WEBVTT
Kind: captions
Language: en
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Okay so let us continue with whatever we were
doing, select me very briefly recall what
00:00:50.779 --> 00:00:57.519
we have been saying, see you were been asking
this question as to what will decide of the
00:00:57.519 --> 00:01:03.819
size of the image of an analytic map be, so
basically you have analytic function, analytic
00:01:03.819 --> 00:01:09.799
is same as holomorphic function on the complex
plane. It is defined on open subset of the
00:01:09.799 --> 00:01:16.299
complex plane and you want to know what is
the image? We namely you want to know
00:01:16.299 --> 00:01:24.670
what is the set of values that the function
takes and you want to know what this in the
00:01:24.670 --> 00:01:32.439
topological sense and also how big the set
is and I told you that there is the to answer
00:01:32.439 --> 00:01:40.130
what the set is topologically, the set will
the image domain will again be a domain,
00:01:40.130 --> 00:01:48.149
the reason being that any non-constant analytic
function will be an open map okay so that
00:01:48.149 --> 00:01:51.289
is the so-called open mapping theorem right.
00:01:51.289 --> 00:01:57.020
So the image of domain which is by definition
an open connected set will again be an open
00:01:57.020 --> 00:02:00.869
connected set and of course you know the image
of a connected set will be connected because
00:02:00.869 --> 00:02:05.319
that is the property of continuous function
an analytic function is of course continuous,
00:02:05.319 --> 00:02:12.031
so at least you know that the images the
image of the domain is a domain and the image
00:02:12.031 --> 00:02:18.819
is certainly an open set, now the question
is the next question is that we asked was
00:02:18.819 --> 00:02:29.250
how big is the image okay so the answer to
that at least in the case of let us say analytic
00:02:29.250 --> 00:02:33.569
functions which are analytic on the whole
plane which are called as entire functions.
00:02:33.569 --> 00:02:38.579
So in that case we have the so-called little
Picard theorem or the small Picard theorem
00:02:38.579 --> 00:02:44.459
which says that functions which is analytic
on the whole plane okay its image will be
00:02:44.459 --> 00:02:49.590
either the whole plane or it will be a punctured
plane namely it will be… the whole plane
00:02:49.590 --> 00:02:55.489
minus 1 value so that is the only if at all
it misses a value if at all it misses value,
00:02:55.489 --> 00:03:00.049
so it will miss only one value okay or it
will miss no values okay and the case when
00:03:00.049 --> 00:03:07.420
it misses one value is in that case the image
is a complex plane minus 1 value and that
00:03:07.420 --> 00:03:12.200
is called the punctured plane okay and the
standard example is the exponential function
00:03:12.200 --> 00:03:17.231
Z going to E power Z which misses the value
0 and takes every other value which you can
00:03:17.231 --> 00:03:22.439
verify because any non-zero complex number
has a logarithm okay.
00:03:22.439 --> 00:03:28.469
So and of course you know if you take functions
like polynomials okay then you will see that
00:03:28.469 --> 00:03:35.889
the image of the whole plane again be the
whole plane and this is one example is possible
00:03:35.889 --> 00:03:42.640
to reduce this using the fundamental theorem
of algebra that any polynomial equation in
00:03:42.640 --> 00:03:49.129
one variable in one complex variable with
complex coefficients always has all its roots
00:03:49.129 --> 00:03:58.400
as complex numbers okay, so well so see the
idea is that you know in the 1st course in
00:03:58.400 --> 00:04:02.419
complex analysis the little Picard theorem
straight stated okay but since we are since
00:04:02.419 --> 00:04:07.159
this is advanced complex analysis we would
like to see a proof of the little Picard theorem
00:04:07.159 --> 00:04:12.939
and interestingly I was telling you last time
that the key to that the key to the proof
00:04:12.939 --> 00:04:18.959
of the Little Picard theorem that you are
going to see is actually having it deduce
00:04:18.959 --> 00:04:23.220
from the so-called big Picard theorem or the
great Picard theorem and that is interestingly
00:04:23.220 --> 00:04:25.750
a theorem which involves singularity okay.
00:04:25.750 --> 00:04:32.800
So we have the so-called great Picard theorem,
the great Picard theorem says that you know
00:04:32.800 --> 00:04:39.710
you take you take an analytic function when
you take a point which is an essential singularity
00:04:39.710 --> 00:04:47.639
of the function okay and then you take a small
disk about the essential singularity, small
00:04:47.639 --> 00:04:53.260
open disk about the essential singularity
such that in that disk the function is analytic
00:04:53.260 --> 00:04:58.639
of course leaving out the singular point and
then the image of that disk no matter how
00:04:58.639 --> 00:05:05.699
small is going to be again the whole complex
plane or the whole complex plane minus a single
00:05:05.699 --> 00:05:10.990
value namely the punctured plane and that
is what the great Picard theorem says and
00:05:10.990 --> 00:05:19.379
that is an amazing fact, so so what I want
to tell you is that…so this leads us to
00:05:19.379 --> 00:05:21.360
understand what singularities are?
