WEBVTT
Kind: captions
Language: en
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Okay so welcome again to lectures on advance
complex Analysis, so what we are going to
00:00:50.660 --> 00:01:00.930
do today is ask very basic question okay so
let me switch to the writing board.
00:01:00.930 --> 00:01:18.960
So suppose suppose f from D to C is an analytic
function, of course the analytic means the
00:01:18.960 --> 00:01:35.470
same as holomorphic, so analytic is the same
as holomorphic and always as usual we assume
00:01:35.470 --> 00:01:47.280
D is a domain in the complex plane okay so
D domain in the complex plane so that means
00:01:47.280 --> 00:02:03.530
that D is a subset of the complex plane D
is open, D is connected and of course you
00:02:03.530 --> 00:02:13.230
know D is certainly non-empty I mean because
by definition you know the empty set is also
00:02:13.230 --> 00:02:19.980
open okay, so of course we are not interested
in looking at the empty set, so D is connected
00:02:19.980 --> 00:02:27.400
and of course you know in this context that
for an open set connectedness is equivalent
00:02:27.400 --> 00:02:34.610
to path connectedness, so D is also part connected
so so you have a function f, f is analytic,
00:02:34.610 --> 00:02:42.330
f is a function which is complex value function,
it is a complex value function of 1 complex
00:02:42.330 --> 00:02:45.980
variable and that one complex variable you
might call it as Z.
00:02:45.980 --> 00:02:54.030
So you can think of the function as f of the
Z and Z varies over D okay and what is the
00:02:54.030 --> 00:02:55.840
fundamental question that we are asking?
00:02:55.840 --> 00:03:00.440
The fundamental question that we are asking
is what is the image of f?
00:03:00.440 --> 00:03:13.190
So that is the question so here is the
so let me change color to something else so
00:03:13.190 --> 00:03:25.220
here is the question, what is the image of
f?
00:03:25.220 --> 00:03:33.510
So this is the question so in other words
so that is you take you look at f of D okay,
00:03:33.510 --> 00:03:34.510
what is f of D?
00:03:34.510 --> 00:03:42.030
It is set of values of f okay so this is equal
to the set of all f of Z where Z varies in
00:03:42.030 --> 00:03:54.699
D okay this is the set of values of f, this
is all the values that f takes on D okay and
00:03:54.699 --> 00:03:59.370
obviously it takes complex values, so f of
D is a subset of the complex parent, the question
00:03:59.370 --> 00:04:11.280
is what kind of a subset is this okay so here
yes so what is the image of f that is
00:04:11.280 --> 00:04:25.810
that is if f of D is equal to this then what
is the nature, what is the nature of f of
00:04:25.810 --> 00:04:27.810
D.
00:04:27.810 --> 00:04:35.430
So when I say what is the nature of f of D
you know you ask what do you mean by nature?
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Nature of course one can ask a lot of things
one is of course topological nature okay,
00:04:41.440 --> 00:04:46.130
so f of D is subset of the complex parent
you know complex plane is a topological space
00:04:46.130 --> 00:04:52.460
so you can ask whether f of D is open, whether
f of D has any one of these properties that
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substance of a topological space satisfy okay.
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Properties that you know are open sets, sets
being opened, sets being closed sets being
00:05:00.720 --> 00:05:05.060
connected, sets being part connected, sets
being compact and so on okay.
00:05:05.060 --> 00:05:11.560
So you can is what is the nature of this set
of f of D is in the topological okay then
00:05:11.560 --> 00:05:15.669
you can ask another question, how big is f
of D okay what is the…
00:05:15.669 --> 00:05:22.460
How much of it or how big it is when compared
to the whole complex plane okay, so so when
00:05:22.460 --> 00:05:35.960
I say nature I can ask topological and the
other thing is how big, how big is it?
00:05:35.960 --> 00:05:47.960
So and it so happens that complex Analysis
gives you several nice theorems which answers
00:05:47.960 --> 00:05:52.680
go along way and answering these questions
okay.
00:05:52.680 --> 00:06:02.800
So let me you know so let me go ahead and
look at in just a minute let me resize
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the screen so that I get…
00:06:08.340 --> 00:06:14.039
So and so let me look at the following
let us ask them simple questions let us ask
00:06:14.039 --> 00:06:22.360
them simple questions so there are 1st few
obvious things that you can say, see f is
00:06:22.360 --> 00:06:30.629
an analytical function so you know analytical
functions are in fact infinitely differentiable
00:06:30.629 --> 00:06:35.479
that is what you learn in the 1st course in
complex analysis once differentiability implies
00:06:35.479 --> 00:06:40.169
infinite differentiability on an open set
and that is one of the characteristics properties
00:06:40.169 --> 00:06:45.249
of an analytic function and therefore in particular
they are certainly continuous and you know
00:06:45.249 --> 00:06:49.830
continuity preserve certain properties or
example the continues image of connected set
00:06:49.830 --> 00:06:54.639
is connected, continues image of a path connected
set is path connected, the continues image
00:06:54.639 --> 00:07:02.310
of compact set is compact so you can say immediately
that f of D is certainly a connected set okay
00:07:02.310 --> 00:07:07.499
so that is very very basic topology it just
uses the fact that the continues image of
00:07:07.499 --> 00:07:09.099
a connected set is connected.
