Okay so welcome again to lectures on advance complex Analysis, so what we are going to
do today is ask very basic question okay so let me switch to the writing board.
So suppose suppose f from D to C is an analytic function, of course the analytic means the
same as holomorphic, so analytic is the same as holomorphic and always as usual we assume
D is a domain in the complex plane okay so D domain in the complex plane so that means
that D is a subset of the complex plane D is open, D is connected and of course you
know D is certainly non-empty I mean because by definition you know the empty set is also
open okay, so of course we are not interested in looking at the empty set, so D is connected
and of course you know in this context that for an open set connectedness is equivalent
to path connectedness, so D is also part connected so so you have a function f, f is analytic,
f is a function which is complex value function, it is a complex value function of 1 complex
variable and that one complex variable you might call it as Z.
So you can think of the function as f of the Z and Z varies over D okay and what is the
fundamental question that we are asking?
The fundamental question that we are asking is what is the image of f?
So that is the question so here is the so let me change color to something else so
here is the question, what is the image of f?
So this is the question so in other words so that is you take you look at f of D okay,
what is f of D?
It is set of values of f okay so this is equal to the set of all f of Z where Z varies in
D okay this is the set of values of f, this is all the values that f takes on D okay and
obviously it takes complex values, so f of D is a subset of the complex parent, the question
is what kind of a subset is this okay so here yes so what is the image of f that is
that is if f of D is equal to this then what is the nature, what is the nature of f of
D.
So when I say what is the nature of f of D you know you ask what do you mean by nature?
Nature of course one can ask a lot of things one is of course topological nature okay,
so f of D is subset of the complex parent you know complex plane is a topological space
so you can ask whether f of D is open, whether f of D has any one of these properties that
substance of a topological space satisfy okay.
Properties that you know are open sets, sets being opened, sets being closed sets being
connected, sets being part connected, sets being compact and so on okay.
So you can is what is the nature of this set of f of D is in the topological okay then
you can ask another question, how big is f of D okay what is the…
How much of it or how big it is when compared to the whole complex plane okay, so so when
I say nature I can ask topological and the other thing is how big, how big is it?
So and it so happens that complex Analysis gives you several nice theorems which answers
go along way and answering these questions okay.
So let me you know so let me go ahead and look at in just a minute let me resize
the screen so that I get…
So and so let me look at the following let us ask them simple questions let us ask
them simple questions so there are 1st few obvious things that you can say, see f is
an analytical function so you know analytical functions are in fact infinitely differentiable
that is what you learn in the 1st course in complex analysis once differentiability implies
infinite differentiability on an open set and that is one of the characteristics properties
of an analytic function and therefore in particular they are certainly continuous and you know
continuity preserve certain properties or example the continues image of connected set
is connected, continues image of a path connected set is path connected, the continues image
of compact set is compact so you can say immediately that f of D is certainly a connected set okay
so that is very very basic topology it just uses the fact that the continues image of
a connected set is connected.
So so let me say the following thing of course f is continues so you know I will use some
abbreviations CTS assistance for continuous, so f is continuous so f of D is is connected
this is of course very very basic well then you want to know more so you know let us let
us try to see lets ask a few questions, so for example we ask a question like can f take
values on a line or can f take all values only on the curve okay, so so let us ask this
question, so so let me again change color, can f take values only on a curve?
Of course with the curve I mean any simple curve that can think of like a parabola or
a circle or something like that.
I mean particularly also it could very well be a straight line this is also curve okay
so can f take values on the curve, so say for example you know suppose I take, so example
so let me look at a few examples can f takes values on the real line?
I mean take only values on the real line this means you are saying that the image of f is
the subset of the real line okay and you know if f takes values only on the real line this
is equivalent to saying that the imaginary part of f is 0 okay because if f is a complex
valued function, normally we write f is equal to U plus IV where U is the real part of f
and V is the imaginary part of f and you know very well from the 1st course in complex analysis
that you and we have to you know be harmonic and in fact they will satisfy the Cauchy Riemann
equations okay because f is analytic but the point is that if you say that f takes only
values on the real line it means you are saying that V is always 0 that means the imaginary
part of f is 0.
So this equivalent to saying that you know imaginary part of f is 0 okay and you know
I can also ask can f take a values on the imaginary axis
okay that is the y-axis consider it as there imaginary axis on the complex plane okay and
that is equal in to saying that the real part of f is 0 okay, so there is another case of
f taking values only on a line then of course they can ask can f takes values on a circle
okay, so let me ask that also.
