So you see we are looking at compactness of families of functions, okay we want to look
at Meromorphic functions alright and so you know what we did last time was Montel’s
Theorem, okay which was for analytic functions, alright. So let me recall this so that I will
give you the background for the formulation of the version of Montel's Theorem for Meromorphic
functions which goes by the name of Marty's Theorem, okay and once you have that then
we can go ahead and try to we get closer to the proof of the Picard Theorems, alright.
So you see so you can recall so maybe I will put the tittle as Marty's Theorem so let me
put here version of Montel's Theorem for Meromorphic functions, okay. So let us recall see so the
idea is the following so you have D inside the complex plane its domain it is an open
connected set non-empty ofcourse, okay and you have script F is family of analytic functions
on D so this is this is subset of holomorphic functions on D, okay and mind you these are
ofcourse certainly continuous functions on D with complex values, alright.
So you have this family of holomorphic functions on D and then the whole idea is you want to
worry about the compactness of these familiar functions, okay and compactness in what sense
compactness can be you know in general compactness is the same as sequential compactness, okay
as a general philosophy and so and you should expect therefore you want conditions for which
given a sequence of functions in F in the family script F you want to pick out a convergent
subsequence, okay.
Now the point is that you know ofcourse since you are working with holomorphic functions,
okay just convergence is not a very useful thing you would like to have a kind of convergence
that will reserve properties of original functions which are converging to the limit function
and therefore the best kind of convergence you can expect is uniform convergence but
ofcourse uniform convergence is too much to expect what you will normally get is only
normal convergence, okay that is you will get uniform convergence restricted only to
compact subsets, alright.
So that is the background and so what is it that you have? See on the one hand you want
F is normally sequentially compact and what do I mean by this? I mean that given any sequence
in F, I can find subsequence which converges normally that means which converges uniformly
on compact subsets, okay so let me write that any sequence in F admits a subsequence that
converges uniformly on compact subsets on compact subsets
of D, okay so this is this is compactness for us, okay that and that given any sequence
you can extract a convergent subsequence but the convergence is only with respect to the
normal convergence, okay so this is what you want.
Now how do you what what are the necessary and sufficient conditions for this to happen
what are the necessary and sufficient conditions for this to happen? So you know if you want
to go from here to here if you want to go from here to here alright so you will get
immediately that F is normally uniformly bounded, okay so that means that you given this family
of functions in scripts F and if you restrict it to any compact set then that family will
have uniform bound.
So there is some positive number such that the modulus of the functions is bounded by
that number and this will work for all functions and for all points on that compact set, okay.
So this is something that you will get and the other thing you will get is you will also
get that F is the family is equicontinuous, okay so plus so let me write this F is so
for some reason let me write it at a distance so I will write here F is equicontinuous.
So how does this come? These two implications come because of the they basically come because
of the Arzela-Ascoli Theorem, okay so this is so let me write this let me put AA here
where you know I will put AA here also where let me write here as a legend AA stands for
Arzela-Ascoli Theorem you know this is just the Arzela-Ascoli Theorem. See what Arzela-Ascoli
Theorem says is that you know if you are looking at continuous complex or real valued functions
on a compact metric space, okay.
Then the condition that such a family of functions is compact, okay is equivalent to that family
of functions being uniformly bounded and equicontinuous that is Arzela-Ascoli Theorem. So if you now
look at it if you now look at this thing that I have assumed on the this property on the
left side which says that this family is sequentially compact if I restrict to any compact subset,
okay. So if I take capital K a compact subset of D and I restrict this family to that, okay
then I am looking at a family of continuous functions on a compact metric space, okay
any compact subset of D is also a metric space it is a compact metric space metric is just
metric on D restricted to that subset, okay.
And then by the Arzela-Ascoli Theorem that the family script F will become actually compact
you know with respect to the topology given by the supremum norm, okay the supremum norm
is defined and you have the metric induced by that norm and the topology induced by the
metric so with respect to that this family script F actually becomes a compact family,
okay it becomes a compact subset of points, okay and you are now considering this family
inside C K, C namely the set of all continuous functions on the compact set K with values
in C, okay because analytic functions are ofcourse continuous, okay.
