WEBVTT
Kind: captions
Language: en
00:00:43.960 --> 00:00:50.820
Okay so let us continue with what we are trying
to do, so you know we are trying to prove
00:00:50.820 --> 00:01:01.010
the final aims to prove the Picard theorems
okay and the little Picard theorem will be
00:01:01.010 --> 00:01:08.280
deduced as corollary of the great Picard theorem
okay and but the great Picard… The little
00:01:08.280 --> 00:01:14.580
Picard theorem is about the image of an entire
function, the great Picard theorem is about
00:01:14.580 --> 00:01:23.100
the image of a deleted neighbourhood of an
isolated essential singularity okay and what
00:01:23.100 --> 00:01:31.230
we are going to now 1st prove is the easiest
thing to prove to begin with to get you a
00:01:31.230 --> 00:01:37.679
feel of things is the Casorati Weierstrass
theorem which says that if you take any neighbourhood
00:01:37.679 --> 00:01:44.930
of an isolated essential singularity the image
of that neighbourhood of course with the singularity
00:01:44.930 --> 00:01:52.780
omitted of course the image of that neighbourhood
will always be a dense subset of the complex
00:01:52.780 --> 00:02:00.030
plane okay namely its closer will be the whole
complex plane okay.
00:02:00.030 --> 00:02:06.430
What this means is that it at any complex
number you can always find a sequence of points
00:02:06.430 --> 00:02:11.090
in any neighbourhood of an isolated essential
singularity such that the function values
00:02:11.090 --> 00:02:16.150
at those points approaches that complex number
okay, so let me write that down let us prove
00:02:16.150 --> 00:02:21.261
it, the key to proving the Casorati Weierstrass
theorem is the Riemann removable singularity
00:02:21.261 --> 00:02:26.760
theorem which we proved last time okay, so
let me write that down.
00:02:26.760 --> 00:03:05.329
So Casorati Weierstrass theorem, so let Z
naught be an isolated essential singularity
00:03:05.329 --> 00:03:44.340
of the analytic function f of Z given any
complex number W naught we can find a sequence
00:03:44.340 --> 00:04:07.180
of points in a neighbourhood of Z naught such
that if you take the function values at those
00:04:07.180 --> 00:04:17.220
points fZn that tends to W naught okay, so
this is the this is the Casorati Weierstrass
00:04:17.220 --> 00:04:24.780
theorem okay. You take an isolated essential
singularity Z naught of an analytic function
00:04:24.780 --> 00:04:32.419
f of Z and then you can always find a sequence
of points in the neighbourhood of Z naught
00:04:32.419 --> 00:04:40.000
such as the function value at those points
tends to the limit W naught okay so in this
00:04:40.000 --> 00:04:48.830
way, so what you are what does this mean,
it means see W naught can be…f of Z n tends
00:04:48.830 --> 00:04:54.870
to W naught means that you can no matter how
close you go to W naught confined function
00:04:54.870 --> 00:04:56.669
values okay.
00:04:56.669 --> 00:05:04.090
So that means that the W naught is in the
closure of the set of function values that
00:05:04.090 --> 00:05:10.660
what it means, so W naught is the limit point
of the image set of f, image set of f is just
00:05:10.660 --> 00:05:17.590
the set of values of f okay and what this
says is that every complex number is in the
00:05:17.590 --> 00:05:27.710
closure of the image set, the set of values
that f takes okay and this is this very deep
00:05:27.710 --> 00:05:33.479
theorem because what it says is that it tells
you that therefore the if you take the image
00:05:33.479 --> 00:05:39.310
of a deleted neighbourhood of the isolated
essential singularity then the image is going
00:05:39.310 --> 00:05:46.750
to be huge a set which is tenses huge okay,
the image is going to be dense in the complex
00:05:46.750 --> 00:05:47.750
plane.
00:05:47.750 --> 00:05:51.530
So that means the image closure is the whole
complex plane so it is the image is huge and
00:05:51.530 --> 00:05:58.300
this is the this is kind of much weaker when
compared to the the great Picard theorem which
00:05:58.300 --> 00:06:03.600
says that the images actually the whole complex
plane or at most a punctured plane namely
00:06:03.600 --> 00:06:09.050
it can at most omit a point okay but you must
remember that that omitted point is also in
00:06:09.050 --> 00:06:15.539
the closure because it can be approached by
the points okay, so so this Casorati Weierstrass
00:06:15.539 --> 00:06:24.919
theorem is weaker version of you know the
great Picard theorem but it tells you it answers
00:06:24.919 --> 00:06:30.719
this question that we have been worried about
namely what is the image of an analytic function
00:06:30.719 --> 00:06:38.370
okay, so so let us prove this, the prove of
this is is going to involve it is just
00:06:38.370 --> 00:06:45.050
going to involve Riemann removable singularity
theorem, the use of Riemann removable singularity
00:06:45.050 --> 00:07:06.490
theorem, so so let me write this also in other
words, the image of f is dense in the complex
00:07:06.490 --> 00:07:11.759
plane, this is another way of saying it okay.