Can f take only values on a circle so you know if you think that the circle is centred
at the point w naught and has the radius R naught, this is equivalent to saying that
the modulus of f of modulus of f minus w naught is equal to R naught this is what it means,
this is the condition that f takes values on a circle okay.
Now surprisingly, not surprisingly in fact there is something you should have seen if
you just recall that these are all the conditions that will ensure that the derivatives of f
vanishes and therefore f has to be constant, so if the imaginary part of f is 0 at is the
same as saying the imaginary part as a constant okay.
It is a special case of the fact that the imaginary part of the constant and if the
constant is 0 this is equivalent to saying that amounts to saying that the imaginary
part of f is 0 which means it takes for the real valued and if the real part of f is 0
that is a special case of the real part being constant okay and the f taking values on a
circle is the same as saying that the function f minus w naught which is f added to minus
w naught which is a constant function okay.
That has constant modulus that modulus is R naught okay now you have done this in the
1st course in complex analysis probably by using Cauchy Riemann equations that you know
if the function has imaginary part constant or the real part constant or the modulus constant
in the function has to be constant okay so all these things can happen only if f is constant
okay so all these all these can happen only when f is a constant, f is a constant function
okay takes the same value, so of course therefore the question is we are not certainly we are
not interested in studying constant functions because there is nothing special about these
constant functions they constant function maps the whole plane onto a single point which
is the value of the function of that constant, we are not interested in…so we are not interested
in such constant functions, we are worried about non-constant functions okay.
So what this will tell you immediately is that if you have a non-constant analytic function
okay it cannot take values at least on a line or something like a circle okay but then so
what is it so what this tells you is that the either you have the case that you are
looking at a constant function in which case the images a single point okay it is that
single constant value that it takes or the image cannot be a set of just the real
line or it cannot be a subset of only the imaginary axis, it can be a subset of the
circle.
Such things cannot happen that is what this says okay but what is the that you have more
generally, so more generally we have you know we have very nice theorem, so here is
the theorem.
So here is the theorem and it is called the open mapping theorem and is a very very fundamental
theorem, what it says is that non-constant analytic map is always an open map okay, so
if f from D to C is a non-constant analytic map then f is an open map okay it is an open
map, so so let us try to understand what this means?
It means that see what is an open map?
An open map is a map which for which the image of any open set is again an open set okay,
so in particular what this will tell you is that f of D is an open set because D is already
a domain, the D is the domain so D is already an open connected set, so f of D will become
open okay and it is already connected so it is the same as it being path connected so
f of D is again a domain okay so what this what, so let us try to understand what this
means?
If f of D is open so it is a domain so that is something that comes immediately okay and
mind you f is not a constant function, so it takes more than one value so f of D is
non-empty of course okay and the other important thing is the following thing.
What is the condition of open mapping?
If you take U a subset of D which is an open subset then f of U would also continue to
be open okay so if U is subset of D is open then f of U is open, this is what an open
map means it maps open sets to open sets.
The image of an open set under an open map is again an open set that is the definition
of what an open map is?
Okay and…so let us go a little bit more into this and you know try to see what it
really means, what is the meaning of an open set?
An open set is a set which every point is an integer point okay that means you take
any point in the open set then there is a small disk open disk surrounding that point
which is also in that set that is what an integer point means okay, so so what is f
being open mean suppose f takes a certain value w let us say takes a value w naught
okay then there is a it means you are saying w naught belongs to the image of f okay if
f takes the value w naught okay that means w naught is in the image of f because image
of f just consist of the values of f okay and then but since the image of f is open
w naught is a point of an open set therefore there is a small open disk centered at w naught
which is also in the image.
So it means that if f takes a certain value will take all values in a small disk surrounding
that value okay so this should immediately tell you that f cannot simply take values
on the curve because the moment f takes values at the point will take all values in a small
disk surrounding that point and you know no curve can accommodate a small disk however
small okay therefore you immediately get this idea that you know the image of an analytic
mapping, non-constant analytic mapping cannot go into a curve, we saw special cases, we
saw that it cannot go on into a real axis, it cannot go into the imaginary axis, it cannot
be a circle okay and go into the circle but is more generally the reason is the image
is open okay and of course curves are closed sets okay.