So then by the Arzela-Ascoli Theorem what will happen is that on that compact set K
what will happen is that this family will be uniformly bounded, okay that is bounded
with respect to the sup norm which is uniform boundedness and you will get equicontinuity.
So and since this and equicontinuity is something that is needs to be checked at every point
so you will get equicontinuity for the family the whole family throughout all of D, okay
to check equicontinuity at all of D at every point of D I just have to check equicontinuity
at each point of D and to check at each point of D it is enough to check on each compact
subset of D even a point is a compact subset if you want, okay.
So equicontinuity will fall out and restricted to that compact subset I also have uniform
boundedness, okay therefore restricted to compact subsets I have uniform boundedness
and that is normal uniform boundedness so that is how I get these two implications,
okay but the serious thing is to go so that the serious thing is to go the other way round,
okay so starting with starting with you know the Arzela-Ascoli Theorem in another direction
tells you that is you are on a compact metric space and you are looking at continuous functions
complex valued or real valued, if you know these collection of functions is uniformly
bounded and it is equicontinuous then your family is compact, okay.
Now so if I start go from this direction suppose I assume F is normally uniformly bounded script
F is normally uniformly bounded namely this condition so I will purposely change colour
because I want to emphasize something else. So you know I take this I take this condition
script F is normally uniformly bounded alright then the beautiful thing is we do not have
to add equicontinuity that is the big deal, the big deal is you can go from here to here
directly as a theorem and this is the this is Montel's Theorem so this is Montel's Theorem
that is you start with a normal uniformly bounded family of analytic functions, okay
then it is normally sequentially compact that means you can given given a sequence you can
extract a subsequence which converges normally, alright.
And mind you so I have I have to tell you few things here this is this Montel Theorem
is stronger than Arzela-Ascoli, okay in the following sense. See what Arzela-Ascoli Theorem
will say is that you know if you restrict yourself to a compact subset of D if you take
a particular compact subset of D, okay and suppose I have this condition that this family
script F is normally uniformly bounded then family script F will become uniformly bounded
on that compact subset and again I will get equicontinuity and I will apply the Arzela-Ascoli
Theorem and I will be able to pick a subsequence which converges uniformly on that compact
subset, okay.
And the ofcourse equicontinuity comes because of the derivatives being bounded which comes
as a result of the Cauchy integral formulas and some estimations, okay and the Cauchy
estimates of the first derivative, okay but let us forget that for the moment basically
what happens is you are able to for every compact subset if I start with the sequence
in the family F for every compact subset I am able to get a convergent subsequence I
am able to pick a subsequence such that on that compact subset the convergence is uniform.
But the point is if I change the compact subset the subsequence can change, okay if I apply
only the Arzela-Ascoli Theorem if I change the compact subset then the subsequence can
change but the Montel Theorem is very strong what it says is that I can uniformly find
a single subsequence of the original sequence which will converge uniformly on every compact
subset it will work for every compact subset, okay that is the power in the Montel Theorem
and if you remember this we got this by diagonalization argument, okay we covered the domain D by
an increasing sequence of compact sets, okay which fill out the domain and on each member
of this sequence of compact sets we picked out a convergent subsequence using Arzela-Ascoli
Theorem, okay and then we wrote down this metric of convergent subsequences by you know
for the first compact set in the sequence we picked out a subsequence in the sequence
of sets covering the space D we picked out one sequence from the original sequence we
from Arzela-Ascoli, okay.
Then from this sequence we picked out another subsequence which will work on the next bigger
compact subsets and then we went on like this and all these compact subsets eventually filled
their union filled the whole of D, okay and the diagonal sequence gave us a sequence of
the original sequence which will converge uniformly on every compact subset of D because
every compact subset of D is contained in one of the members of this sequence of compact
sets that we increasing sequence of compact sets that we constructed to cover D, okay.