00:07:11.759 --> 00:07:21.790
So so let us go on to the the proof of
the theorem, so you know so you know the idea
00:07:21.790 --> 00:07:28.510
of the proof is very very simple, what you
do in the proof is that you prove by contradiction
00:07:28.510 --> 00:07:36.240
okay, so you assume that there is a complex
value which is not in the closure of the
00:07:36.240 --> 00:07:42.539
image that means there is a complex value
which is not in the limit of values in the
00:07:42.539 --> 00:07:49.889
image okay that means that the image is bounded
away from a certain complex value okay you
00:07:49.889 --> 00:07:56.360
assume that and then you show that if this
is the case then your function cannot have
00:07:56.360 --> 00:08:00.539
an isolated essential… Essential singularity
at Z naught but the singularity has to either
00:08:00.539 --> 00:08:05.719
be removable or it has to be a pole and this
is where we will be using Riemann removable
00:08:05.719 --> 00:08:07.669
singularity theorem, so this is the technique
of the proof.
00:08:07.669 --> 00:08:13.240
The taking of the proof is just by contradiction
okay. You you assume that a certain value
00:08:13.240 --> 00:08:20.150
is not in the limit okay and then you sure
that this will imply that either Z naught
00:08:20.150 --> 00:08:24.050
is a removable singularity or it is a pole
and both of these are not possible they will
00:08:24.050 --> 00:08:27.810
give contradiction because we have assumed
that Z naught is an essential singularity
00:08:27.810 --> 00:08:33.010
which by definition is something that neither
removable singularity nor a pole okay so that
00:08:33.010 --> 00:08:39.880
is how the proof works, so let me write down
the proof, so prove so let me go to a different
00:08:39.880 --> 00:09:07.810
color. Proof by contradiction, assume
that there existed W naught complex value
00:09:07.810 --> 00:09:29.880
such that such that the conclusion of the
theorem does not hold so you assume.
00:09:29.880 --> 00:09:35.260
Conclusion of the theorem is that given any
complex value W naught okay you can find the
00:09:35.260 --> 00:09:42.090
sequence of point Z n such that the function
values at Z n approaches W naught, now you
00:09:42.090 --> 00:09:46.130
assume that that is not the case assumed that
there is at least one W naught or which this
00:09:46.130 --> 00:09:51.750
does not happen okay and we will try to get
a contradiction, so so what does this mean?
00:09:51.750 --> 00:10:01.640
What it means is that it means that W naught
cannot be approached by image points okay
00:10:01.640 --> 00:10:06.790
the another way of saying this is that you
are trying to say that there is a neighbourhood
00:10:06.790 --> 00:10:11.590
of W naught okay which has nothing to do with
the image okay which does not intersect the
00:10:11.590 --> 00:10:12.590
image.
00:10:12.590 --> 00:10:18.720
So that means there is an Epsilon greater
than 0 such that this open disk centred at
00:10:18.720 --> 00:10:25.950
W naught radius Epsilon is disjoint from the
image of f okay that is the that is what it
00:10:25.950 --> 00:10:37.720
means okay so let me write that down so thus
that exists Epsilon greater than 0 such that
00:10:37.720 --> 00:10:52.220
mod W minus W, W naught less than Epsilon
does not meet the image of f, image of f Im
00:10:52.220 --> 00:10:58.810
f is just the set of values of f and set of
values of f includes all values at f takes
00:10:58.810 --> 00:11:08.200
in a also in the deleted neighbourhood
of the isolated singularity Z naught which
00:11:08.200 --> 00:11:15.530
we have assumed is essential, okay so so what
does this mean, so what do I do now?
00:11:15.530 --> 00:11:27.580
Now you see so let us rewrite that thus for
all Z not equal to Z naught of course okay,
00:11:27.580 --> 00:11:40.650
we have mod f of Z minus W naught is greater
than equal to Epsilon this is what it means
00:11:40.650 --> 00:11:49.990
okay. So so let me again tell you what I wrote
down before what I wrote down before is that
00:11:49.990 --> 00:11:58.550
the disk centred at W naught radius Epsilon
does not meet the image okay, it means that
00:11:58.550 --> 00:12:02.980
there is no point in the image whose distance
from W naught is less than Epsilon, so it
00:12:02.980 --> 00:12:11.170
means that if you take any point in the image
the distance from that point to W naught is
00:12:11.170 --> 00:12:15.630
at least Epsilon greater than Epsilon that
is what I have written down, so when you write
00:12:15.630 --> 00:12:19.350
mod f when I write mod f Z minus W naught
is greater than not equal to Epsilon I am
00:12:19.350 --> 00:12:25.440
actually saying that the distance between
f of Z which is the point to the image of
00:12:25.440 --> 00:12:32.110
f the value of f at Z and W naught is at least
Epsilon.
00:12:32.110 --> 00:12:38.550
Now so you know the advantage with having
this kind of thing is that see whenever you
00:12:38.550 --> 00:12:47.560
have a function which is bounded away
from 0 okay, what is the advantage of having
00:12:47.560 --> 00:12:52.260
a function on bounded away from 0? The advantage
of having a function bounded away from 0 is
00:12:52.260 --> 00:12:59.680
that you can invert it okay it is reciprocal
make sense okay, so here f Z minus W naught
00:12:59.680 --> 00:13:04.780
mind you is also function f of Z is a function,
f Z minus W naught is just a function f of
00:13:04.780 --> 00:13:09.690
Z added to the constant minus W naught and
adding this does not change the analyticity
00:13:09.690 --> 00:13:13.450
of the function except at the point Z naught
where which of course we are not going to
00:13:13.450 --> 00:13:16.790
worry about okay.