00:07:09.099 --> 00:07:18.659
So so let me say the following thing of course
f is continues so you know I will use some
00:07:18.659 --> 00:07:33.050
abbreviations CTS assistance for continuous,
so f is continuous so f of D is is connected
00:07:33.050 --> 00:07:39.629
this is of course very very basic well then
you want to know more so you know let us let
00:07:39.629 --> 00:07:46.710
us try to see lets ask a few questions, so
for example we ask a question like can f take
00:07:46.710 --> 00:07:54.539
values on a line or can f take all values
only on the curve okay, so so let us ask this
00:07:54.539 --> 00:08:11.810
question, so so let me again change color,
can f take values only on a curve?
00:08:11.810 --> 00:08:16.460
Of course with the curve I mean any simple
curve that can think of like a parabola or
00:08:16.460 --> 00:08:18.360
a circle or something like that.
00:08:18.360 --> 00:08:23.419
I mean particularly also it could very well
be a straight line this is also curve okay
00:08:23.419 --> 00:08:34.560
so can f take values on the curve, so say
for example you know suppose I take, so example
00:08:34.560 --> 00:08:51.410
so let me look at a few examples can f takes
values on the real line?
00:08:51.410 --> 00:08:57.780
I mean take only values on the real line this
means you are saying that the image of f is
00:08:57.780 --> 00:09:08.230
the subset of the real line okay and you know
if f takes values only on the real line this
00:09:08.230 --> 00:09:14.360
is equivalent to saying that the imaginary
part of f is 0 okay because if f is a complex
00:09:14.360 --> 00:09:18.760
valued function, normally we write f is equal
to U plus IV where U is the real part of f
00:09:18.760 --> 00:09:23.550
and V is the imaginary part of f and you know
very well from the 1st course in complex analysis
00:09:23.550 --> 00:09:30.670
that you and we have to you know be harmonic
and in fact they will satisfy the Cauchy Riemann
00:09:30.670 --> 00:09:36.980
equations okay because f is analytic but the
point is that if you say that f takes only
00:09:36.980 --> 00:09:40.839
values on the real line it means you are saying
that V is always 0 that means the imaginary
00:09:40.839 --> 00:09:41.839
part of f is 0.
00:09:41.839 --> 00:09:52.380
So this equivalent to saying that you know
imaginary part of f is 0 okay and you know
00:09:52.380 --> 00:10:07.860
I can also ask can f take a values on the
imaginary axis
00:10:07.860 --> 00:10:14.570
okay that is the y-axis consider it as there
imaginary axis on the complex plane okay and
00:10:14.570 --> 00:10:20.970
that is equal in to saying that the real part
of f is 0 okay, so there is another case of
00:10:20.970 --> 00:10:25.250
f taking values only on a line then of course
they can ask can f takes values on a circle
00:10:25.250 --> 00:10:27.940
okay, so let me ask that also.
00:10:27.940 --> 00:10:39.250
Can f take only values on a circle so
you know if you think that the circle is centred
00:10:39.250 --> 00:10:43.680
at the point w naught and has the radius R
naught, this is equivalent to saying that
00:10:43.680 --> 00:10:50.110
the modulus of f of modulus of f minus w naught
is equal to R naught this is what it means,
00:10:50.110 --> 00:10:53.910
this is the condition that f takes values
on a circle okay.
00:10:53.910 --> 00:11:01.089
Now surprisingly, not surprisingly in fact
there is something you should have seen if
00:11:01.089 --> 00:11:06.420
you just recall that these are all the conditions
that will ensure that the derivatives of f
00:11:06.420 --> 00:11:12.410
vanishes and therefore f has to be constant,
so if the imaginary part of f is 0 at is the
00:11:12.410 --> 00:11:15.670
same as saying the imaginary part as a constant
okay.
00:11:15.670 --> 00:11:20.500
It is a special case of the fact that the
imaginary part of the constant and if the
00:11:20.500 --> 00:11:24.399
constant is 0 this is equivalent to saying
that amounts to saying that the imaginary
00:11:24.399 --> 00:11:30.089
part of f is 0 which means it takes for the
real valued and if the real part of f is 0
00:11:30.089 --> 00:11:38.850
that is a special case of the real part being
constant okay and the f taking values on a
00:11:38.850 --> 00:11:43.860
circle is the same as saying that the function
f minus w naught which is f added to minus
00:11:43.860 --> 00:11:48.130
w naught which is a constant function okay.
