say that
you know we entire function the functions
00:29:57.200 --> 00:30:06.870
which are either polynomials or quotient of
polynomials okay they are called Meromorphic
00:30:06.870 --> 00:30:12.610
functions they are all called algebraic okay
and the non-algebraic functions are the transcendental
00:30:12.610 --> 00:30:14.380
ones okay.
00:30:14.380 --> 00:30:34.740
So so let me write that in general, the
polynomials and more generally the Meromorphic
00:30:34.740 --> 00:31:00.450
functions in the extended plane
are called algebraic okay at least in
00:31:00.450 --> 00:31:34.180
the sense of algebraic geometry okay in the
sense of complex algebraic the non-algebraic
00:31:34.180 --> 00:31:53.980
ones are called transcendental, okay. Examples,
examples of transcendental functions, this
00:31:53.980 --> 00:32:07.260
is something that one should look ahead, so
so you take for example E power Z this is
00:32:07.260 --> 00:32:14.360
transcendental because Z equal to infinity
is an essential singular point and why is
00:32:14.360 --> 00:32:21.090
that so because basically if you write the
expansion for E power Z Taylor expansion or
00:32:21.090 --> 00:32:25.220
MacLaurin expansion which is a Taylor expansion
in the origin you write the usual expansion
00:32:25.220 --> 00:32:28.920
that we all know and you know it has infinitely
many terms okay.
00:32:28.920 --> 00:32:35.870
So E power Z and then similarly you can take
the trigonometric functions you can take sin
00:32:35.870 --> 00:32:44.870
Z you can take cos Z and so on okay, so these
are all transcendental functions and the way
00:32:44.870 --> 00:32:51.670
of course you know you can also see that infinity
is an essential singular point because if
00:32:51.670 --> 00:32:58.530
you change if you invert the variable you
will get origin as the essential singular
00:32:58.530 --> 00:33:03.460
points, so if you take E power 1 by Z origin
will be an essential singular point, if you
00:33:03.460 --> 00:33:09.059
take sin 1 by Z origin will be an essential
singular point, if you take cos 1 by Z origin
00:33:09.059 --> 00:33:13.490
will be an essential singular point okay,
so these are examples of transcendental functions
00:33:13.490 --> 00:33:21.950
and the other thing that I want to tell you
is that I want to also recall the big Picard
00:33:21.950 --> 00:33:30.881
theorem in this connection okay take now you
see you know take an entire function, take
00:33:30.881 --> 00:33:37.170
an entire function which is not constant okay
and of course if infinity is a infinity cannot
00:33:37.170 --> 00:33:44.631
be a removable okay because if infinity is
a removable singularity then it will reduce
00:33:44.631 --> 00:33:48.260
to a constant. Since I have taken a nonconstant
entire function, infinity is not a removable
00:33:48.260 --> 00:33:49.260
singularity.
00:33:49.260 --> 00:33:54.730
The next possibility is infinity is a pole
in which case the entire function is a polynomial
00:33:54.730 --> 00:34:00.300
and you know the image of a polynomial map
is the whole plane okay because a polynomial
00:34:00.300 --> 00:34:06.710
can assume we will assume all values okay
you take any value you can equate to the polynomial
00:34:06.710 --> 00:34:11.320
and you can solve for it and you will get
solutions that is because of the fundamental
00:34:11.320 --> 00:34:15.510
theorem of algebra, so if you take a polynomial
mapping it will be the image of the whole
00:34:15.510 --> 00:34:20.990
plane under a polynomial mapping will be the
whole plane then you look at the 3rd case
00:34:20.990 --> 00:34:25.639
namely when infinity is an essential singular
point okay. Now if infinity is an essential
00:34:25.639 --> 00:34:35.260
singular point okay you see what does the
big Picard theorem say?
00:34:35.260 --> 00:34:42.050
The big Picard theorem says that you take
any neighbourhood of an essential singular
00:34:42.050 --> 00:34:48.510
point no matter how small, the image will
be the whole plane and or it may be a punctured
00:34:48.510 --> 00:34:53.470
plane it might miss at one point okay. Now
you see this is the so what you are saying
00:34:53.470 --> 00:34:58.040
is that if I take a transcendental entire
function okay For example E power Z or sin
00:34:58.040 --> 00:35:05.420
Z or cos Z okay then the big Picard theorem
tell you that even if you take the image of
00:35:05.420 --> 00:35:12.160
not the whole plane but the exterior of a
circle no matter of how large radius you will
00:35:12.160 --> 00:35:18.070
still get the whole plane or the punctured
plane and you can compare it to the little
00:35:18.070 --> 00:35:25.300
Picard theorem which tells you that the image
of whole plane is either the whole plane or
00:35:25.300 --> 00:35:28.160
the plane minus a point, so I want you to
understand the significance.
00:35:28.160 --> 00:35:36.839
See if I take for example take E power Z okay
the little Picard theorem will tell you that
00:35:36.839 --> 00:35:41.539
the image of E power Z is either the whole
plane or it is the whole plane minus a point
00:35:41.539 --> 00:35:46.460
because you know it is E power Z we know it
is the plane minus the origin okay but the
00:35:46.460 --> 00:35:52.559
little Picard theorem never tells you what
is the image of anything other than the whole
00:35:52.559 --> 00:35:59.970
plane okay but if you apply the big Picard
theorem to E power Z okay you are applying
00:35:59.970 --> 00:36:05.760
mind you when I apply the big Picard theorem
to E power Z I am trying to apply I have to
00:36:05.760 --> 00:36:09.910
apply it only to an essential singularity
and where does E power Z have an essential
00:36:09.910 --> 00:36:11.460
singularity at infinity.