00:13:16.790 --> 00:13:21.290
Mind you when I write f of Z I am of course
assuming Z is in the domain of analysis the
00:13:21.290 --> 00:13:28.500
of f it is implicit and of course Z is not
Z naught because at Z naught is not a point
00:13:28.500 --> 00:13:33.230
where f is analytic to begin with I have assume
that Z naught is an essential singularity
00:13:33.230 --> 00:13:39.870
it is a singular point okay, so so the point
is that since I have mod f minus W naught
00:13:39.870 --> 00:13:47.490
is greater than Epsilon 1 by f minus W naught
makes sense as function okay, so and what
00:13:47.490 --> 00:13:54.740
it tells you is that that function in a deleted
neighbourhood of Z naught is bounded by 1
00:13:54.740 --> 00:14:05.560
by Epsilon okay that is what it says, so now
look at g of Z defined to be 1 by f of Z minus
00:14:05.560 --> 00:14:24.940
W naught okay in in a deleted neighbourhood
of Z naught. Now look at this function, now
00:14:24.940 --> 00:14:32.700
you see this function mod g Z is greater than
or equal to is less than or equal to 1 by
00:14:32.700 --> 00:14:42.230
Epsilon you have that that is just inverting
mod f Z minus W naught greater than equal
00:14:42.230 --> 00:14:43.230
to Epsilon.
00:14:43.230 --> 00:14:53.120
So what you have is now you have 2 things
see g of Z makes sense as an analytic function
00:14:53.120 --> 00:15:00.170
okay it is because it is a reciprocal of an
analytic function and it is defined where
00:15:00.170 --> 00:15:08.730
the denominator does not vanish, so f Z minus
W naught never vanishes because if it vanishes
00:15:08.730 --> 00:15:14.420
then it is modulus will be 0 and vice a versa
but the modulus is always bounded away from
00:15:14.420 --> 00:15:19.040
0 it is always greater than equal to Epsilon,
so f Z minus W naught never vanishes okay
00:15:19.040 --> 00:15:26.820
and mind you wherever f is analytic, f Z minus
W naught is also analytic okay because it
00:15:26.820 --> 00:15:31.990
is just f of Z added to the constant minus
W naught adding a constant does not change
00:15:31.990 --> 00:15:35.840
the analyticity of a function because or constant
function is also analytic and sum of analytic
00:15:35.840 --> 00:15:44.650
functions is analytic okay, so f Z minus W
naught is also analytic in a deleted neighbourhood
00:15:44.650 --> 00:15:46.810
of Z naught okay and it is non-zero.
00:15:46.810 --> 00:15:53.920
So it is reciprocal 1 by f Z minus W naught
is as well analytic in a deleted neighbourhood
00:15:53.920 --> 00:16:00.800
of Z naught okay, so what I have now is I
have this function g Z it is analytic in the
00:16:00.800 --> 00:16:09.430
deleted neighbourhood of Z naught okay Z naught
is a singular point but look at the last inequality,
00:16:09.430 --> 00:16:14.191
this function is bounded in a neighbourhood
of Z naught. Now Riemann removable singularity
00:16:14.191 --> 00:16:23.260
theorem will tell you that Z naught has to
be a removable singularity for g okay, so
00:16:23.260 --> 00:16:35.470
since so let me write that down, since g of
Z is analytic in a deleted neighbourhood of
00:16:35.470 --> 00:17:02.040
Z naught, Z naught is also a singular point
of g of g and further is a removable singularity
00:17:02.040 --> 00:17:22.650
by Riemann’s theorem on removable singularities
since it is bounded okay, so here is where
00:17:22.650 --> 00:17:27.150
we are using the Riemann’s removable singularity
theorem okay.
00:17:27.150 --> 00:17:37.309
So what it means is that so it means that
g can be redefined at Z naught so that you
00:17:37.309 --> 00:17:45.680
get function which is analytic at Z naught
as well okay and the fact that you can define
00:17:45.680 --> 00:17:57.070
g redefine g at Z naught okay should tell
you that f has to have at Z naught either
00:17:57.070 --> 00:18:03.399
a removable singularity or a pole, it cannot
have an essential singularity that is the
00:18:03.399 --> 00:18:09.000
conclusion of all this and that is the contradiction
to the hypothesis that Z naught is actually
00:18:09.000 --> 00:18:12.990
an essential singularity and that is how we
get the proof of the theorem of the Casorati
00:18:12.990 --> 00:18:22.330
Weierstrass theorem okay, so let me write
that down. Thus limit Z tends to Z naught
00:18:22.330 --> 00:18:46.289
g of Z exist call it g of Z naught and then
g extends to an analytic function at Z naught
00:18:46.289 --> 00:18:54.990
okay, so this is what removable singularity
means you can take the you can take the so
00:18:54.990 --> 00:18:58.590
so this is probably the right time for me
to recall the Riemann removable singularity
00:18:58.590 --> 00:19:00.110
theorem, what does it say?
00:19:00.110 --> 00:19:07.059
It gives you an equivalence of 4 statements,
the first statement is that the point in concern
00:19:07.059 --> 00:19:15.470
the isolated singularity concern is is removable
okay which is equivalent to saying by definition
00:19:15.470 --> 00:19:19.610
that the function can be extended to an analytic
function at that point. The 2nd condition
00:19:19.610 --> 00:19:25.720
is slightly weaker that you can extend the
function continuously to that point okay namely
00:19:25.720 --> 00:19:31.470
the condition is that the limit of the function
as you approach that point exist okay as a
00:19:31.470 --> 00:19:39.070
finite complex number and then the 3rd condition
was the condition equivalent condition that
00:19:39.070 --> 00:19:43.730
involves the Laurent expansion and that condition
was that if you write out the Laurent expansion
00:19:43.730 --> 00:19:48.220
about removable singularity you actually get
Taylor expansion namely there are no negative
00:19:48.220 --> 00:19:52.260
powers there is no principle part, there is
no singular apart okay.