00:11:48.130 --> 00:11:53.480
That has constant modulus that modulus is
R naught okay now you have done this in the
00:11:53.480 --> 00:11:58.160
1st course in complex analysis probably by
using Cauchy Riemann equations that you know
00:11:58.160 --> 00:12:06.160
if the function has imaginary part constant
or the real part constant or the modulus constant
00:12:06.160 --> 00:12:10.630
in the function has to be constant okay so
all these things can happen only if f is constant
00:12:10.630 --> 00:12:26.970
okay so all these all these can happen only
when f is a constant, f is a constant function
00:12:26.970 --> 00:12:33.029
okay takes the same value, so of course therefore
the question is we are not certainly we are
00:12:33.029 --> 00:12:37.180
not interested in studying constant functions
because there is nothing special about these
00:12:37.180 --> 00:12:42.130
constant functions they constant function
maps the whole plane onto a single point which
00:12:42.130 --> 00:12:46.950
is the value of the function of that constant,
we are not interested in…so we are not interested
00:12:46.950 --> 00:12:51.180
in such constant functions, we are worried
about non-constant functions okay.
00:12:51.180 --> 00:12:56.730
So what this will tell you immediately is
that if you have a non-constant analytic function
00:12:56.730 --> 00:13:06.070
okay it cannot take values at least on a line
or something like a circle okay but then so
00:13:06.070 --> 00:13:13.139
what is it so what this tells you is that
the either you have the case that you are
00:13:13.139 --> 00:13:16.920
looking at a constant function in which case
the images a single point okay it is that
00:13:16.920 --> 00:13:25.660
single constant value that it takes or
the image cannot be a set of just the real
00:13:25.660 --> 00:13:30.300
line or it cannot be a subset of only the
imaginary axis, it can be a subset of the
00:13:30.300 --> 00:13:31.300
circle.
00:13:31.300 --> 00:13:35.860
Such things cannot happen that is what this
says okay but what is the that you have more
00:13:35.860 --> 00:13:43.720
generally, so more generally we have you
know we have very nice theorem, so here is
00:13:43.720 --> 00:13:44.910
the theorem.
00:13:44.910 --> 00:13:55.370
So here is the theorem and it is called the
open mapping theorem and is a very very fundamental
00:13:55.370 --> 00:14:06.480
theorem, what it says is that non-constant
analytic map is always an open map okay, so
00:14:06.480 --> 00:14:36.460
if f from D to C is a non-constant analytic
map then f is an open map okay it is an open
00:14:36.460 --> 00:14:39.360
map, so so let us try to understand what this
means?
00:14:39.360 --> 00:14:41.210
It means that see what is an open map?
00:14:41.210 --> 00:14:47.529
An open map is a map which for which the image
of any open set is again an open set okay,
00:14:47.529 --> 00:14:54.130
so in particular what this will tell you is
that f of D is an open set because D is already
00:14:54.130 --> 00:15:00.160
a domain, the D is the domain so D is already
an open connected set, so f of D will become
00:15:00.160 --> 00:15:05.320
open okay and it is already connected so it
is the same as it being path connected so
00:15:05.320 --> 00:15:10.420
f of D is again a domain okay so what this
what, so let us try to understand what this
00:15:10.420 --> 00:15:11.420
means?
00:15:11.420 --> 00:15:25.019
If f of D is open so it is a domain so that
is something that comes immediately okay and
00:15:25.019 --> 00:15:28.810
mind you f is not a constant function, so
it takes more than one value so f of D is
00:15:28.810 --> 00:15:37.980
non-empty of course okay and the other important
thing is the following thing.
00:15:37.980 --> 00:15:41.509
What is the condition of open mapping?
00:15:41.509 --> 00:15:47.360
If you take U a subset of D which is an open
subset then f of U would also continue to
00:15:47.360 --> 00:16:03.029
be open okay so if U is subset of D is open
then f of U is open, this is what an open
00:16:03.029 --> 00:16:05.170
map means it maps open sets to open sets.
00:16:05.170 --> 00:16:08.410
The image of an open set under an open map
is again an open set that is the definition
00:16:08.410 --> 00:16:11.709
of what an open map is?
00:16:11.709 --> 00:16:22.029
Okay and…so let us go a little bit more
into this and you know try to see what it
00:16:22.029 --> 00:16:25.329
really means, what is the meaning of an open
set?
00:16:25.329 --> 00:16:30.730
An open set is a set which every point is
an integer point okay that means you take
00:16:30.730 --> 00:16:35.430
any point in the open set then there is a
small disk open disk surrounding that point
00:16:35.430 --> 00:16:41.209
which is also in that set that is what an
integer point means okay, so so what is f
00:16:41.209 --> 00:16:48.639
being open mean suppose f takes a certain
value w let us say takes a value w naught
00:16:48.639 --> 00:16:55.769
okay then there is a it means you are saying
w naught belongs to the image of f okay if
00:16:55.769 --> 00:17:01.639
f takes the value w naught okay that means
w naught is in the image of f because image
00:17:01.639 --> 00:17:08.720
of f just consist of the values of f okay
and then but since the image of f is open
00:17:08.720 --> 00:17:14.069
w naught is a point of an open set therefore
there is a small open disk centered at w naught
00:17:14.069 --> 00:17:16.060
which is also in the image.