00:36:11.460 --> 00:36:18.460
So if I apply the big Picard theorem to E
power Z at infinity okay then you see that
00:36:18.460 --> 00:36:24.690
you take any neighbourhood of infinity okay
which is you take the outside the reason exterior
00:36:24.690 --> 00:36:31.140
to a circle of any radius, no matter how large,
the image of that itself will be the whole
00:36:31.140 --> 00:36:39.049
plane or a punctured plane okay and in fact
every value is taken in finitely many times,
00:36:39.049 --> 00:36:46.089
so you see you see in that sense how the big
Picard theorem is far far stronger than the
00:36:46.089 --> 00:36:51.509
little Picard theorem in the case of entire
functions you can see that of course for polynomials
00:36:51.509 --> 00:36:55.999
there is nothing special because your fundamental
theorem of algebra which will tell you things
00:36:55.999 --> 00:36:58.849
very clearly but if you are looking at non-polynomials.
00:36:58.849 --> 00:37:05.380
If you are looking at transcendental entire
functions you see you can really see the amount
00:37:05.380 --> 00:37:16.380
of difference in the conclusion and you can
see the strength of the theorem, you see the
00:37:16.380 --> 00:37:21.920
great Picard theorem is you much more than
the little Picard theorem okay, so if you
00:37:21.920 --> 00:37:27.150
take E power Z and you take the image of the
exterior of a circle matter how large radius
00:37:27.150 --> 00:37:35.470
it will still be the punctured plane okay
that is what the big Picard theorem applied
00:37:35.470 --> 00:37:39.550
to E power Z will tell you okay whereas if
you try to apply the little Picard theorem
00:37:39.550 --> 00:37:45.880
you will not get anything it will give you
only the image of the whole plane but it will
00:37:45.880 --> 00:37:51.890
not tell you anything about the image of the
exterior of a circle okay, so in that sense
00:37:51.890 --> 00:37:55.920
you see how powerful the big Picard theorem
is okay.
00:37:55.920 --> 00:38:03.119
Now what I want to do next is tell you about
tell you about Meromorphic functions okay,
00:38:03.119 --> 00:38:10.050
so you see so I want to concentrate on Meromorphic
functions and you know we need to study Meromorphic
00:38:10.050 --> 00:38:15.130
functions in order to get to our main aim
which is the proof of the Picard theorems
00:38:15.130 --> 00:38:26.420
alright, so the 1st fact about Meromorphic
functions is that you see if you take domain
00:38:26.420 --> 00:38:30.900
either the domain may be in the plane or it
may be a domain in the extended plane which
00:38:30.900 --> 00:38:36.230
means it could include the point at infinity
as an interior point. On a domain if you look
00:38:36.230 --> 00:38:42.349
at the set of Meromorphic functions on the
domain that is automatically a field okay,
00:38:42.349 --> 00:38:46.400
so you get an algebraic structure called a
field and it will be an extension field of
00:38:46.400 --> 00:38:57.180
the field of complex numbers okay. So and
this field extension its algebraic properties
00:38:57.180 --> 00:39:05.739
are deeply connected with a geometric properties
of the domain you are studying okay.
00:39:05.739 --> 00:39:11.220
So so let me make that statement so let me
go to the next thing the field of the Meromorphic
00:39:11.220 --> 00:39:30.869
functions, so let me recall what is a Meromorphic
function? So it is basically a function which
00:39:30.869 --> 00:39:38.109
is analytic and has only isolated singularities
which can be at most okay and of course they
00:39:38.109 --> 00:39:42.239
could be removable singularities but if they
are removable singularity is you really do
00:39:42.239 --> 00:39:45.920
not consider them because they are actually
points where the function is analytic okay
00:39:45.920 --> 00:39:55.769
otherwise the only singularities that it has
are poles okay, so so the moment am talking
00:39:55.769 --> 00:40:02.911
about a Meromorphic function I have to remember
that 1st of all what is allowed is ů what
00:40:02.911 --> 00:40:06.640
kind of singularities are allowed us are only
poles which means there are only isolated
00:40:06.640 --> 00:40:08.780
singularities there are no non-isolated singularities
okay.
00:40:08.780 --> 00:40:14.390
There are only isolated singularities and
they these isolated singularities are actually
00:40:14.390 --> 00:40:24.460
only poles okay and the point is that you
see if you look at a Meromorphic function
00:40:24.460 --> 00:40:33.769
in the extended plane okay then you see that
since an isolated set in the extended plane
00:40:33.769 --> 00:40:40.920
has to have only finitely many points okay
because the extended plane is compact okay
00:40:40.920 --> 00:40:46.489
therefore there will be only finitely many
poles okay, so the moral of the story is that
00:40:46.489 --> 00:40:50.559
if you are looking at Meromorphic functions
on the extended plane it will have only finitely
00:40:50.559 --> 00:40:54.700
many poles okay. I think probably I will stop
here.