00:19:52.260 --> 00:19:57.379
And then the 4th condition which was the most
amazing was the condition that the function
00:19:57.379 --> 00:20:05.320
is bounded in a neighbourhood of the singularity
okay bounded of course bounded means bounded
00:20:05.320 --> 00:20:15.649
in modulus okay and I told you that that is
the weakest of all the 4 conditions and that
00:20:15.649 --> 00:20:22.080
boundedness in the neighbourhood of a singularity
can only happen if the singularity is a removable
00:20:22.080 --> 00:20:25.779
singularity that is essentially Riemann’s
removable singularity theorem.
00:20:25.779 --> 00:20:33.940
So so you know so because of that the limit
Z tends to Z naught g of Z exist let us call
00:20:33.940 --> 00:20:38.450
it as g of Z naught so g extends to analytic
function at Z naught, so here I am using several
00:20:38.450 --> 00:20:43.429
equivalent version of the Riemann removable
singularity theorem you must realize that.
00:20:43.429 --> 00:20:51.730
Now the question is it all depends on
what is the value of g of Z naught is, the
00:20:51.730 --> 00:21:04.289
point is that if g of Z naught is 0 okay then
f has a pole at Z naught okay and if g of
00:21:04.289 --> 00:21:10.690
Z naught is not 0 then f has a removable singularity
at Z naught okay and then thus we have manifest
00:21:10.690 --> 00:21:15.429
a contradiction to what we have assumed okay
so let me write that down.
00:21:15.429 --> 00:21:33.000
Now if g of Z naught is 0 we see that limit
Z tends to Z naught f of Z has to be infinity
00:21:33.000 --> 00:21:40.590
this has to happen because you know the limit
as I said tends to Z naught g of Z is
00:21:40.590 --> 00:21:46.889
0 so the limit is Z tends to Z naught 1 by
f Z minus W naught to 0 that means the denominator
00:21:46.889 --> 00:21:52.499
has to become unbounded, so that means and
if f Z minus W naught has to become unbounded
00:21:52.499 --> 00:21:57.529
in modulus then f has… because W naught
is just a constant f has to become unbounded
00:21:57.529 --> 00:22:05.059
in modulus and that is a condition for a pole
okay, so so if g of Z naught is 0 then limit
00:22:05.059 --> 00:22:17.210
Z tends to Z naught f of Z is infinity and
Z naught is a pole, is a pole of f and you
00:22:17.210 --> 00:22:30.860
see Z naught is a pole of f which is not possible
so that is ruled out so you have ruled out
00:22:30.860 --> 00:22:36.590
the case that g of Z naught is 0 the only
other case is when g of Z naught is not zero
00:22:36.590 --> 00:22:37.590
okay.
00:22:37.590 --> 00:22:51.950
If g of Z naught is not equal to 0 then limit
as Z tends to Z naught f of Z is actually
00:22:51.950 --> 00:23:07.970
W naught plus 1 by g of Z naught will get
this okay which means again by Riemann’s
00:23:07.970 --> 00:23:38.169
removable singularity theorem that f has a
removable singularity at Z naught okay
00:23:38.169 --> 00:23:53.179
which is again not possible, so so in both
cases you get a contradiction and we are done
00:23:53.179 --> 00:24:00.940
okay so that that is the that brings you do
the end of the proof okay so so you see we
00:24:00.940 --> 00:24:05.090
are applying the you must see that we have
applied the Riemann’s removable singularity
00:24:05.090 --> 00:24:10.269
theorem twice we have applied it once to g
and then we have applied it in one of the
00:24:10.269 --> 00:24:23.919
cases to f itself okay, fine. So this so this
theorem is a very nice theorem and so what
00:24:23.919 --> 00:24:29.710
it tells you is that you take neighbourhood
of an isolated essential singularity and take
00:24:29.710 --> 00:24:34.610
its image you are going to more or less fill
up the whole complex plane.
00:24:34.610 --> 00:24:41.120
You are going to get the images that is dense
and we have to move towards the proof the
00:24:41.120 --> 00:24:48.360
the great Picard theorem okay. Now what I
am going to do next is going to deal with
00:24:48.360 --> 00:24:56.730
the point at infinity okay, so you see you
see in this proof itself for example when
00:24:56.730 --> 00:25:03.130
I wrote down limit Z instead to Z naught f
of Z is infinity okay I am using the point
00:25:03.130 --> 00:25:09.139
at infinity, so you would have seen the point
at infinity as the extra point that is added
00:25:09.139 --> 00:25:14.129
to get 1 point compactification of the complex
plane and you would also have seen it as Riemann’s
00:25:14.129 --> 00:25:18.230
sphere in the 1st course and but anyway I
want to recall these things because you see
00:25:18.230 --> 00:25:24.710
there is very important from now on to be
able to think of infinity both in the domain
00:25:24.710 --> 00:25:28.529
of definition of the function as well as in
the range of values of the function.