So let me write that now if w naught is equal to f of Z naught for Z naught in D that
is the same as saying that w naught is an image of f which is f of D then f of D being
open implies that there exist small open disk in f of D containing w naught and that implies
that f takes all values in a small disk centred at w naught, so this is what is very very
important if f takes certain value it will take all values in a disk about that value
okay this is a very very important property and this is true for of course for a non-constant
analytic function okay.
So so this is about this is about at the moment this is about the topological property of
f of D this theorem tells you that f of D the image of f is certainly a domain it is
an open connected set.
It is very very important that it is an open set and in fact going into a higher geometric
point of view okay what actually happens is this so let me tell it to you in words, what
actually happens is that the mapping f from D to f of D becomes what is called ramified
cover of Riemann surfaces okay so it means that it there are set of points these are
the points where the derivatives of f vanishes okay these are called the points of the ramification
and outside those points in the complement of those points this is actually a covering
map okay it is covering map in the topological and is and also in the analytical or holomorphic
sense okay so this open mapping theorem this so important that tells you that essentially
every analytic non-constant analytic mapping is ramified cover okay fine so now what I
am going to do is I am going to go to ask a more specific question.
So we are trying to look at the image of a domain under analytic function, so let us
look at the cases where 1st at the case where you know the function is analytic on the whole
plane so these are the entire functions, so what is an entire function?
An entire function is a complex valued function which is analytic on the whole plane okay
then the question is what is the what is the image of sets of functions?
So that is a very deep theorem namely it is the so-called little Picard theorem which
says that the image is either the whole complex plane or it is the punctured plane, it is
the punctured plane namely it is the complex plane minus single point that means an entire
function okay we will take all values except for omitting at most one value okay and this
is called the Little Picard theorem okay.
So let me state that so here is the Little Picard theorem, sometimes
people also use the adjective Small Picard theorem, so what is this?
If f from C to C is analytic that is so let me write it here f is entire
then either f of C is equal to C or f of C is equal to minus w naught for some w naught
C so this is a little Picard theorem, so you know it is the it is a very tremendous theorem
it says that you take an entire function, you take the image, the image is huge I mean
the image is literally everything at the worst if the image omits it can omit only one
value okay and the case where the image omits a single value is of course the simplest example
is that of the exponential function, you know if you take the function Z going to E power
Z that is an entire function okay and the image will not it will be the whole punctured.
It will be the punctured plane it will be the complex plane minus the origin because
exponential function will never take the value 0 because 0 does not have a logarithm okay
so if you take any non-zero complex number you can find the logarithm and exponential
of that logarithm give you back that complex number of course you will get many logarithms
okay but you can find at least one for a nonzero complex number, so it means that the exponential
function will take all values except 0 okay and that is the…so in that case it is an
example that illustrates Picard theorem if you take f of Z equal to E power Z then the
image of f is actually C minus 0 which is a punctured plane.
Normally if you take the whole complex plane and remove single point that is called the
punctured plane okay with puncher at that point because that point is being removed
and of course there is also the case when function an entire function can take all
values, the simplest case is that of a polynomial, so if you take a polynomial if you take
f of Z equal to P of Z where P of Z is a polynomial then it will take all values because I can
always solve for P of Z equal to w naught for any w naught and that is because of the
fundamental theorem of algebra namely that the complex numbers of algebra closed so I
can always solve a polynomial equation in one variable okay so a polynomial is also
an entire function and it is it gives the case the 1st case namely the image of the
whole complex plane is the whole complex plane okay fine so this is the little Picard theorem.
Now somehow what I want to do is I want to really to prove this okay it is a deep theorem
normally this theorem is only stated in the 1st course in complex analysis but since this
is advanced was in complex analysis I think it is fitting to look at a proof of
this.
Now well you know interestingly it is very interesting that the proof of this that I
am going to present is actually gotten by deriving this as a corollary were much more
deeper theorem, it is called the big Picard theorem and the funny thing is that the big
Picard theorem is a theorem which deals which again asked the same kind of questions, it
answers the same kind of questions namely what is the image of a domain under an analytic
map okay but the point is that the domain you are looking at is a disk around an essential
singularity of an analytic function.
So you know so let me state that so here is so let me use something else this will be
deduced from the from the big or great Picard theorem and that is let so here is the statement
of the theorem, let Z naught be an isolated essential singularity
of an analytic function f okay then f of…so let me write this 0 less than mod Z minus
Z naught less than Epsilon is equal to C or C minus single value for every Epsilon greater
than 0 in the domain of analyticity of f okay so I have just stated a part of the theorem
there is still more to it, so I want you to look at this.