00:17:16.060 --> 00:17:23.540
So it means that if f takes a certain value
will take all values in a small disk surrounding
00:17:23.540 --> 00:17:29.940
that value okay so this should immediately
tell you that f cannot simply take values
00:17:29.940 --> 00:17:35.940
on the curve because the moment f takes values
at the point will take all values in a small
00:17:35.940 --> 00:17:40.490
disk surrounding that point and you know no
curve can accommodate a small disk however
00:17:40.490 --> 00:17:47.700
small okay therefore you immediately get this
idea that you know the image of an analytic
00:17:47.700 --> 00:17:51.610
mapping, non-constant analytic mapping cannot
go into a curve, we saw special cases, we
00:17:51.610 --> 00:17:55.670
saw that it cannot go on into a real axis,
it cannot go into the imaginary axis, it cannot
00:17:55.670 --> 00:18:01.590
be a circle okay and go into the circle but
is more generally the reason is the image
00:18:01.590 --> 00:18:09.000
is open okay and of course curves are closed
sets okay.
00:18:09.000 --> 00:18:22.230
So let me write that now if w naught is
equal to f of Z naught for Z naught in D that
00:18:22.230 --> 00:18:35.790
is the same as saying that w naught is an
image of f which is f of D then f of D being
00:18:35.790 --> 00:18:58.080
open implies that there exist small open disk
in f of D containing w naught and that implies
00:18:58.080 --> 00:19:24.730
that f takes all values in a small disk centred
at w naught, so this is what is very very
00:19:24.730 --> 00:19:32.120
important if f takes certain value it will
take all values in a disk about that value
00:19:32.120 --> 00:19:37.260
okay this is a very very important property
and this is true for of course for a non-constant
00:19:37.260 --> 00:19:40.470
analytic function okay.
00:19:40.470 --> 00:19:49.500
So so this is about this is about at the moment
this is about the topological property of
00:19:49.500 --> 00:19:56.050
f of D this theorem tells you that f of D
the image of f is certainly a domain it is
00:19:56.050 --> 00:19:57.050
an open connected set.
00:19:57.050 --> 00:20:05.780
It is very very important that it is an open
set and in fact going into a higher geometric
00:20:05.780 --> 00:20:12.070
point of view okay what actually happens is
this so let me tell it to you in words, what
00:20:12.070 --> 00:20:18.680
actually happens is that the mapping f from
D to f of D becomes what is called ramified
00:20:18.680 --> 00:20:29.350
cover of Riemann surfaces okay so it means
that it there are set of points these are
00:20:29.350 --> 00:20:34.660
the points where the derivatives of f vanishes
okay these are called the points of the ramification
00:20:34.660 --> 00:20:39.810
and outside those points in the complement
of those points this is actually a covering
00:20:39.810 --> 00:20:45.400
map okay it is covering map in the topological
and is and also in the analytical or holomorphic
00:20:45.400 --> 00:20:52.950
sense okay so this open mapping theorem this
so important that tells you that essentially
00:20:52.950 --> 00:21:00.120
every analytic non-constant analytic mapping
is ramified cover okay fine so now what I
00:21:00.120 --> 00:21:06.530
am going to do is I am going to go to ask
a more specific question.
00:21:06.530 --> 00:21:12.680
So we are trying to look at the image of
a domain under analytic function, so let us
00:21:12.680 --> 00:21:17.760
look at the cases where 1st at the case where
you know the function is analytic on the whole
00:21:17.760 --> 00:21:21.420
plane so these are the entire functions, so
what is an entire function?
00:21:21.420 --> 00:21:26.190
An entire function is a complex valued function
which is analytic on the whole plane okay
00:21:26.190 --> 00:21:33.190
then the question is what is the what is the
image of sets of functions?
00:21:33.190 --> 00:21:41.760
So that is a very deep theorem namely it is
the so-called little Picard theorem which
00:21:41.760 --> 00:21:48.210
says that the image is either the whole complex
plane or it is the punctured plane, it is
00:21:48.210 --> 00:21:52.510
the punctured plane namely it is the complex
plane minus single point that means an entire
00:21:52.510 --> 00:22:01.580
function okay we will take all values except
for omitting at most one value okay and this
00:22:01.580 --> 00:22:03.680
is called the Little Picard theorem okay.
00:22:03.680 --> 00:22:21.420
So let me state that
so here is the Little Picard theorem, sometimes
00:22:21.420 --> 00:22:34.930
people also use the adjective Small Picard
theorem, so what is this?