00:25:28.529 --> 00:25:35.480
So you want to have a situation where you
can talk of a function variable, an independent
00:25:35.480 --> 00:25:41.259
variable going to infinity and its value at
for example the value of function at infinity
00:25:41.259 --> 00:25:47.400
you want to say that and you also want a function
to take the value infinity okay you want to
00:25:47.400 --> 00:25:53.149
include infinity into your set of values of
the independent variable and the set of values
00:25:53.149 --> 00:25:57.440
of the independent variable, so you have to
deal with infinity carefully and usually in
00:25:57.440 --> 00:26:03.730
the 1st course probably this is sometimes
not covered very thoroughly, so I want to
00:26:03.730 --> 00:26:08.909
just revise these things so that you are comfortable
about thinking about limits at infinity and
00:26:08.909 --> 00:26:13.080
infinite limits okay, so that is what I am
going to do next.
00:26:13.080 --> 00:26:21.070
So and I need that because because of the
following reason, see I want to be able to
00:26:21.070 --> 00:26:25.879
think of infinity as one of the values of
a function okay and I also want to be able
00:26:25.879 --> 00:26:31.470
to think of infinity as singularity okay see
For example if you take an entire function
00:26:31.470 --> 00:26:41.059
okay then the point at infinity is of course
approachable by infinity is of course approachable
00:26:41.059 --> 00:26:48.320
by any by any curve on the complex plane which
is not bounded okay so you can always approach
00:26:48.320 --> 00:26:53.929
the point at infinity and then the question
is whether the function is analytic at infinity
00:26:53.929 --> 00:26:58.700
or it is not analytic at infinity, so you
want to think of infinity as a singular point
00:26:58.700 --> 00:27:03.489
okay and then the question is what kind of
singularity is it because you know we have
00:27:03.489 --> 00:27:09.989
already classified singularity as either removable
or pole or essential, so the question is I
00:27:09.989 --> 00:27:16.220
want be able to think of infinity the point
at infinity as a singularity and question
00:27:16.220 --> 00:27:25.080
or study when they singularity is either removable
singularity or a pole or essential singularity,
00:27:25.080 --> 00:27:26.770
so the point at infinity is very very important.
00:27:26.770 --> 00:27:54.559
So so let me go to that so I will take another
color the point at infinity infinite limits,
00:27:54.559 --> 00:28:09.350
infinite values so this is what I am going
to I am going to tell you about okay,
00:28:09.350 --> 00:28:18.369
so well so the idea is as follows, so what
we do is let us so 1st of all let me you know
00:28:18.369 --> 00:28:23.429
the approach to everything is since we are
doing calculus approach is always through
00:28:23.429 --> 00:28:32.239
limits, so let me recall what finite limit
is okay as we as an independent variable approaches
00:28:32.239 --> 00:28:39.669
finite value, so you know see limit so when
I write limit Z tends to Z naught okay when
00:28:39.669 --> 00:28:47.539
I write this what does it mean? For Z naught
in C for let Z naught be a complex number,
00:28:47.539 --> 00:28:48.539
okay.
00:28:48.539 --> 00:28:54.090
What does limit Z tends to Z naught mean,
see it means that you going closer and closer
00:28:54.090 --> 00:29:00.429
to the point Z naught okay so basically what
it means is that and what does it mean to
00:29:00.429 --> 00:29:05.850
say that you are going closer and closer to
Z naught, if you think of Z as a moving point
00:29:05.850 --> 00:29:10.549
a variable point then you are saying that
the distance Z to Z naught is becoming smaller
00:29:10.549 --> 00:29:18.609
and smaller okay so the limit Z tends to Z
naught can be interpreted as limit mod Z minus
00:29:18.609 --> 00:29:28.009
Z naught tends to 0 okay that is how you can
interpret it okay so this means so let me
00:29:28.009 --> 00:29:44.620
write that here means mod Z minus Z naught
tends to 0 okay means we let so let me write
00:29:44.620 --> 00:29:50.629
that we are letting mod Z minus Z naught tends
to 0 and mod Z minus Z naught mind you is
00:29:50.629 --> 00:29:59.440
the distance between Z and Z naught and what
it means is that, if you go to definitions
00:29:59.440 --> 00:30:03.940
what is the business of letting something
go to 0.
00:30:03.940 --> 00:30:10.759
In analysis trying to let something go to
0 is the same as making it as small as possible,
00:30:10.759 --> 00:30:17.480
so that is where your Epsilon comes in, so
usually we use Epsilon for the values of the
00:30:17.480 --> 00:30:23.730
function and use delta for the values of the
variables, so let me use delta so the point
00:30:23.730 --> 00:30:29.970
is that you are putting you are choosing delta
as small as you want and you are letting mod
00:30:29.970 --> 00:30:39.100
Z minus Z naught less than delta okay so let
me write that down which in turn means
00:30:39.100 --> 00:30:55.379
means let delta be small consider mod Z minus
Z naught less than delta and let delta tends
00:30:55.379 --> 00:31:08.580
to 0 this is what it means you are making
something small means you are actually taking
00:31:08.580 --> 00:31:15.929
values of that which are getting closer and
closer to 0 okay and you know therefore you
00:31:15.929 --> 00:31:22.019
know of course all this conveys very clearly
what is happening topologically mod Z minus
00:31:22.019 --> 00:31:27.470
Z naught less than delta is actually a disk,
it is an open disk centred at Z naught radius
00:31:27.470 --> 00:31:28.720
delta.