So you see we have got this very strong statement from Montel's Theorem, okay so that is one
point you have to remember, okay and the other important point about Montel's Theorem is
that you do not worry about you really do not worry about equicontinuity, okay and this
is basically because you are working with analytic functions. So what is happening is
that this condition that the family is normally uniformly bounded tells you that if you take
the family of derivatives of these functions then that family of derivatives is also uniform
normally uniformly bounded, okay.
And so let me write this so I will write this as F prime is normally uniformly bounded
so there is this thing here in between, okay so when I write ofcourse when I write F prime
I mean the set of all I mean all those derivatives of functions f which are in small f which
are in script F, okay so so script F prime is just the derivatives of the functions in
script F, okay and mind you the functions in script F are all analytic therefore the
derivatives you know if you take a function which is analytic then all orders of derivatives
of that function exist and they are also analytic, okay.
So script F prime is also a bonafied family of holomorphic functions on the same domain,
okay and the point is that that is normally uniformly bounded, okay and that is because
that is simply because of the fact that the derivative of a function can be expressed
in terms of the function using the Cauchy integral formula, okay and therefore if the
original functions are uniformly bounded then the derivatives are also uniformly bounded
on closed on sufficiently small closed disks, okay.
So the moral of the story is that the derivatives are normally uniformly bounded and because
the derivatives are normally uniformly bounded you know this is the philosophy that I told
you last time whenever the derivatives are bounded uniformly then the original family
is equicontinuous, okay because you just have to integrate, okay so this so there is another
implication that is going like this whenever the derivatives whenever the family of derivatives
is uniformly bounded then the original family is equicontinuous.
So what happens is that because I assumed that the family is normally uniformly bounded
I am also getting equicontinuity the way I am getting equicontinuity is because I am
getting actually normal uniform boundedness of the family of derivatives that is the whole
point and the reason I am able to get this is because of the Cauchy integral formula
because of the Cauchy estimates, okay. So this is how this is how everything works.
Now what is that so what is that we want to do with meromorphic functions okay so if you
now if you are see so far we are working here in the set of all holomorphic functions I
mean analytic functions, alright but you know you want to work with meromorphic functions
the problem with that is that if you are working with meromorphic functions then you are going
to allow the value infinity, okay.
So you are going to take values not if you are going to take a meromorphic function you
cannot just considered it as a function into complex numbers because then at a pole you
cannot define it. Whereas if you considered it as a function into the extended complex
plane, okay then at a pole you can define the function value to be infinity and still
keep the function continuous even at a pole, okay.
So if you are working with meromorphic functions you want to do a same you want to have the
same kind of theorem, okay then you know it is little it is little troublesome somehow
you can see that in this whole game you have to pass through this red box that I have put
here which is that the derivatives are normally uniformly bounded, okay and that see will
work for analytic functions as it is not work for meromorphic functions because the problem
will be at the poles at a pole I cannot apply any kind of Cauchy integral formula I cannot
express in fact even derivative at a pole is not defined it is a singular point, okay
so I am in trouble.
And you know in order to overcome this we had introduced this concept of spherical derivatives,
okay so that is what we are going to use. So in fact we will get this theorem that now
you again take a domain in the complex plane, you take family of (holo) not holomorphic
but meromorphic functions on the domain but mind you you are now considering this as functions
not into the complex plane but functions into the extended complex plane, okay.
And when you consider it as functions into the extended complex plane mind you the target
plane is not complex plane it is the extended complex plane and the extended complex plane
has been made into a metric space by putting the spherical metric and with respect to the
spherical metric it is a compact metric space, it is a beautiful metric space, it is just
the Riemann sphere with the spherical metric on that, okay alright.