00:22:34.930 --> 00:22:57.940
If f from C to C is analytic that is so let
me write it here f is entire
00:22:57.940 --> 00:23:20.040
then either f of C is equal to C or f of C
is equal to minus w naught for some w naught
00:23:20.040 --> 00:23:24.950
C so this is a little Picard theorem, so you
know it is the it is a very tremendous theorem
00:23:24.950 --> 00:23:31.070
it says that you take an entire function,
you take the image, the image is huge I mean
00:23:31.070 --> 00:23:38.500
the image is literally everything at the worst
if the image omits it can omit only one
00:23:38.500 --> 00:23:45.090
value okay and the case where the image omits
a single value is of course the simplest example
00:23:45.090 --> 00:23:49.040
is that of the exponential function, you know
if you take the function Z going to E power
00:23:49.040 --> 00:23:57.630
Z that is an entire function okay and the
image will not it will be the whole punctured.
00:23:57.630 --> 00:24:01.010
It will be the punctured plane it will be
the complex plane minus the origin because
00:24:01.010 --> 00:24:05.790
exponential function will never take the value
0 because 0 does not have a logarithm okay
00:24:05.790 --> 00:24:15.250
so if you take any non-zero complex number
you can find the logarithm and exponential
00:24:15.250 --> 00:24:19.460
of that logarithm give you back that complex
number of course you will get many logarithms
00:24:19.460 --> 00:24:26.370
okay but you can find at least one for a nonzero
complex number, so it means that the exponential
00:24:26.370 --> 00:24:34.120
function will take all values except 0 okay
and that is the…so in that case it is an
00:24:34.120 --> 00:24:40.550
example that illustrates Picard theorem if
you take f of Z equal to E power Z then the
00:24:40.550 --> 00:24:44.740
image of f is actually C minus 0 which is
a punctured plane.
00:24:44.740 --> 00:24:48.880
Normally if you take the whole complex plane
and remove single point that is called the
00:24:48.880 --> 00:24:54.880
punctured plane okay with puncher at that
point because that point is being removed
00:24:54.880 --> 00:24:59.850
and of course there is also the case when
function an entire function can take all
00:24:59.850 --> 00:25:07.890
values, the simplest case is that of a polynomial,
so if you take a polynomial if you take
00:25:07.890 --> 00:25:14.800
f of Z equal to P of Z where P of Z is a polynomial
then it will take all values because I can
00:25:14.800 --> 00:25:23.640
always solve for P of Z equal to w naught
for any w naught and that is because of the
00:25:23.640 --> 00:25:28.920
fundamental theorem of algebra namely that
the complex numbers of algebra closed so I
00:25:28.920 --> 00:25:34.930
can always solve a polynomial equation in
one variable okay so a polynomial is also
00:25:34.930 --> 00:25:40.420
an entire function and it is it gives the
case the 1st case namely the image of the
00:25:40.420 --> 00:25:45.670
whole complex plane is the whole complex plane
okay fine so this is the little Picard theorem.
00:25:45.670 --> 00:25:53.350
Now somehow what I want to do is I want to
really to prove this okay it is a deep theorem
00:25:53.350 --> 00:25:57.410
normally this theorem is only stated in the
1st course in complex analysis but since this
00:25:57.410 --> 00:26:03.520
is advanced was in complex analysis I
think it is fitting to look at a proof of
00:26:03.520 --> 00:26:04.550
this.
00:26:04.550 --> 00:26:12.070
Now well you know interestingly it is very
interesting that the proof of this that I
00:26:12.070 --> 00:26:19.880
am going to present is actually gotten by
deriving this as a corollary were much more
00:26:19.880 --> 00:26:27.271
deeper theorem, it is called the big Picard
theorem and the funny thing is that the big
00:26:27.271 --> 00:26:36.270
Picard theorem is a theorem which deals which
again asked the same kind of questions, it
00:26:36.270 --> 00:26:40.440
answers the same kind of questions namely
what is the image of a domain under an analytic
00:26:40.440 --> 00:26:49.900
map okay but the point is that the domain
you are looking at is a disk around an essential
00:26:49.900 --> 00:26:54.120
singularity of an analytic function.
00:26:54.120 --> 00:27:07.950
So you know so let me state that so here is
so let me use something else this will be
00:27:07.950 --> 00:27:40.710
deduced from the from the big or great Picard
theorem and that is let so here is the statement
00:27:40.710 --> 00:27:57.000
of the theorem, let Z naught be an isolated
essential singularity
00:27:57.000 --> 00:28:20.820
of an analytic function f okay then f of…so
let me write this 0 less than mod Z minus
00:28:20.820 --> 00:28:35.140
Z naught less than Epsilon is equal to C or
C minus single value for every Epsilon greater
00:28:35.140 --> 00:28:59.580
than 0 in the domain of analyticity of f okay
so I have just stated a part of the theorem
00:28:59.580 --> 00:29:05.790
there is still more to it, so I want you to
look at this.
00:29:05.790 --> 00:29:10.660
What I want you to appreciate is I wanted
to appreciate the following things to reduce
00:29:10.660 --> 00:29:15.510
the little Picard theorem which is theorem
about a function is analytic on the whole
00:29:15.510 --> 00:29:20.930
plane if function is mind you if a function
is analytic on the whole plane it has no singularities
00:29:20.930 --> 00:29:21.930
okay.