00:05:21.360 --> 00:05:28.599
And eventually let me tell you that you know
when we try to prove the big Picard theorem
00:05:28.599 --> 00:05:39.770
we have 2 study families of analytic functions
with singularities and in particular singularities
00:05:39.770 --> 00:05:45.800
which are poles and such functions are called
Meromorphic functions so we have to study
00:05:45.800 --> 00:05:51.840
families of Meromorphic functions we have
to do study of topology of the space consisting
00:05:51.840 --> 00:05:57.150
of elements which are actually Meromorphic
functions that is the generality in which
00:05:57.150 --> 00:06:02.240
we will have to go to understand the great
Picard theorem okay but to begin with we need
00:06:02.240 --> 00:06:08.340
to worry about singularity so what I am going
to do now is I am going to tell you I mean
00:06:08.340 --> 00:06:11.949
I am going to recall events about singularities
which you have probably seen the 1st course
00:06:11.949 --> 00:06:19.120
in complex analysis okay but anyway it is
good to recall them.
00:06:19.120 --> 00:06:37.270
So let me, so let me write here, so singularities
of an analytic function, so singularities
00:06:37.270 --> 00:06:45.479
of an analytic function is what we are going
to worry about, so 1st of all let me recall
00:06:45.479 --> 00:06:50.449
so what is a singular point of an analytic
function? So you see by definition a singular
00:06:50.449 --> 00:06:55.900
point is defined only for a function which
is analytic okay if the function is not analytic
00:06:55.900 --> 00:07:01.650
then there is no question of talking about
singularities, so the idea behind defining
00:07:01.650 --> 00:07:07.539
the singular point is that the singular point
should be approachable by points where the
00:07:07.539 --> 00:07:12.090
function is analytic. A singular point should
always be a limit of good points, good points
00:07:12.090 --> 00:07:17.889
for the function means good point I mean for
points where the function is actually analytic
00:07:17.889 --> 00:07:18.889
okay.
00:07:18.889 --> 00:07:28.280
So you know in a way when I say singularity
of an analytic function it seems to be you
00:07:28.280 --> 00:07:37.520
know a misnomer or you know on the one hand
an analytic function is supposed to be analytic
00:07:37.520 --> 00:07:42.439
at all the points where it is defined and
then I say singularity of an analytic function
00:07:42.439 --> 00:07:48.360
it sometime looks odd but that is not the
point, the point is that you see a function
00:07:48.360 --> 00:07:55.229
which is analytic is usually analyticity is
defined on open set okay that is an open set
00:07:55.229 --> 00:08:00.150
of points at each of which the function is
analytic and of course you know for the definition
00:08:00.150 --> 00:08:04.069
of analyticity you need an open set you need
every point to be an interior point of the
00:08:04.069 --> 00:08:10.720
domain of analyticity okay. So but the question
is that by move to the boundary of this open
00:08:10.720 --> 00:08:16.009
set, if I go to a point in the boundary of
this open set then how is it that the function
00:08:16.009 --> 00:08:17.639
is going to be here?
00:08:17.639 --> 00:08:21.689
That point may be a point where the function
might continue to be analytic or it may fail
00:08:21.689 --> 00:08:29.319
to be analytic okay and it is only about these
boundary points, in the boundary points of
00:08:29.319 --> 00:08:34.190
the domain or the open set where the analytic
function is defined these are the points that
00:08:34.190 --> 00:08:43.409
we have to study for singularities okay. So
the very 1st thing is that you know a singular
00:08:43.409 --> 00:08:48.320
point is defined only for an analytic function
okay and by definition it is a point such
00:08:48.320 --> 00:08:57.490
that it is approachable or this limit of points
where the function is analytic okay, so let
00:08:57.490 --> 00:09:15.949
me write that down a singular point point
Z naught of an analytic function
00:09:15.949 --> 00:09:56.839
f of Z is a limit of points where f is analytic
but such that a priori f may not be defined
00:09:56.839 --> 00:10:01.910
at Z naught.
00:10:01.910 --> 00:10:07.160
So you see so look at this definition very
carefully what it says that you know point
00:10:07.160 --> 00:10:14.019
is it not in the complex plane is a singular
point for analytic function, if you can approach
00:10:14.019 --> 00:10:23.420
that point it is a limit of points where the
function is analytic okay but at that point
00:10:23.420 --> 00:10:31.589
itself the function may not be defined okay
and force the phrase a priori means is that
00:10:31.589 --> 00:10:38.260
to begin with or in advance you do not know
whether f is defined at Z naught or where
00:10:38.260 --> 00:10:43.550
the f can be defined at Z naught these are
things you do not know okay so it is a point
00:10:43.550 --> 00:10:48.380
which is outside, so singular point is a point
who is outside the domain of definition, it
00:10:48.380 --> 00:10:54.380
is outside the open set that the function
is defined and your question is whether the
00:10:54.380 --> 00:10:57.580
function, how does a function behave close
to that point?