00:31:28.720 --> 00:31:37.119
It means that you are going infinitesimally
small very very small neighbourhood of Z naught
00:31:37.119 --> 00:31:42.799
and the smaller the delta is the smaller the
neighbour is, so you are basically looking
00:31:42.799 --> 00:31:46.950
at you are just concentrating attention at
a very small neighbourhood of Z naught that
00:31:46.950 --> 00:32:01.940
is what it means okay and now you know in
the same way… so now this this can be
00:32:01.940 --> 00:32:13.549
used to also define what limit Z tends to
infinity means okay so, so limit Z tends to
00:32:13.549 --> 00:32:20.320
infinity, what we make of this? What you make
of limit as Z tends to infinity, okay so now
00:32:20.320 --> 00:32:25.669
this is the point where you will have to have
a little bit of imagination okay, so the idea
00:32:25.669 --> 00:32:30.389
is that 1st of all you should be able to think
of infinity as a point okay as a concrete
00:32:30.389 --> 00:32:36.580
point and the 2nd thing is that once you think
of it as a concrete point in a space then
00:32:36.580 --> 00:32:43.350
you can think of limit Z going to infinity
just as you thought of limit Z going to Z
00:32:43.350 --> 00:32:48.649
naught you know the limit as Z going to Z
naught meant that you were going to a small
00:32:48.649 --> 00:32:51.090
neighbourhood of Z naught.
00:32:51.090 --> 00:32:55.539
Now if you can think of infinity as a point
in the same way limit Z tends to infinity
00:32:55.539 --> 00:33:00.070
means at your going to a small neighbourhood
of infinity okay, so all you need is the way
00:33:00.070 --> 00:33:06.379
of thinking of infinity is a point on a space
where you can think of a small neighbourhood
00:33:06.379 --> 00:33:10.600
around that point okay and the key to this
is as you would have seen in the 1st course
00:33:10.600 --> 00:33:14.330
in complex analysis, the key to this is the
so-called Riemann’s stereographic projection
00:33:14.330 --> 00:33:27.340
okay, so let me explain that, so let me write
this down as go to a small neighbourhood neighbourhood
00:33:27.340 --> 00:33:40.840
of infinity, now what does that mean? What
does that mean? So the key is the Riemann’s
00:33:40.840 --> 00:33:53.400
stereographic projection
that is the key.
00:33:53.400 --> 00:34:02.029
So let me recall what that this, so basically
so the idea is you should have seen this,
00:34:02.029 --> 00:34:13.280
so what you do is the following. So let me
draw a diagram so here is well this is the
00:34:13.280 --> 00:34:23.840
three-dimensional space, so this is my
so let me draw it like this, so this is my
00:34:23.840 --> 00:34:32.609
usual x y plane, this is the origin okay and
this is the x axis and this is the y axis
00:34:32.609 --> 00:34:42.230
okay or rather this is if you want to be right-handed
then that is the y-axis as the negative y-axis
00:34:42.230 --> 00:34:52.710
right, so so if you want the 0.1 is here 0.1
on the y-axis is here, so this point is 1,
00:34:52.710 --> 00:35:06.010
0 this point is 0, 1. So if you think of this
x y plane
00:35:06.010 --> 00:35:15.349
as usual complex plane then you have 2 coordinates
x and y and of course 1, 0 is the 0.1 complex
00:35:15.349 --> 00:35:23.030
number 1, 0, 1 is the complex number I okay
the point is I want to put in a 3rd axis which
00:35:23.030 --> 00:35:26.650
are normally in three-dimensional you would
call the Z axis but you do not want to use
00:35:26.650 --> 00:35:29.329
Z because Z is already supposed to be x plus
I y.
00:35:29.329 --> 00:35:34.819
So you use let us see with something else
for it, you use u if you want okay some books
00:35:34.819 --> 00:35:39.410
use u so let me also use it, so what we do
is now you take the three-dimensional space
00:35:39.410 --> 00:35:45.230
now the snore xyz but it is xyu there is a
positive u axis then what you do is that you
00:35:45.230 --> 00:35:52.660
draw this you draw this circle I mean you
draw this sphere centred at the origin and
00:35:52.660 --> 00:36:00.770
radius 1 okay so so let me rub these coordinates
off because it will make things easier for
00:36:00.770 --> 00:36:09.400
me to draw, so you know draw this, so I have
this circle I have this circle, this is the
00:36:09.400 --> 00:36:20.780
unit circle on the complex plane and then
I have this 0.1 with u coordinate 1 okay and
00:36:20.780 --> 00:36:29.359
x and y coordinates 0 okay, so it will be
the North pole of a sphere of this sphere
00:36:29.359 --> 00:36:37.000
centred at the origin and radius 1, so what
I am going to get is am going to get something,
00:36:37.000 --> 00:36:45.980
so I am going to get something like this,
so here is my sphere.
00:36:45.980 --> 00:36:50.930
So this is this sphere is, this is the sphere
that is called the Riemann’s sphere, it
00:36:50.930 --> 00:36:58.310
is the sphere centred at the origin radius
one unit and this point here which has coordinates
00:36:58.310 --> 00:37:06.569
0, 0, 1 or x y and u is called the North pole
so I will use the word I will put the symbol
00:37:06.569 --> 00:37:12.059
N okay and of course even if you project it
all the way down you are going to get the
00:37:12.059 --> 00:37:19.400
point with ordinate 0, 0, minus 1 which is
the South pole so-called South pole okay you
00:37:19.400 --> 00:37:23.950
can think of the Earth as a sphere and you
have the North pole and the South pole it
00:37:23.950 --> 00:37:35.900
is just like that and so what is stereographic
projection? So what it does is that so let
00:37:35.900 --> 00:37:40.250
me call this sphere let me give you a name
for the sphere, so let me call this as I will
00:37:40.250 --> 00:37:47.569
put this S like a dollar symbol and put S
2 and I will put a subscript R, so this is
00:37:47.569 --> 00:37:49.440
the standard topological notation.