What I want you to appreciate is I wanted to appreciate the following things to reduce
the little Picard theorem which is theorem about a function is analytic on the whole
plane if function is mind you if a function is analytic on the whole plane it has no singularities
okay.
It has no singular points okay so the little Picard theorem is a theorem about a function
which has no singular points okay and it says that the image of the whole plane under
such a function is either the whole plane or a punctured plane okay but we are deducing
it from a theorem about the image of function with a singularity, so that is the funny thing
so it is like even to answer question about an entire function you are forced to study
singularities this is the point I want you to understand okay.
See normally we would not like to dirty our hands with singularities, why study singularities
when there are functions without singularities but the point is you know sometimes mathematics
and theory teaches us that even to study good things we have to study bad things okay so
if you want to prove the little Picard theorem which is a theorem about good things I mean
the function is analytic entire you have to still study functions which are having singularities
and so here is the big Picard theorem and obviously you know the adjective great or
big should tell you that this big Picard theorem has to be a Big Brother of the little Picard
theorem and therefore you know the little Picard theorem can be reduce from the support
of the Big Brother and what is this big Picard theorem, what does it say?
See you are looking at an analytic function okay and you were looking at a singularity
of an analytic function okay now so I come later to what a singularity is?
Okay because that is motivation for me to recall these things okay so you look at a
function f which at a point has isolated singularity, isolated means there is a whole his surrounding
that point where there are no other singularities okay and a deleted disk surrounding that point
is given in this form as I have written here in the on my board zero strictly less than
mod Z minus Z naught strictly less than Epsilon is actually the disk centred at Z naught the
radius Epsilon is an open disk but I have thrown out Z naught that is the reason for
putting 0 strictly less than I am not allowing Z equal to Z naught that means it is a punctured
disk.
It is a punctured disk centred at Z naught and the punctured is exactly at Z naught I
have thrown out Z naught okay and on this disk the I am assuming that this disk is full
of points where function is analytic okay and that will be through at least for
small values of Epsilon greater than 0 because the point Z naught is an isolated singularity
okay and look at what the theorem says, it says you take the image of this when I write
f of something okay it means f of this set which is the punctured disk that is the whole
complex plane or it is a complex plane minus a single point and this is true for Epsilon
sufficiently small and therefore it will be true for even larger Epsilon so long as this
deleted disk is in the domain of analyticity of f because larger disk, larger deleted disk
will contain smaller deleted disk and therefore their images will contain images of smaller
deleted disk okay so.
So you see this is again you see the result of the conclusions of the theorem both the
big Picard theorem and the little Picard theorem they are the same I mean the conclusion always
says that the image under analytic function of a certain domain okay is either the whole
complex plane or it is the complex plane minus a point okay and in case of little Picard
theorem you are looking at the domain is the whole complex plane but in the case of great
Picard theorem the domain is a very small neighbourhood deleted neighbourhood of an
essential singularity of an analytic function and what is really amazing is in fact there
is more to this Picard theorem what it says is you see so I want you all to observe is
the following thing it is a very very deep results.
It says take a very small neighbourhood of the essential singularity okay deleted neighbourhood
that means of course you do not take the neighbourhood that you take should be a domain weather function
is analytic so it cannot include the singularity, so when I say take a neighbourhood of essential
singularity of course I mean delete that essential singularity so you are taking a deleted neighbourhood
of the essential singularity and mind you you a neighbourhood as small as I want, you
see this Epsilon can be extremely small okay and the theorem is amazing it says you take
no matter how smaller neighbourhood you take, the image of that neighbourhood is still the
whole plane no matter how small your neighbourhood is.
The image of the very small neighbourhood no matter how small is still the whole plane
it still takes all those values so what this tells you is you know it tells you that it
tells you how the values of analytic functions change in a neighbourhood of an essential
singularity, in our neighbourhood of an essential singularity this analytic function is taking
all values at the worst it can omit one value okay and of course you know the example for
this is just as in the case of the little Picard theorem where you where the example
of an entire function omitting a value is exponentially E power Z a which omits the
value 0, here you can take E power 1 by Z okay you can take the function E power 1 by
Z and this E power 1 by Z and E power 1 by Z at Z equal to 0 has an essential singularity
and if you take any small deleted neighbourhood of 0 however small and you take the image
under E power 1 by Z you will get the whole plane except the origin because exponential
function will never take the value 0.