00:10:57.580 --> 00:11:02.120
You see what you must understand is reason
why we define a singularity like this is because
00:11:02.120 --> 00:11:08.960
you see I want to study a function at singularity
okay and I told you what is the motivation,
00:11:08.960 --> 00:11:14.339
why should be at all worry about functions
with singularities, the answers because singularities
00:11:14.339 --> 00:11:19.589
occur, not all functions are going to be entire
okay not all functions that you are going
00:11:19.589 --> 00:11:26.861
to study are going to be defined on the whole
complex plane. A lot of functions they come
00:11:26.861 --> 00:11:29.237
out naturally with singularity so for example
you take the identity function f of Z equal
00:11:29.237 --> 00:11:32.660
to Z that is the identity function f of Z
equal to Z at is the identity function is
00:11:32.660 --> 00:11:40.170
of course entire okay but the moment you invert
it if I take f of Z equal to 1 by Z okay then
00:11:40.170 --> 00:11:43.519
you see immediately at 0 it is not defined
okay.
00:11:43.519 --> 00:11:50.000
So the problem is that there are it is very
easy you know to get hold of functions which
00:11:50.000 --> 00:11:56.350
you cannot define a point at that point was
surrounded by points at the functions analytic
00:11:56.350 --> 00:12:01.870
okay therefore such a point is a singular
point okay, so singular point will come when
00:12:01.870 --> 00:12:09.269
naturally. They are the most natural things
that you have to come across you have to study
00:12:09.269 --> 00:12:16.100
and of course I told you the the 1st motivation
was that you know we are trying to prove the
00:12:16.100 --> 00:12:22.490
big Picard theorem which is actually a theorem
about the mapping properties of a function
00:12:22.490 --> 00:12:27.910
around an essential singularity, so that is
also a motivation as to why worry about singularities
00:12:27.910 --> 00:12:34.139
okay, so let me come back to this definition
of singularity see the point is that I have
00:12:34.139 --> 00:12:39.269
a point that there is a point Z naught where
the functions not defined okay but I would
00:12:39.269 --> 00:12:46.500
like to study the function close to that point
okay and why should I study function close
00:12:46.500 --> 00:12:51.149
to that point because that is the only way
I can study out the function behave as I go
00:12:51.149 --> 00:12:53.380
closer and closer to that point okay.
00:12:53.380 --> 00:12:57.709
So it means that no matter how close I get
to Z naught I should be able to study the
00:12:57.709 --> 00:13:03.660
function okay that means the function should
be defined no matter how close I get to Z
00:13:03.660 --> 00:13:07.740
naught that is the only only if the function
is defined can I study it okay so that is
00:13:07.740 --> 00:13:16.540
the reason why our singular point is always
defined as a limit of good points okay, so
00:13:16.540 --> 00:13:21.090
what I want to tell you is that there are
functions for which you know singularities
00:13:21.090 --> 00:13:29.970
per se do not exists, so for example take
the example of f of Z equal to let say mod
00:13:29.970 --> 00:13:36.139
Z the whole square okay and if you take f
of Z equal to mod Z the whole square this
00:13:36.139 --> 00:13:42.199
is of course defined on the whole complex
plane and if you check mod Z the whole square
00:13:42.199 --> 00:13:49.420
is Z into Z bar where Z bar is a conjugative
Z okay and you will see that the Cauchy Riemann
00:13:49.420 --> 00:13:52.850
equation I satisfied only at the origin okay.
00:13:52.850 --> 00:13:58.009
So if at all this function is differentiable
will be only at the origin okay, so certainly
00:13:58.009 --> 00:14:02.199
the function cannot be you can find a single
point where this function is analytic okay
00:14:02.199 --> 00:14:06.399
because analyticity means the function should
not only be that point it should also be in
00:14:06.399 --> 00:14:09.579
the whole neighbourhood around that point
but there is no such point, the only point
00:14:09.579 --> 00:14:14.600
where this function f of Z equal to Z square
mod Z the whole square differentiable is the
00:14:14.600 --> 00:14:19.730
origin and at that point is not analytic because
at no other point it is differentiable so
00:14:19.730 --> 00:14:24.410
if you take this function what is a set of
singularity which the empty set, I mean singularities
00:14:24.410 --> 00:14:29.190
not even defined because it is not even analytic
okay so what you must understand is the singularity
00:14:29.190 --> 00:14:33.910
are defined for only analytic functions okay
we are not worried about functions which are
00:14:33.910 --> 00:14:36.230
not analytic in the 1st place okay.
00:14:36.230 --> 00:14:41.980
So that is one thing that is 1 point then
the 2nd thing is that you know singularities
00:14:41.980 --> 00:14:50.160
comes in 2 categories if you want or 2 types
okay and one is friendly one is less friendly
00:14:50.160 --> 00:14:55.600
and other is more friendly okay, see the more
friendly ones are called the so-called isolated
00:14:55.600 --> 00:15:01.310
singularities okay, what is an isolated singularity?