00:37:49.440 --> 00:37:56.890
The 2 on top S is supposed to denote as sphere
okay the 2 on top this supposed to denote
00:37:56.890 --> 00:38:01.349
the dimensions okay it is the surface of the
sphere okay and mind you I am not taking the
00:38:01.349 --> 00:38:05.339
solids sphere I am only taking the surface
of the sphere which is surface okay, so it
00:38:05.339 --> 00:38:10.710
is two-dimensional and by dimension I mean
real dimension, so it is real two-dimensional
00:38:10.710 --> 00:38:16.490
okay and the subscript R is to remind
you that this is being done in real space
00:38:16.490 --> 00:38:31.480
okay, this is being done in real space, real
3 space so the ambience space here is R 3
00:38:31.480 --> 00:38:37.890
, so the ambience space here is R 3 in 3 space
the only thing is that I am treating the x
00:38:37.890 --> 00:38:44.420
y plane as the complex plane and instead of
the usual Z axis I am calling it the U axis
00:38:44.420 --> 00:38:50.559
because Z is already reserved for x plus i
y of now what you do is well this is a stereographic
00:38:50.559 --> 00:38:57.390
projection there is very very simple projection
what it does is that it goes from the sphere
00:38:57.390 --> 00:39:02.240
I will call this sphere the Riemann’s sphere
okay.
00:39:02.240 --> 00:39:12.020
So it is called the Riemann’s sphere okay
it goes from the Riemann’s sphere minus
00:39:12.020 --> 00:39:25.170
the North pole to the complex plane
to the complex plane okay and what is a map?
00:39:25.170 --> 00:39:31.510
So what you do is you take any point on the
sphere okay and mind you you are taking a
00:39:31.510 --> 00:39:36.069
point on the surface of the sphere okay and
you are not taking the point at infinity I
00:39:36.069 --> 00:39:45.410
mean you not taking the point N you are not
taking the point N, so as I just said inadvertently
00:39:45.410 --> 00:39:51.799
the point and will be the missing point at
infinity okay so that will be the analogy
00:39:51.799 --> 00:39:59.250
so we will see that, so you take any point
P here on this sphere other than the North
00:39:59.250 --> 00:40:03.670
pole and then what you do is you joined this
thing the straight line passing through N
00:40:03.670 --> 00:40:09.940
and in P okay that straight line will go down
and hit the plane at some point and that point
00:40:09.940 --> 00:40:14.010
will give you complex number because for me
any point on the plane is a complex number.
00:40:14.010 --> 00:40:17.799
I have thought of the x y plane as a complex
plane and that is the complex number to which
00:40:17.799 --> 00:40:23.070
I am going to send P to okay and you can see
clearly it is a project map okay, so projection
00:40:23.070 --> 00:40:27.940
map and that is why this called the stereographic
projection okay. So so basically what I do
00:40:27.940 --> 00:40:37.260
is that I take this line from N passing through
P and then it will go and hit the let
00:40:37.260 --> 00:40:44.170
me call this as phi of P, so this is the map
phi which sends P to phi of P and phi of P
00:40:44.170 --> 00:40:51.200
is a complex number, phi of P is a complex
number and this is a stereographic projection,
00:40:51.200 --> 00:40:58.970
so you have if you want to think of it as
a projection it is like this okay.
00:40:58.970 --> 00:41:08.450
Now now the beautiful thing is that so the
beautiful thing is that this map phi is actually
00:41:08.450 --> 00:41:16.770
a Bijective map you can very will see that
if P changes then phi P will change so it
00:41:16.770 --> 00:41:22.970
is an map okay and conversely give me any
point on the complex plane it is so the form
00:41:22.970 --> 00:41:28.230
phi P for a unique point on this on the Riemann
sphere because I can get that point by simply
00:41:28.230 --> 00:41:33.760
joining that point to the North pole that
will hit the sphere at a certain point and
00:41:33.760 --> 00:41:38.760
that will be the point which will be mapped
to the given point under phi okay so it is
00:41:38.760 --> 00:41:45.770
very clear that this map is Bijective okay
it is very clear that this map is Bijective
00:41:45.770 --> 00:41:53.549
and in fact you can try it out as an exercise
this map is actually a homeomorphism there
00:41:53.549 --> 00:41:55.510
is a topological isomorphism.
00:41:55.510 --> 00:42:02.980
See the complex plane as a topology and this
topology is also a topology that it is a same
00:42:02.980 --> 00:42:08.910
as a topology that the plane inherits as a
subspace of three-dimensional space okay you
00:42:08.910 --> 00:42:16.059
take the x y plane take the complex plane
x y plane and treated as plane in 3 space
00:42:16.059 --> 00:42:21.779
and you take the natural topology on R 3 you
restrict that topology to the subset that
00:42:21.779 --> 00:42:27.869
is the same as the topology on the complex
plane okay, so the topology on the complex
00:42:27.869 --> 00:42:34.380
plane is same as the topology that it inherits
as a subspace from 3 space and this sphere
00:42:34.380 --> 00:42:42.589
is also living in 3 space, so it is also subset
of the 3 space so it also has inherits a topology
00:42:42.589 --> 00:42:51.300
okay and the fact is that this map from this
sphere to the plane is in fact continues Bijective
00:42:51.300 --> 00:42:57.130
map which is open and therefore you know its
inverse is also continuous so it is homeomorphism
00:42:57.130 --> 00:43:00.160
it is a topological isomorphism there is a
map which is continuous whose inverse is also
00:43:00.160 --> 00:43:04.529
continuous okay and of course the inverse
being defined because the map is Bijective
00:43:04.529 --> 00:43:06.549
okay.