00:29:21.930 --> 00:29:26.870
It has no singular points okay so the little
Picard theorem is a theorem about a function
00:29:26.870 --> 00:29:32.770
which has no singular points okay and it says
that the image of the whole plane under
00:29:32.770 --> 00:29:37.370
such a function is either the whole plane
or a punctured plane okay but we are deducing
00:29:37.370 --> 00:29:45.890
it from a theorem about the image of function
with a singularity, so that is the funny thing
00:29:45.890 --> 00:29:52.080
so it is like even to answer question about
an entire function you are forced to study
00:29:52.080 --> 00:29:55.020
singularities this is the point I want you
to understand okay.
00:29:55.020 --> 00:30:00.390
See normally we would not like to dirty our
hands with singularities, why study singularities
00:30:00.390 --> 00:30:05.500
when there are functions without singularities
but the point is you know sometimes mathematics
00:30:05.500 --> 00:30:10.450
and theory teaches us that even to study good
things we have to study bad things okay so
00:30:10.450 --> 00:30:14.680
if you want to prove the little Picard theorem
which is a theorem about good things I mean
00:30:14.680 --> 00:30:21.670
the function is analytic entire you have to
still study functions which are having singularities
00:30:21.670 --> 00:30:27.050
and so here is the big Picard theorem and
obviously you know the adjective great or
00:30:27.050 --> 00:30:33.460
big should tell you that this big Picard theorem
has to be a Big Brother of the little Picard
00:30:33.460 --> 00:30:38.350
theorem and therefore you know the little
Picard theorem can be reduce from the support
00:30:38.350 --> 00:30:43.240
of the Big Brother and what is this big Picard
theorem, what does it say?
00:30:43.240 --> 00:30:48.000
See you are looking at an analytic function
okay and you were looking at a singularity
00:30:48.000 --> 00:30:57.740
of an analytic function okay now so I come
later to what a singularity is?
00:30:57.740 --> 00:31:04.380
Okay because that is motivation for me to
recall these things okay so you look at a
00:31:04.380 --> 00:31:10.840
function f which at a point has isolated singularity,
isolated means there is a whole his surrounding
00:31:10.840 --> 00:31:18.220
that point where there are no other singularities
okay and a deleted disk surrounding that point
00:31:18.220 --> 00:31:28.470
is given in this form as I have written here
in the on my board zero strictly less than
00:31:28.470 --> 00:31:35.340
mod Z minus Z naught strictly less than Epsilon
is actually the disk centred at Z naught the
00:31:35.340 --> 00:31:40.810
radius Epsilon is an open disk but I have
thrown out Z naught that is the reason for
00:31:40.810 --> 00:31:46.060
putting 0 strictly less than I am not allowing
Z equal to Z naught that means it is a punctured
00:31:46.060 --> 00:31:47.060
disk.
00:31:47.060 --> 00:31:52.040
It is a punctured disk centred at Z naught
and the punctured is exactly at Z naught I
00:31:52.040 --> 00:31:58.890
have thrown out Z naught okay and on this
disk the I am assuming that this disk is full
00:31:58.890 --> 00:32:05.600
of points where function is analytic okay
and that will be through at least for
00:32:05.600 --> 00:32:10.370
small values of Epsilon greater than 0 because
the point Z naught is an isolated singularity
00:32:10.370 --> 00:32:17.480
okay and look at what the theorem says, it
says you take the image of this when I write
00:32:17.480 --> 00:32:25.830
f of something okay it means f of this set
which is the punctured disk that is the whole
00:32:25.830 --> 00:32:35.360
complex plane or it is a complex plane minus
a single point and this is true for Epsilon
00:32:35.360 --> 00:32:40.140
sufficiently small and therefore it will be
true for even larger Epsilon so long as this
00:32:40.140 --> 00:32:46.470
deleted disk is in the domain of analyticity
of f because larger disk, larger deleted disk
00:32:46.470 --> 00:32:50.150
will contain smaller deleted disk and therefore
their images will contain images of smaller
00:32:50.150 --> 00:32:53.880
deleted disk okay so.
00:32:53.880 --> 00:33:02.110
So you see this is again you see the result
of the conclusions of the theorem both the
00:33:02.110 --> 00:33:06.380
big Picard theorem and the little Picard theorem
they are the same I mean the conclusion always
00:33:06.380 --> 00:33:12.000
says that the image under analytic function
of a certain domain okay is either the whole
00:33:12.000 --> 00:33:17.280
complex plane or it is the complex plane minus
a point okay and in case of little Picard
00:33:17.280 --> 00:33:24.220
theorem you are looking at the domain is the
whole complex plane but in the case of great
00:33:24.220 --> 00:33:31.700
Picard theorem the domain is a very small
neighbourhood deleted neighbourhood of an
00:33:31.700 --> 00:33:36.030
essential singularity of an analytic function
and what is really amazing is in fact there
00:33:36.030 --> 00:33:40.570
is more to this Picard theorem what it says
is you see so I want you all to observe is
00:33:40.570 --> 00:33:45.840
the following thing it is a very very
deep results.