It is a point where the function has a singularity
00:15:01.310 --> 00:15:06.410
but there is small open disk surrounding that
point but the function has no other singularities,
00:15:06.410 --> 00:15:10.110
so it means that there is a deleted neighbourhood
of the point that the function is analytic
00:15:10.110 --> 00:15:18.069
okay such singularities are called isolated
singularities okay and then the less friendlier
00:15:18.069 --> 00:15:24.420
singularities are the so-called non-isolated
singularities okay and these are more difficult
00:15:24.420 --> 00:15:27.639
to study okay.
00:15:27.639 --> 00:15:37.519
The standard example of non-isolated singularity
is that of the log function okay if you take
00:15:37.519 --> 00:15:45.899
f of Z equal to log Z to be the principal
brands of the logarithm okay you know that
00:15:45.899 --> 00:15:50.680
to make it analytic you have to make it happen
throughout the negative real axis along with
00:15:50.680 --> 00:15:54.600
the origin of course origin will now come
into the picture because you cannot define
00:15:54.600 --> 00:16:00.550
log 0 okay and then you will have to cut out
the negative real axis okay and then you get
00:16:00.550 --> 00:16:09.339
the so-called slit plane, it is the plane
minus the real axis from the origin to the
00:16:09.339 --> 00:16:14.339
going to minus infinity that whole line segment
that whole ray is cut off okay.
00:16:14.339 --> 00:16:18.450
So this is the function which this is the
domain the slit plane is the domain where
00:16:18.450 --> 00:16:23.829
the principal brands of the logarithm be defined
and is analytic there and every point on the
00:16:23.829 --> 00:16:28.750
negative real axis is a singularity by definition
because the function is not defined there
00:16:28.750 --> 00:16:35.759
and it is not analytic at those points okay,
so well in fact the truth is that function
00:16:35.759 --> 00:16:39.170
can be defined at each of those points but
you cannot define it in such a way as it becomes
00:16:39.170 --> 00:16:50.440
analytic okay on the whole punctured plane
okay, so the negative real axis in the case
00:16:50.440 --> 00:16:59.610
of along with the origin is all the points
on this ray they are all examples of non-isolated
00:16:59.610 --> 00:17:00.880
singularities okay.
00:17:00.880 --> 00:17:31.250
So let me write that down, so singularities
are of 2 types isolated and non-isolated well
00:17:31.250 --> 00:17:49.970
so let me write that down Z naught is an isolated
singularity of f of Z if there exists an open
00:17:49.970 --> 00:18:02.640
disk 0 less than mod Z minus Z naught less
than Epsilon for some Epsilon greater than
00:18:02.640 --> 00:18:12.920
0 where f is analytic. This is just another
way of saying that there is a small neighbourhood
00:18:12.920 --> 00:18:22.710
around Z naught where the function f is analytic
okay and so and what about non-isolated singularity
00:18:22.710 --> 00:18:32.720
well singularities which are not isolated
are non-isolated singularities okay as the
00:18:32.720 --> 00:18:58.430
name says so singularities that are not isolated
are non-isolated okay and of course the examples
00:18:58.430 --> 00:19:17.190
well you take f of Z equal to 1 by Z then
Z equal to 0 yes of course an isolated singularity.
00:19:17.190 --> 00:19:29.760
Then I can take f of Z equal to if you want
sin Z over Z again Z equal to 0 is an isolated
00:19:29.760 --> 00:19:36.220
singularity okay but you will recognize immediately
that the lemmatize Z tends to zero sign Z
00:19:36.220 --> 00:19:43.920
by Z is one, so is a singularity that can
be really removed okay we will see about that
00:19:43.920 --> 00:19:51.720
very soon. Then let me give you the principal
branch of the logarithm f of Z equal to principal
00:19:51.720 --> 00:20:04.830
branch of log Z which is mod Z plus i times
principal argument of Z with principal argument
00:20:04.830 --> 00:20:15.220
of Z varying from minus pi to pi, minus pi
included pi not included, plus pi not included
00:20:15.220 --> 00:20:38.200
and this is so so let me write here the negative
real axis including 0 or let me say points
00:20:38.200 --> 00:20:55.760
on the negative real axis including 0 are
non-isolated singularities, okay.
00:20:55.760 --> 00:21:05.690
So the logarithm of course you know you have
2 so keep in mind the domain of definition
00:21:05.690 --> 00:21:11.300
so in the case f of Z the 1st example f of
Z equal to 1 by Z the domain of definition
00:21:11.300 --> 00:21:15.400
is the punctured plane, the complex plane
minus the origin and origin is the isolated
00:21:15.400 --> 00:21:21.740
singularity. In the 2nd case also it is positive
plane is a complex plane minus the origin
00:21:21.740 --> 00:21:25.120
and the 3rd case of force the principal branch
of the logarithm it is the slit plane, so
00:21:25.120 --> 00:21:33.350
it is the plane minus the negative real axis,
with the origin removed okay fine, so that
00:21:33.350 --> 00:21:34.350
is that.