And now what you do is you get this version of the Montel the correct version of the Montel's
Theorem for meromorphic functions will tell you that now you again take a family of meromorphic
functions the condition that it is normally sequentially compact is equivalent to saying
that the spherical derivatives are bounded that is it, okay. So what I want you to understand
is that it is rather funny when you move from the holomorphic version of the Montel Theorem
to the meromorphic version of the Montel’s Theorem which is called Marty's Theorem, okay
your condition changes from the normal boundedness of the of the family of functions to the normal
boundedness of the derivatives but what derivatives spherical derivatives that is the big change,
okay and with that everything works, okay so that is what I am going to state next.
So here is Marty's Theorem. Let D in C be a domain ofcourse non-empty as usual and script
F is a family of meromorphic functions on D, okay mind you this is a subspace of the
continuous functions on D with values in the extended complex plane, okay you have to remember
this this is very very important we are considering meromorphic when you say meromorphic function
you are allowing the value infinity, okay otherwise you will not continuity at the pole
at poles, okay that is very very important, okay.
Then F is sequentially is normally sequentially compact compact i.e, every sequence in script
F in script F admits a subsequence that converges uniformly normally that is uniformly on compact
subsets of D if and only if
so I will write F hash, what is F hash? This is the collection of spherical derivatives
of the functions in F so we use prime for derivative when it is an analytic function
when it is not an analytic function but it is a meromorphic function we use hash, okay
which is which is the notation we introduced earlier so this is set of f hash such that
f belongs to script F is normally uniformly bounded that is uniformly bounded on on compact
subsets, okay so this is Marty's Theorem so this is meromorphic version of Montel's Theorem
meromorphic version of Montel's Theorem.
So let me write that here
and the big deal in this statement is essentially to say that instead of requiring that the
original family of functions is normally uniformly bounded which is what the original Montel
Theorem want you know needed you shift to the spherical derivatives of these functions
that is the difference, okay.
Now you see what does this say in retrospect I mean what it says in retrospect is that
in principle it says that if you take a family of even if you take a family of analytic functions,
okay even if you take a family of analytic functions the condition that the the usual
derivatives are normally uniformly bounded is also equivalent to the normal sequential
compactness of the family that is the big deal, the big deal is you know if you go back
to this diagram, okay we had this we had this red box here which said that the derivatives
the usual derivatives which in this case are they make sense because functions are analytic
the usual derivatives are normally uniformly bounded they are uniformly bounded on compact
subsets, okay.
Now this itself this itself is good enough to give you normal sequential compactness,
okay but there is only there is only one small issue since the compactness is I mean since
the convergence is with respect to functions which can take the value infinity the convergence
point wise convergence is with respect to the spherical metric that is the difference,
okay. So what it means is it means the following suppose I have a family of analytic functions
on a domain how do I decide that this family is compact, okay that is it is normally sequentially
compact one direct way is use the usual Montel Theorem for which I need all the functions
in the family to be normally uniformly bounded to be I must be able to find uniform bound
for this family on every compact subset.
There is another way the other way is verify that the derivatives the family of derivatives
of these functions that is normally uniformly bounded if you verify that okay then what
happens because of this meromorphic version of Montel's Theorem that family of see if
the usual derivatives are uniformly bounded on a set then the spherical derivatives are
also be uniformly bounded on the set, okay that is because of the way in which the spherical
derivatives are defined.
See how is the spherical derivative defined see it is defined like this for for f meromorphic
function, recall that the spherical derivative of f at z you know it is defined as 2 times
mod f dash of z divided by 1 plus mod f z the whole square this is the definition of
spherical derivative this is how spherical derivative is defined, okay this is how we
define spherical derivative.
And ofcourse you know there is an issue I have used a mod f dash in the numerator that
makes sense only if f dash exist and therefore I can write this only at point z which are
not poles but at poles what happens at poles you know we did this in an earlier lecture
at poles you extend the spherical derivative by continuity and what happens is that the
spherical derivative will become 0 at a pole of higher order, okay and at a pole of order
1 namely at a simple pole the spherical derivative is 2 divided by the modulus of the residue
of the function at that pole, okay.