00:33:45.840 --> 00:33:51.540
It says take a very small neighbourhood of
the essential singularity okay deleted neighbourhood
00:33:51.540 --> 00:33:57.110
that means of course you do not take the neighbourhood
that you take should be a domain weather function
00:33:57.110 --> 00:34:02.210
is analytic so it cannot include the singularity,
so when I say take a neighbourhood of essential
00:34:02.210 --> 00:34:06.980
singularity of course I mean delete that essential
singularity so you are taking a deleted neighbourhood
00:34:06.980 --> 00:34:11.890
of the essential singularity and mind you
you a neighbourhood as small as I want, you
00:34:11.890 --> 00:34:17.829
see this Epsilon can be extremely small okay
and the theorem is amazing it says you take
00:34:17.829 --> 00:34:23.540
no matter how smaller neighbourhood you take,
the image of that neighbourhood is still the
00:34:23.540 --> 00:34:27.129
whole plane no matter how small your neighbourhood
is.
00:34:27.129 --> 00:34:32.390
The image of the very small neighbourhood
no matter how small is still the whole plane
00:34:32.390 --> 00:34:39.500
it still takes all those values so what this
tells you is you know it tells you that it
00:34:39.500 --> 00:34:46.590
tells you how the values of analytic functions
change in a neighbourhood of an essential
00:34:46.590 --> 00:34:51.480
singularity, in our neighbourhood of an essential
singularity this analytic function is taking
00:34:51.480 --> 00:35:01.910
all values at the worst it can omit one value
okay and of course you know the example for
00:35:01.910 --> 00:35:09.079
this is just as in the case of the little
Picard theorem where you where the example
00:35:09.079 --> 00:35:14.579
of an entire function omitting a value is
exponentially E power Z a which omits the
00:35:14.579 --> 00:35:19.599
value 0, here you can take E power 1 by Z
okay you can take the function E power 1 by
00:35:19.599 --> 00:35:25.539
Z and this E power 1 by Z and E power 1 by
Z at Z equal to 0 has an essential singularity
00:35:25.539 --> 00:35:32.190
and if you take any small deleted neighbourhood
of 0 however small and you take the image
00:35:32.190 --> 00:35:36.420
under E power 1 by Z you will get the whole
plane except the origin because exponential
00:35:36.420 --> 00:35:38.500
function will never take the value 0.
00:35:38.500 --> 00:35:43.089
So you know it is an amazing, it is an amazing
result, it is an amazing result and in fact
00:35:43.089 --> 00:35:51.180
there is a stronger version of the Picard
theorem which says that not only does the
00:35:51.180 --> 00:36:02.740
image of any small neighbourhood however small
of an essential singularity under under
00:36:02.740 --> 00:36:09.839
analytic function is a whole plane or plane
minus it says it takes the every complex value
00:36:09.839 --> 00:36:17.380
that it takes it takes infinitely many times,
so there is in fact so that we write that
00:36:17.380 --> 00:36:27.400
down just to tell you how powerful the theorem
is so let me write that.
00:36:27.400 --> 00:36:37.349
For every Epsilon greater than 0 such that
0 less than mod Z minus Z naught less than
00:36:37.349 --> 00:36:55.519
Epsilon is in the domain of analyticity
00:36:55.519 --> 00:37:18.710
of f. f assumes each complex value with at
most one exception w naught infinitely many
00:37:18.710 --> 00:37:30.839
times, so in fact this, this infinitely many
a times part of it which tells you the more
00:37:30.839 --> 00:37:40.230
it tells you with lot of force what is happening
so the 1st part of the great Picard theorem
00:37:40.230 --> 00:37:44.710
says that you take an essential singularity
and take a very smaller deleted neighbourhood
00:37:44.710 --> 00:37:51.440
about that, take a very small disk surrounding
the essential singularity and take its image
00:37:51.440 --> 00:37:58.509
under the analytic function of course you
do not take…the analytic function is not
00:37:58.509 --> 00:38:02.810
defined at the singularity okay so you do
not take the value at the singularity there
00:38:02.810 --> 00:38:03.820
is no such thing.
00:38:03.820 --> 00:38:08.840
So you are actually taking the image of a
deleted neighbourhood but the point is no
00:38:08.840 --> 00:38:12.779
matter how small a deleted neighbourhood is,
your image will be the whole complex plane
00:38:12.779 --> 00:38:17.779
or it may be complex plane minus a single
point that is the 1st part of the theorem
00:38:17.779 --> 00:38:22.779
and in fact what this part of the theorem
says that you know you take any value, any
00:38:22.779 --> 00:38:27.359
of the values in the complex plane except
possibly for that one value w naught which
00:38:27.359 --> 00:38:36.690
it will not take okay, take any other of the
values that it takes that value itself if
00:38:36.690 --> 00:38:41.190
you take the pre image of that value in that
neighbourhood, the pre-majors and infinite
00:38:41.190 --> 00:38:47.180
set okay that means there are infinitely many
points even in that small neighbourhood that
00:38:47.180 --> 00:38:52.690
are infinitely many points at which the function
takes place that prescribed that value that
00:38:52.690 --> 00:38:56.960
you are pointing at and this happens for every
value that it takes.