00:21:34.350 --> 00:21:43.520
Now what are we going to do, so let me tell
you that we are worried only about isolated
00:21:43.520 --> 00:21:47.940
singularities, we will not be worried about
these non-isolated singularities but then
00:21:47.940 --> 00:21:53.580
let me also tell you that what is the way
to study non-isolated singularities, one of
00:21:53.580 --> 00:21:57.970
the theories that helps in the study of non-isolated
singularities is the theory of Riemann surfaces
00:21:57.970 --> 00:22:03.840
okay, so the point is that when you have non-isolated
singularities then you basically have usually
00:22:03.840 --> 00:22:11.670
you have a curve where which is full of points
where there are singularities okay.
00:22:11.670 --> 00:22:14.760
So in the case of the principal branch of
logarithm this curve is actually the negative
00:22:14.760 --> 00:22:24.940
real axis okay and such curve is called a
branch in curve or branch locus okay of your
00:22:24.940 --> 00:22:34.710
function and the way to study that is to do
what is called to go to what is called the
00:22:34.710 --> 00:22:43.380
Riemann surface of the corresponding function
okay so there is a key to study studying simple
00:22:43.380 --> 00:22:50.092
non-isolated singularities which lie on a
curve is a study of…will lead you to a study
00:22:50.092 --> 00:22:55.110
of Riemann surface okay but anyway we are
not going to do that but this is just tell
00:22:55.110 --> 00:23:00.360
you at non-isolated singularities can also
be studied alright and then you can also have
00:23:00.360 --> 00:23:07.720
a very strange situation like there may be
function which has only one singularity and
00:23:07.720 --> 00:23:12.390
that one singularity alone may be non-isolated
and all the other singularities maybe isolated
00:23:12.390 --> 00:23:17.480
you can have all kinds of examples, so here
is so let me give you one example.
00:23:17.480 --> 00:23:32.390
So here is another example you take f of Z
to be 1 by sin of 1 by Z okay, look at this
00:23:32.390 --> 00:23:38.480
function 1 by sin 1 by Z you see the point
is that whenever you take the reciprocal of
00:23:38.480 --> 00:23:43.590
a function your reciprocal is always in trouble
whenever the function vanishes okay so when
00:23:43.590 --> 00:23:51.330
I write 1 by sin 1 by Z this is cosecant of
1 by sin okay and the problem with this function
00:23:51.330 --> 00:23:55.830
is whenever the denominator which is sin 1
by Z vanishes and you know sin 1 by Z vanishes
00:23:55.830 --> 00:24:03.080
when 1 by Z is n pi, so the problem is that
the problem is at point Z equal to 1 by n
00:24:03.080 --> 00:24:15.880
pi where n is an integer okay so this is the
these are the points among these you know
00:24:15.880 --> 00:24:25.870
you can see that if you take function sin
1 by Z that is already already involves 1
00:24:25.870 --> 00:24:29.620
by Z and 1 by Z is not defined it is 0.
00:24:29.620 --> 00:24:34.990
So 0 is already a problem or the function
sin for the function 1 by Z so it is also
00:24:34.990 --> 00:24:42.540
a problem for the function sin 1 by Z okay
therefore you see of course when I write 1
00:24:42.540 --> 00:24:48.160
by n pi I must make sure that n cannot be
0 because it does not make sense, so n cannot
00:24:48.160 --> 00:24:55.850
be 0 but then I should also include Z equal
to 0 because this is a point where the function
00:24:55.850 --> 00:25:00.990
even the function that denominator is not
defined namely sin 1 by Z is not defined okay.
00:25:00.990 --> 00:25:10.700
Now if you look at it carefully see this as
n becomes larger size okay 1 by n pi come
00:25:10.700 --> 00:25:15.950
closer and closer to the origin okay and therefore
you see but all these 1 by n pi for various
00:25:15.950 --> 00:25:21.960
n not 0 they are isolated singularities in
fact they will be simple poles as we will
00:25:21.960 --> 00:25:28.110
see later okay but the origin will be an non-isolated
singularity so here is an example of function
00:25:28.110 --> 00:25:33.810
which helps one singularity which is non-isolated
and all other singularities are isolated okay.
00:25:33.810 --> 00:25:42.750
So so let me write this down Z equal to 1
by n pi, n an integer which is different from
00:25:42.750 --> 00:25:58.610
0 these are all isolated singularities and
Z equal to 0 is non-isolated, so you see you
00:25:58.610 --> 00:26:07.880
can have so this is another example okay,
fine. So what we will do is that we will start
00:26:07.880 --> 00:26:17.260
worrying about only isolated singularities
okay and so we will leave out the case of
00:26:17.260 --> 00:26:26.011
non-isolated singularities and go to the case
of isolated singularities so how do you classify
00:26:26.011 --> 00:26:30.530
isolated singularities, so this is again something
that you should have done in the 1st course
00:26:30.530 --> 00:26:36.890
in complex analysis, the isolated singularities
are classified as removable singularities
00:26:36.890 --> 00:26:44.500
poles and essential singularities, so let
me let me recall what these things are, so
00:26:44.500 --> 00:26:51.860
let me first say in words what is a removable
singularity, removable singularities essentially
00:26:51.860 --> 00:27:01.540
a singularity that can be removed namely that
is an isolated singularity okay but the function
00:27:01.540 --> 00:27:06.500
can be extended to the singularity in a way
that it becomes analytic okay.