So therefore this so in particular you know if you look at it in a very logical kind of
way even if you take the function which is identically infinity, okay mind you that function
is also there in because we are considering functions with you know values possibly being
infinity also. If you take the function which is always infinity that function is also by
definition one for which you have to define the spherical derivative and the spherical
derivative will be 0, okay so this is something that we make as a default definition, okay.
And this so I am just trying to say if you take the function which is uniformly infinity
whose infinity at every point on your domain that function is also included and that function
also spherical derivative is also included spherical derivative is 0, the way you think
about it is a that you know usually the derivative should be 0 if the function is constant after
all the function which is infinity at every point is just the constant function infinity
so you should expect the derivative to be 0 that is one way of looking at it.
The other way of looking at it is what is the spherical derivative? If you take a meromorphic
function you treat it as a map into the Riemann sphere, okay and what it does is it is the
magnification factor of the length of the image I mean it is a magnification factor
that you will have to put in to calculate the length of the image of an arch under this
mapping. So suppose you have an arch on the suppose you have a suppose you have a an arch
or contour on the complex plain in your domain where your meromorphic function is defined,
okay and you take its image under this meromorphic function it will land on the Riemann sphere
where I am thinking of the extended complex plane as the Riemann sphere.
So I am going to get an arch on the Riemann sphere, ofcourse this arch can pass through
infinity, it will pass through infinity if the original arch in the plane pass through
some poles of your meromorphic function wherever original arch in your complex plane hit a
pole the image will hit on the north pole on the Riemann sphere which corresponds to
the point at infinity, okay.
And if you take the image arch how do you get the length of the image arch, what you
will do is you will integrate the meromorphic function not the meromorphic function in fact
you will integrate spherical derivative along the original arch and you will get the length
of the image arch. So the length of the spherical derivative is a magnification factor, okay
and if this if your original function is just the function which is constant function infinity
then it is going to map your whole domain onto a point, okay if you take the function
which is constant function infinity then your whole domain is going to be collapsed to the
point which corresponds to the north pole.
So any arch is going to be collapsed to a single point, okay so what is the magnification
factor 0 and that should be the spherical derivative. So this is another way of saying
that you know if you take if you take the constant function infinity you must think
of the spherical derivative of that to be 0, okay that is another point that you will
have to remember in mind. So but in any case the spherical derivative as it is is always
a continuous function and that is the reason where we are able to integrate integrate it
always even if your path of integration passes through some poles of f that is very very
serious, okay.
But anyway what you see from here is that because that is this mod f dash term here,
okay what it will tell you is that if you are looking at a family of analytic functions
whose derivatives are normally uniformly bounded, okay then these numerators are normally uniformly
bounded, okay but then you know I can forget the factor 1 plus mod f z squared denominator,
okay because that is a factor greater than or equal to 1 and its reciprocal is less than
or equal to 1.
So actually I can write f hash of z is actually is less than or equal to 2 mod f dash of z
I can write this this make sense, okay because the denominator I can forget the denominator,
okay and what does this tell you? This tells you that whenever f dash is defined okay in
particular if you are looking at a family of analytic functions okay and the derivatives
are make sense then if you know the derivatives are bounded then it means that the spherical
derivatives are bounded because the spherical derivative is bounded by 2 times bound for
the usual derivatives.
So if you start with a family of analytic functions such that the derivatives are bounded
then the spherical derivatives are bounded and Marty's Theorem will tell you that these
family of analytic functions considered as a family of meromorphic functions mind you
analytic functions are also meromorphic functions but when you considered it as a family of
meromorphic functions you are allowing the value infinity with that consideration this
family becomes sequentially normally sequentially compact that means given any sequence you
can get hold of a subsequence which converges normally.
So finally what happens is that this box that I wrote down here is the crucial condition
that is crucial for both the original Montel Theorem and also the meromorphic version of
Montel's Theorem which is Marty's Theorem so this is the crucial thing the boundedness
of the derivatives, okay. So but the only thing that can happen is that your sequence
of analytic functions may go to infinity because that is , okay.