00:43:06.549 --> 00:43:13.180
So the beautiful thing is that phi is actually
homeomorphism okay so the fact that phi is
00:43:13.180 --> 00:43:19.950
a homeomorphism so let me write that down
but before that let me tell you something
00:43:19.950 --> 00:43:24.200
the fact that phi is a homeomorphism tells
you that therefore you can think of the whole
00:43:24.200 --> 00:43:32.260
complex plane as a punctured sphere okay see
S2 minus N is a punctured sphere, it is the
00:43:32.260 --> 00:43:37.640
sphere minus the North pole and what this
homeomorphism tells you, what does a homeomorphism
00:43:37.640 --> 00:43:41.300
tells you? It tells you that the 2 spaces
are topologically the same, same means up
00:43:41.300 --> 00:43:45.960
to isomorphism, so when you say phi is an
isomorphism, topological isomorphism namely
00:43:45.960 --> 00:43:49.370
homeomorphism you are actually saying that
you get this is just another way of saying
00:43:49.370 --> 00:43:55.280
that the complex explain can be thought of
topologically as a punctured sphere that is
00:43:55.280 --> 00:43:58.380
the significance of this statement okay.
00:43:58.380 --> 00:44:10.160
So let me write that down, so here let me
take another color phi is check so this check
00:44:10.160 --> 00:44:14.690
is something that is more or less obvious
but you should do is phi is a homeomorphism
00:44:14.690 --> 00:44:25.730
okay I am certain many of you would have done
this in the 1st course in complex analysis
00:44:25.730 --> 00:44:33.670
but it is not very difficult to do if you
have not done it, So phi is a homeomorphism
00:44:33.670 --> 00:44:37.849
which tells you that the complex plane can
be thought of as a punctured sphere okay,
00:44:37.849 --> 00:44:44.880
now the punctured sphere this is only 1 point
namely the North pole and now you know you
00:44:44.880 --> 00:44:51.609
have you wanted a point at infinity you wanted
to attach to the complex plane a point at
00:44:51.609 --> 00:44:53.910
infinity but the point is where do you attach
it?
00:44:53.910 --> 00:44:59.329
I mean you cannot see it okay but then if
you look at this picture you think of the
00:44:59.329 --> 00:45:04.150
complex plane like a punctured sphere now
what is the extra point that you will have
00:45:04.150 --> 00:45:08.950
to add to make it the whole sphere and mind
you when you make it the whole sphere only
00:45:08.950 --> 00:45:15.800
then it becomes compact okay if you remove
a point from a sphere it loses compactness
00:45:15.800 --> 00:45:23.380
okay because it will not be closed since we
are in Euclidean space a subset is compact
00:45:23.380 --> 00:45:27.520
if you know only if it is closed and bounded,
so this sphere this of course any subset of
00:45:27.520 --> 00:45:31.559
the sphere is of course bounded but the problem
is that unless it is closed it is not compact,
00:45:31.559 --> 00:45:37.589
so the only way to make it compact is to hand
that missing point that in this case it is
00:45:37.589 --> 00:45:39.000
the North pole okay.
00:45:39.000 --> 00:45:49.960
So here comes the nice upshot of this what
you do is you think of the complex plane plus
00:45:49.960 --> 00:45:55.510
the point at infinity you denote the point
at infinity as with the symbol of infinity
00:45:55.510 --> 00:45:59.559
and think of it as an extra point that you
add to the set of complex numbers and then
00:45:59.559 --> 00:46:06.400
what the topology you give, you give the topology
which makes the natural extension of this
00:46:06.400 --> 00:46:12.240
homeomorphism phi into a homeomorphism from
this sphere to the complex plane which is
00:46:12.240 --> 00:46:27.520
a complex plane with that extra point at infinity
added okay, so define phi from the Riemann
00:46:27.520 --> 00:46:44.440
sphere to C union infinity
by sending N to the point at infinity okay.
00:46:44.440 --> 00:46:51.730
So the North pole maps to the point at infinity
and once you do this what we have done is
00:46:51.730 --> 00:46:58.570
you have given a Bijection between this sphere
and C union infinity mind you C union infinity
00:46:58.570 --> 00:47:01.990
will be called the extended complex explain,
it is the complex plane plus the point at
00:47:01.990 --> 00:47:07.790
infinity and therefore the complex plane plus
the point at infinity is now nicely identified
00:47:07.790 --> 00:47:16.029
as a sphere okay and the advantage now is
that you have therefore the point at infinity
00:47:16.029 --> 00:47:20.789
being thought of as the North pole on the
sphere and it is a point on the topological
00:47:20.789 --> 00:47:26.510
space you can do topology in a neighbourhood
of the North pole and think of it as doing
00:47:26.510 --> 00:47:32.700
as working in a neighbourhood of infinity
okay that is how you think of infinity the
00:47:32.700 --> 00:47:35.959
point at infinity, so so I will stop here.