00:27:06.500 --> 00:27:16.320
It is like it is the analog of removable discontinuity
that you study in 1st grade analysis okay
00:27:16.320 --> 00:27:26.020
so well then you have these so-called polls,
what kind of singularities are poles these
00:27:26.020 --> 00:27:31.630
are going to be… Poles are supposed to be
thought as the 0 of the denominator is okay
00:27:31.630 --> 00:27:37.220
so the point is that you cannot divide by
0, so whenever the denominator becomes 0 the
00:27:37.220 --> 00:27:41.710
function is not defined so all the places
where the denominator becomes 0 these are
00:27:41.710 --> 00:27:50.080
the polls okay and they should be…and when
I say 0 it should be 0 of a certain order
00:27:50.080 --> 00:28:00.970
okay and in general you think of poles as
zeros of the denominator, the other way of
00:28:00.970 --> 00:28:07.020
saying it is that 0, the poles are actually
zeros of the reciprocal of the function okay.
00:28:07.020 --> 00:28:14.230
So 0 of the reciprocal of the function okay
is exactly what a pole of the function is
00:28:14.230 --> 00:28:19.880
okay and then what are essential singularities
by definition these are the singularities
00:28:19.880 --> 00:28:26.980
which are neither poles nor removable okay
that is the clever way of defining them because
00:28:26.980 --> 00:28:33.170
then you do not have 2 you have to define
them separately, so let me write down these
00:28:33.170 --> 00:28:54.960
definitions. So isolated singularities
are of three types so the 1st one is they
00:28:54.960 --> 00:29:08.760
are called removable singularities the 2nd
ones are called poles and 3rd ones are the
00:29:08.760 --> 00:29:18.600
essentials okay and by definition so so if
you go to definitions, essential is defined
00:29:18.600 --> 00:29:34.951
as not removable not pole okay that is how
you define essential singularity and you may
00:29:34.951 --> 00:29:43.280
be wondering why the name essential singularity
well let me tell you they are really essential
00:29:43.280 --> 00:29:51.630
because they kind of completely distinguish
the function, the behaviour of the function.
00:29:51.630 --> 00:29:55.700
The neighbourhood of the essential singularity
can distinguish the function from other functions,
00:29:55.700 --> 00:30:01.210
so it is an though it is the singularity of
the function it is like it is very essential
00:30:01.210 --> 00:30:05.710
for the function it can distinguish the function,
it holds all the information more or less
00:30:05.710 --> 00:30:10.100
about the function okay that is why it is
call essential. The behaviour of a function
00:30:10.100 --> 00:30:18.010
in a neighbourhood of the essential singularity
completely holds the holds the full information
00:30:18.010 --> 00:30:23.140
about the function okay that is why this call
essential, we will see more about this later
00:30:23.140 --> 00:30:31.340
and of course pole is let me say this is 0
of the reciprocal.
00:30:31.340 --> 00:30:40.120
This is what the pole is and removable is
well to say it in simplest words it can be
00:30:40.120 --> 00:30:59.920
removed so this is this is as simple as it
goes but then you know so there are many ways
00:30:59.920 --> 00:31:10.730
of categorising so-called removable singularities,
poles and essential singularities and one
00:31:10.730 --> 00:31:16.730
of the key is to that doing these things
for that matter one of the key is to studying
00:31:16.730 --> 00:31:22.820
a function around an isolated singularities
so-called Laurent expansion of the function
00:31:22.820 --> 00:31:27.860
okay so this is how you would have you would
have all gone through 1st course in complex
00:31:27.860 --> 00:31:32.390
analysis where you have used Laurent you have
come across Laurent series and then you view
00:31:32.390 --> 00:31:37.840
the residue theorem often trying to find out
the residue at a pole and so on and so forth,
00:31:37.840 --> 00:31:43.330
so did Laurent series is one concrete way
of trying to get formula for the function
00:31:43.330 --> 00:31:51.340
as a series of some of powers both positive
and negative around the isolated singularities,
00:31:51.340 --> 00:31:56.340
so let me state the following thing.
00:31:56.340 --> 00:32:07.880
So this is let me just recall this is Laurent’s
theorem which is kind of a very helpful to
00:32:07.880 --> 00:32:15.240
study functions around there around an isolated
singularity okay, so so here is Laurent’s
00:32:15.240 --> 00:32:40.110
theorem if Z naught is an isolated singularity
of f of Z then f of Z is equal to the Sigma
00:32:40.110 --> 00:32:55.690
a n Z minus Z naught to the power of n, n
equal to minus infinity to plus infinity
00:32:55.690 --> 00:33:17.160
this is the Laurent series of f about Z naught
okay valid in 0 less then mod Z minus Z naught
00:33:17.160 --> 00:33:34.340
less than R okay where R is the distance is
the distance
00:33:34.340 --> 00:33:54.100
from Z naught to the next singularity of f,
so what do I mean by next singularity of f
00:33:54.100 --> 00:34:02.030
what I mean by that is the next nearest singularity
of f, so maybe so let me write next nearest
00:34:02.030 --> 00:34:06.950
okay.