So you know it is an amazing, it is an amazing result, it is an amazing result and in fact
there is a stronger version of the Picard theorem which says that not only does the
image of any small neighbourhood however small of an essential singularity under under
analytic function is a whole plane or plane minus it says it takes the every complex value
that it takes it takes infinitely many times, so there is in fact so that we write that
down just to tell you how powerful the theorem is so let me write that.
For every Epsilon greater than 0 such that 0 less than mod Z minus Z naught less than
Epsilon is in the domain of analyticity
of f. f assumes each complex value with at most one exception w naught infinitely many
times, so in fact this, this infinitely many a times part of it which tells you the more
it tells you with lot of force what is happening so the 1st part of the great Picard theorem
says that you take an essential singularity and take a very smaller deleted neighbourhood
about that, take a very small disk surrounding the essential singularity and take its image
under the analytic function of course you do not take…the analytic function is not
defined at the singularity okay so you do not take the value at the singularity there
is no such thing.
So you are actually taking the image of a deleted neighbourhood but the point is no
matter how small a deleted neighbourhood is, your image will be the whole complex plane
or it may be complex plane minus a single point that is the 1st part of the theorem
and in fact what this part of the theorem says that you know you take any value, any
of the values in the complex plane except possibly for that one value w naught which
it will not take okay, take any other of the values that it takes that value itself if
you take the pre image of that value in that neighbourhood, the pre-majors and infinite
set okay that means there are infinitely many points even in that small neighbourhood that
are infinitely many points at which the function takes place that prescribed that value that
you are pointing at and this happens for every value that it takes.
So what it does it is very funny it looks as if you take the small neighbourhood around
the essential singularity, the function not only match that very small neighbourhood onto
the whole plane or whole plane minus a point but it maps it infinitely many times okay
it is like it maps it thousands and thousands of time okay and that is an amazing thing
okay it is not that for every complex value there is one value here which goes to that,
the fact is you take any complex value other than the exceptional value w naught then there
is infinitely many points in this very small disk however small where that value is taken
by f okay so that small neighbourhood it is really amazing to think of it, think of a
very small infinitesimally small neighbourhood which is being again and again you know it
maps thousands of times I mean probably unaccountably many number of times on to the whole plane
or the whole place minus a point that is how the function behaves in a neighbourhood of
an essential singularity and this is the key to…
This theorem on singularity is the key to proving or reducing as a corollary the little
Picard theorem, so we will try to in the forthcoming lectures will try to give a proof of this
theorem and so I will tell you roughly I will give you an idea of where we are going to
go, so you know 1st of all I want to recall something about singularities, you would have
studied singularities but I would like to recall them and some basic theorems about
singularities especially the Riemann’s theorem on removable singularities and then I want
to reduce from that what is called the weak version of the big Picard theorem which is
called the Casorati–Weierstrass theorem and Casorati–Weierstrass theorem is slightly
weaker what it says is that while the big Picard theorem says that a function assumes
analytical function assumes all values except with possibly one exception in every neighbourhood
of an essential singularity.
What the Casorati–Weierstrass theorem says is that it will come arbitrarily close to
every value okay so Casorati–Weierstrass theorem is a slightly weaker version of the
great Picard theorem and that can be more or less reduced using the Riemann theorem
on removable Singularities which I will prove okay so I have to recall something about singularities
but then as we move towards the proof of big Picard theorem what you have to do is that
we will have to study not one function but we will have to study a space of functions
and we have to study functions with singularities and the kind of functions you going to study
are functions with singularities as and these are called Meromorphic functions.
So what I am going to do I am going to study topology of a space of Meromorphic functions
and prove some fundamental ills like Montel’s theorems and these are the keys to unlocking
the proof of the big Picard theorem okay so what I am going to do in the next few lectures
his 1st recall singularities then tell you something about Riemann removable Singularities
theorem through the Casorati–Weierstrass theorem and then go onto Meromorphic functions
studying Meromorphic functions and then trying to study families of Meromorphic functions
topologically whether that spaces compact and things like that okay, so that is the
that is the direction in which we will be proceeding so at least the 1st part of these
lectures… our aim is to prove the great Picard theorem and you will see on the we
will prove several other important theorems okay.