00:38:56.960 --> 00:39:04.859
So what it does it is very funny it looks
as if you take the small neighbourhood around
00:39:04.859 --> 00:39:09.619
the essential singularity, the function not
only match that very small neighbourhood onto
00:39:09.619 --> 00:39:16.809
the whole plane or whole plane minus a point
but it maps it infinitely many times okay
00:39:16.809 --> 00:39:24.970
it is like it maps it thousands and thousands
of time okay and that is an amazing thing
00:39:24.970 --> 00:39:33.720
okay it is not that for every complex value
there is one value here which goes to that,
00:39:33.720 --> 00:39:38.740
the fact is you take any complex value other
than the exceptional value w naught then there
00:39:38.740 --> 00:39:44.310
is infinitely many points in this very small
disk however small where that value is taken
00:39:44.310 --> 00:39:50.440
by f okay so that small neighbourhood it is
really amazing to think of it, think of a
00:39:50.440 --> 00:39:55.839
very small infinitesimally small neighbourhood
which is being again and again you know it
00:39:55.839 --> 00:40:03.770
maps thousands of times I mean probably unaccountably
many number of times on to the whole plane
00:40:03.770 --> 00:40:08.680
or the whole place minus a point that is how
the function behaves in a neighbourhood of
00:40:08.680 --> 00:40:13.730
an essential singularity and this is the key
to…
00:40:13.730 --> 00:40:20.869
This theorem on singularity is the key to
proving or reducing as a corollary the little
00:40:20.869 --> 00:40:30.790
Picard theorem, so we will try to in the forthcoming
lectures will try to give a proof of this
00:40:30.790 --> 00:40:36.970
theorem and so I will tell you roughly I will
give you an idea of where we are going to
00:40:36.970 --> 00:40:41.790
go, so you know 1st of all I want to recall
something about singularities, you would have
00:40:41.790 --> 00:40:47.180
studied singularities but I would like to
recall them and some basic theorems about
00:40:47.180 --> 00:40:51.859
singularities especially the Riemann’s theorem
on removable singularities and then I want
00:40:51.859 --> 00:40:59.850
to reduce from that what is called the weak
version of the big Picard theorem which is
00:40:59.850 --> 00:41:04.720
called the Casorati–Weierstrass theorem
and Casorati–Weierstrass theorem is slightly
00:41:04.720 --> 00:41:14.430
weaker what it says is that while the big
Picard theorem says that a function assumes
00:41:14.430 --> 00:41:20.450
analytical function assumes all values except
with possibly one exception in every neighbourhood
00:41:20.450 --> 00:41:22.070
of an essential singularity.
00:41:22.070 --> 00:41:26.160
What the Casorati–Weierstrass theorem says
is that it will come arbitrarily close to
00:41:26.160 --> 00:41:33.089
every value okay so Casorati–Weierstrass
theorem is a slightly weaker version of the
00:41:33.089 --> 00:41:41.009
great Picard theorem and that can be more
or less reduced using the Riemann theorem
00:41:41.009 --> 00:41:46.999
on removable Singularities which I will prove
okay so I have to recall something about singularities
00:41:46.999 --> 00:41:51.250
but then as we move towards the proof of big
Picard theorem what you have to do is that
00:41:51.250 --> 00:41:55.420
we will have to study not one function but
we will have to study a space of functions
00:41:55.420 --> 00:42:03.650
and we have to study functions with singularities
and the kind of functions you going to study
00:42:03.650 --> 00:42:07.049
are functions with singularities as and these
are called Meromorphic functions.
00:42:07.049 --> 00:42:12.420
So what I am going to do I am going to study
topology of a space of Meromorphic functions
00:42:12.420 --> 00:42:19.859
and prove some fundamental ills like Montel’s
theorems and these are the keys to unlocking
00:42:19.859 --> 00:42:26.640
the proof of the big Picard theorem okay so
what I am going to do in the next few lectures
00:42:26.640 --> 00:42:33.190
his 1st recall singularities then tell you
something about Riemann removable Singularities
00:42:33.190 --> 00:42:38.450
theorem through the Casorati–Weierstrass
theorem and then go onto Meromorphic functions
00:42:38.450 --> 00:42:43.539
studying Meromorphic functions and then trying
to study families of Meromorphic functions
00:42:43.539 --> 00:42:48.619
topologically whether that spaces compact
and things like that okay, so that is the
00:42:48.619 --> 00:42:55.079
that is the direction in which we will be
proceeding so at least the 1st part of these
00:42:55.079 --> 00:42:58.880
lectures… our aim is to prove the great
Picard theorem and you will see on the we
00:42:58.880 --> 00:43:00.569
will prove several other important theorems
okay.