00:34:06.950 --> 00:34:12.040
So this is the Laurent’s theorem okay I
have stated Laurent theorems for an isolated
00:34:12.040 --> 00:34:19.500
singularity okay but Laurent theorem is also
valid in an annulus actually okay, so and
00:34:19.500 --> 00:34:25.429
you know a deleted a punctured disk is a special
case of an annulus with the inner radius 0
00:34:25.429 --> 00:34:34.950
you know an annulus about a point is the open
region between 2 circles centred at that point
00:34:34.950 --> 00:34:42.000
of different radii okay and if you make the
inner radius 0 okay then you get a punctured
00:34:42.000 --> 00:34:47.290
disk which is also a special case of the annulus,
so and in that if you make the outer radius
00:34:47.290 --> 00:34:49.000
infinity then you get the punctured plane.
00:34:49.000 --> 00:34:54.220
So a punctured plane is also a special case
of an annulus okay so for example if you take
00:34:54.220 --> 00:35:01.710
the function e power 1 by Z okay and you are
right out the use simply take the what is
00:35:01.710 --> 00:35:06.569
the Laurent expansion, the Lauren expansion
is you know e powers Z has a Taylor expansion
00:35:06.569 --> 00:35:12.930
which is valid for all Z and that Taylor expansion
simply replace Z by 1 by Z and that continues
00:35:12.930 --> 00:35:18.740
to be valid the whole plane except the origin
and the whole plane except the origin is is
00:35:18.740 --> 00:35:24.930
again an annulus with inner radius 0 out
radius infinity okay punctured plane is also
00:35:24.930 --> 00:35:26.690
a special case of an annulus okay.
00:35:26.690 --> 00:35:33.990
So so this is Laurent’s theorem and when
I write f of Z is equal to Sigma n equal to
00:35:33.990 --> 00:35:40.960
minus infinity to a n Z minus Z naught power
n, notice 1st of all that what this is supposed
00:35:40.960 --> 00:35:47.839
to mean is that the series on the right converges
and it converges to f that is what it means
00:35:47.839 --> 00:35:57.789
okay and technically what are these a n, the
Laurent coefficient they are given by integral
00:35:57.789 --> 00:36:11.869
so here a n is 1 by 2 pi i integral over gamma
fw dw by w minus Z naught to the power of
00:36:11.869 --> 00:36:21.539
n plus 1. This is the these are the values
of these a n and what is this gamma, see gamma
00:36:21.539 --> 00:36:28.380
is a simple so here is a Z naught and
gamma is some simple closed curve, gamma is
00:36:28.380 --> 00:36:39.759
some simple closed curve going one around
Z naught okay and of course in the region
00:36:39.759 --> 00:36:45.589
enclosed by gamma Z naught is the only
singularity and there is no other singularities
00:36:45.589 --> 00:36:47.420
on gamma for the function okay.
00:36:47.420 --> 00:36:56.190
So so this is the Laurent theorem okay and
of course you know you must keep in mind that
00:36:56.190 --> 00:37:01.299
whenever you write an integral like this when
you write integral over gamma you know it
00:37:01.299 --> 00:37:06.640
is very important that or such an integral
to be defined of the curve should be contour
00:37:06.640 --> 00:37:12.460
so it should be piecewise smooth which means
piecewise continuously differentiable curve
00:37:12.460 --> 00:37:17.890
okay and the integral should be piecewise
continuous at least on the contour for the
00:37:17.890 --> 00:37:25.320
integral to be defined okay so of course I
can deform gamma little bit and the integral
00:37:25.320 --> 00:37:30.369
will not change that is because of cautious
theorem, so in particular if you want to make
00:37:30.369 --> 00:37:35.779
calculations you can take this gamma to be
a small circle centred at Z naught okay with
00:37:35.779 --> 00:37:39.030
sufficiently small radius okay.
00:37:39.030 --> 00:37:43.839
Really the shape of gamma does not matter
okay there is only the fact that gamma should
00:37:43.839 --> 00:37:48.660
be simple closed curve, simple means that
it does not cross itself okay and it goes
00:37:48.660 --> 00:37:57.839
around 1 exactly one around Z naught and this
is Laurent’s theorem and the point, the
00:37:57.839 --> 00:38:02.830
important thing about Laurent theorem is that
as you would have learnt in the 1st course
00:38:02.830 --> 00:38:06.880
in complex analysis is the most important
thing about Laurent’s theorem is the coefficient
00:38:06.880 --> 00:38:13.920
A minus 1 okay when I put n equal to minus
1 what I get a is a minus 1is 1 by 2 pi i
00:38:13.920 --> 00:38:20.980
integral over gamma fw dw okay and that is
important that is the residue of f at Z naught
00:38:20.980 --> 00:38:26.329
it is the A minus 1 and it is important because
it gives you it tells you what the integral
00:38:26.329 --> 00:38:30.009
of the function this around a singularity
okay.