See if you go back and think we proved the following thing you take a sequence of analytic
functions on a domain, okay if you take convergence with respect to the spherical metric either
they will converge to an analytic function or they will converge to a function which
is identically infinity, okay and the same kind the of thing happens in meromorphic functions
you take a sequence of meromorphic functions which converges normally on a domain, okay
then either the limit is a meromorphic function or it is the function which is you know identically
infinity you do not get bad behaviour you do not get a sequence of holomorphic functions
or analytic functions going to a function which is meromorphic strictly meromorphic
or you do not get a sequence of meromorphic functions which goes to a function which has
funny singularities namely it may have non isolated singularities or it may have isolated
essential singularities such these kind of horrible things do not happen, okay.
So if you take this thing that I have put down which I have now rounded in a long ellipse
as the important condition, okay then that is the condition for sequential compactness
that is what I want to say and see these conditions work if the functions are analytic, okay and
the analogous condition namely the derivatives replaced by the spherical derivatives that
works if the functions are meromorphic, okay so this is the very very important point in
our theory that we are finally managed to translate compactness of a family of meromorphic
functions or analytic functions to just uniform boundedness of derivatives that is all, okay.
And if they are usual analytic functions use the usual derivatives, if they are meromorphic
functions use spherical derivatives that is all, okay. So bringing in the derivatives
is the big deal here, okay. So we will now need to see a look at a proof of this theorem
and the proof is pretty except that you will have to worry about all these issues there
are little little things that need to be checked, okay.
So I will try to write down the proof pretty short steps and I will ask you to do some
small verifications so let me say the following thing so you see let me write the so here
is what I have to proof I have domain in the complex plane, I have a family of meromorphic
functions I assume if I have to assume first that the family is normally sequentially compact
and I have to show that the spherical derivatives are bounded, okay normally uniformly bounded
and I have to do the other way round.
And what does the proof actually involve it involves the few simple results
so let me write this so here the few lemmas that I want to worry about or rather let me
say lemma if a sequence f k of meromorphic functions
converges to f, okay in m D okay then the same thing happens to the sequence of spherical
derivatives, okay ofcourse here again I must so I have written in very very simple words
but I must again insist when I say converges I mean converges normally, okay it means it
is uniform of compact subsets.
So if f k is a sequence of meromorphic functions on D it converges uniformly on compact subsets
to a function f we have already seen that this f can either be meromorphic or it can
be the function which is identically infinity that we have already seen then this normal
convergence preserves derivatives, okay. So this is something that we have seen with analytic
functions if a sequence of analytic functions converges normally to a given function then
the limit function is also analytic and you know you can the nth order derivatives of
the original sequence of functions will converge normally to the nth order derivative of the
limit function this is all just because of normal convergence uniform convergence of
compact subsets, okay.
So the same thing happens with spherical derivatives this is one fact that we will have to use,
okay. So you can check this and well if you want to go back to the let me tell you about
the let me tell you atleast in words about the proof of this theorem ya atleast one way
is very clear, okay. Suppose your family is normally sequentially compact, okay and suppose
contrary to what is required the family of derivatives spherical derivatives is not normally
uniformly bounded then you know there is a compact subset on which these derivatives
will go to infinity, okay.
And so there is a compact subset and a sequence of functions where the spherical derivatives
will go to infinity, okay that is what you get if you contradict a normal uniform boundedness,
okay of the spherical derivatives but if that happens then the original family could not
have be normal because if the original family were normal then what would happen is that
the original from every sequence you can pick out a normally convergent subsequence if the
subsequence is normally convergent then the spherical derivative is also normally convergent,
okay but then we have already obtained a sequence of spherical derivatives which does not converge,
okay.
So the point you will have to remember here is that when you are considering spherical
derivatives the convergence is with respect to the usual distance function on the real
line, okay mind you that is another important point the spherical derivative is a positive
non-negative real valued function, okay and whenever you talk about convergence of the
spherical derivatives you are working with convergence on the real line that is something
that you should not forget, okay and therefore you get a contradiction. So this is one way
of the theorem the other way I will prove in the next lecture, okay.