00:38:30.009 --> 00:38:36.450
If I put n equal to minus 1 I get a minus
1 is equal to 1 by 2 pi i integral over gamma
00:38:36.450 --> 00:38:43.140
fw dw, so integral of fw dw over gamma where
gamma is going around singularity that is
00:38:43.140 --> 00:38:49.240
a very important thing okay. Cautious theorem
tells you that if you go around a point where
00:38:49.240 --> 00:38:59.029
the function is analytic okay if you try to
integrate a function around a closed curve
00:38:59.029 --> 00:39:03.460
so the function is analytic inside and on
the curve then you are going to get 0 that
00:39:03.460 --> 00:39:08.480
is what cautious theorem says it says you
will not get anything so but then even now
00:39:08.480 --> 00:39:14.119
a question, what will happen if you integrate
around that singularity if you take a function
00:39:14.119 --> 00:39:17.970
and you integrate it along the singularity
around the singularity what will you get?
00:39:17.970 --> 00:39:22.660
The answer is the residue so that is why residue
is important, they help you to calculate the
00:39:22.660 --> 00:39:31.190
integral of a function around singularity
okay so and that is so essentially that the
00:39:31.190 --> 00:39:40.970
residue is 2 pi i, so a minus 1 is the residue
and 2 pi i times a minus 1 is equal to integral
00:39:40.970 --> 00:39:46.089
over gamma fw dw that is exactly the residue
theorem okay. Residue theorem actually says
00:39:46.089 --> 00:39:50.990
that the integral around a…if you go ones
around if there is a curve which goes once
00:39:50.990 --> 00:39:56.509
around a singular point then the integral
of the function along that curve is going
00:39:56.509 --> 00:40:00.839
to give you 2 pi i times the residue that
is if you are going around 1 singularity and
00:40:00.839 --> 00:40:03.900
then the residue theorem in general says that
if you have several singularities and you
00:40:03.900 --> 00:40:07.610
have to take sum of all those residue it will
be 2 pi i times sum of all the residue.
00:40:07.610 --> 00:40:11.140
So this is the residue this is the residue
theorem okay which helps us to compute lot
00:40:11.140 --> 00:40:15.260
of integrals even real integrals which
you would have seen the 1st course in complex
00:40:15.260 --> 00:40:21.380
analysis okay so very well. This is Laurent’s
theorem, now what I am going to do is I am
00:40:21.380 --> 00:40:29.700
going to you know go back to our study which
is the study of singularities and I am going
00:40:29.700 --> 00:40:36.440
to tell you you know we saw 1st that they
were that there are 3 types of singularities
00:40:36.440 --> 00:40:42.849
there are the removable singularities, there
are the poles and then there are the essential
00:40:42.849 --> 00:40:44.430
singularities.
00:40:44.430 --> 00:40:52.119
Now let me say something about poles which
is something that you would have which you
00:40:52.119 --> 00:40:57.680
would have come across but you should try
to now do this is an exercise will help you
00:40:57.680 --> 00:41:09.829
to revise your basic knowledge of complex
analysis, so here is the theorem, so here
00:41:09.829 --> 00:41:30.369
is the theorem. So this is theorem about poles,
let Z naught be an isolated singularity of
00:41:30.369 --> 00:41:56.460
f of Z then a following conditions
are equivalent number 1 f of Z has a pole
00:41:56.460 --> 00:42:19.040
of order n greater than 0 at Z naught, so
and that is 1 by f of Z has 0 of order n greater
00:42:19.040 --> 00:42:24.059
than 0 at Z naught, so the definition of a
pole this is one of the definitions of a pole.
00:42:24.059 --> 00:42:31.359
In fact a pole can be defined in many ways
and what this theorem says is that it gives
00:42:31.359 --> 00:42:33.440
you various equivalent conditions.
00:42:33.440 --> 00:42:39.170
So the 1st thing is the definition of a pole
which is the which is as 0 of the reciprocal,
00:42:39.170 --> 00:42:48.009
so f has a pole of order n if one by f which
is reciprocal of f has 0 of order n okay and
00:42:48.009 --> 00:42:57.450
so what is the 2nd one, the 2nd one is limit
Z tends to Z naught f of Z is infinity okay,
00:42:57.450 --> 00:43:04.080
so this is this is another condition for a
pole, the function becomes arbitrarily large
00:43:04.080 --> 00:43:18.210
in modellers as you approach a pole okay and
here is the 3rd one the Laurent expansion
00:43:18.210 --> 00:43:42.470
of f at Z naught has only finitely many negative
powers of Z minus . So this is the way you
00:43:42.470 --> 00:43:49.810
define a this is the way you define a
pole using Laurent expansion, see a Laurent
00:43:49.810 --> 00:43:56.019
expansion helps you also classify singularities,
so what I want you to do now is that you should
00:43:56.019 --> 00:44:01.369
you need to I want you to go back and as an
exercise of that all the 3 statements are
00:44:01.369 --> 00:44:05.890
equal okay at least you should have seen this
in the 1st course but I want you to recall
00:44:05.890 --> 00:44:11.769
the proof that is an exercise and so let me
stop here and we will continue in the next
00:44:11.769 --> 00:44:12.469
lecture.