Okay, so we have seen in the last lecture the famous theorem of montel or normality
of families and what it says is that suppose we have a family of meromorphic functions
defined on a domain, the domain can be extended complex plane. To decide that the family is
normal, all you have to do is to ensure that all the functions in the family do not take
3 distinct values in the extended plane, okay. And you know because you are working with
meromorphic functions, you have to allow the value infinity because that is the value that
a meromorphic function at a pole will take, okay.
And but of course you are looking at a family of analytic functions, okay, then the condition
is much more simpler, you have to just find 2 complex values which the functions in the
family do not take and if that is true then the family is normal, okay, there is this
great theorem of montel. And you see it is the key to proving the picard’s theorem
which we will do, right.
So, so, so here is picard’s big or great theorem, okay. So you know what the theorem
is. The theorem is that if you take a function which has an isolated essential singularity,
then the image of any neighbourhood of that singularity is either the whole complex plane
or it is a complex plane minus the single value, okay. That means, it means, it means
that it can at most omit one complex value, all right. And what is the restatement, the
restatement is that if it omits more than one value, if it omits 2 values, that
is something cannot be, that is not possible, okay.
So what we will do is we will assume that it omits 2 values and then use the montel
criterion that the resulting family of zoomed functions is normal, okay. And examine the
limit of the zoomed functions and that will give you the proof, okay. So let me state
it, let f of z, let f be, let f have an isolated, let f of z have an isolated singularity at
z equal to z0 , then the image under f of any deleted neighbourhood of z0 is either
all of complex plane or omits at most one value in c. So this is the picard’s theorem,
okay.
And of course this is valid for every neighbourhood, it means that you know it will take except
for one value which it might omits, all other complex values it will take infinitely many
times because you, if you find the point where it will take that value, then you can find
a smaller neighbourhood, you can take a smaller neighbourhood, deleted neighbourhood of z0
and in that neighbourhood also again it has to take that value and can go on like this,
therefore it will take every value except one value infinitely many times, okay. That
is amazing behaviour of function, analytics function around an isolated essential singularity,
okay.
So this is the theorem and, so what is the proof? The proof is, by contradiction, by
contradiction, by contradiction using the zooming process. So you know the zooming process
has been, we had been using it right from zaltzmann lemma, okay and you know what is
the main idea behind the zooming process? The idea behind the zooming process is, as
you zoom into a normal point, then all those zoomed functions will converge normally to
a constant function. And if you zoom to a non-normal point, then all the zoomed functions
will converge to a nonconstant meromorphic function. Okay that is the, basically the
principle, all right.
So what we will do is, so assume f omits 2 values in f omits 2 values in c in some deleted
neighbourhood 0 less than mod z minus z0 less than say rho, okay. So you have to
show that f takes either take all values or it will take all values except 1. So if you
want to contradict that, you have to assume that it omits at least 2 values, okay. So
let us assume that, all right. So you, what you do is, so here is my diagram. So i have,
so this is the, this is the complex plane, this is z plane and i have this point z0.
And you know i, there is this small disc surrounding z0, radius rho, this disc with radius rho
and on this disc f does not take 2 values, 2 complex values.
That means there are 2 distinct complex values which f will take, it may not take many more
values also but at least 2 values of it misses, okay. And what am i going to do, i am going
to construct a sequence of zoomed functions. So what is the sequence of zoomed functions,
what you do is, well take any, take any, so before that let me write it ideologically.
Zoom into the function f at z0 itself, okay. Mind you the function in, z0 is an isolated
singular point, it is an essential single point, therefore the deleted neighbourhood
of z0 in this deleted disc, this punctured disc centred at z0 and radius rho, you throw
out the point z0.
In the punctured disc it is the function is analytic, mind you, okay. And what i am going
to do, i am actually going to zoom in to z0, okay, i am just going to zoom in to z0 and
how do i zoom in to z0, by taking smaller and smaller, discs of smaller and smaller
radii which are centred at z0. And of course i have to exclude z0 because z0 is a point
of singularity of f. So what you do is, zoom in to z0 and so let me say that is take a
sequence epsilon n tending to 0, 0 less than epsilon n lesser than rho, okay. So you take
a sequence of smaller and smaller radii, okay.
And let gn of zeta be the zooming of f the same function f centred at z0 , the scaling
factor is 1 by epsilon n and the variable is that, okay. So this means that you are
just taking f of z0 plus epsilon n times theta, this is a function. So this is what my gn
is, so i am using that single function and constructing a family of functions. Using
the single function f i am constructing a family of functions gn, okay. And where are
these gns defined? So you see, this is the z plane and then if you look at correspondingly,
we have the, you have, this is a complex which is the z plane, then i also have complex thing
which is zeta plane, okay.
And in the zeta plane what happens is that, you know if i take, if i take this disc centred
at the origin and radius, if i take the radius to be rho by epsilon n, okay, that is for
mod zeta less than epsilon n, mod, epsilon and theta will be less than rho. And therefore
this is the, so this is the disc in which gn is defined by the only thing is it is not
defined as the origin because at the origin the origin corresponds to z0. And f at z0
is not defined because z0 is a singularity of f, okay.
So let us write that down, gn is defined, so let me write this here, gn of zeta is defined
in a 0 less than mod zeta less than rho by epsilon n. And of course you know the point
about this whole business is that this epsilon n tends to 0plus, rho by epsilon n tends to
infinity and therefore gn zeta is going to be, you can talk about convergence, normal
convergence of gn zeta on the whole punctured plane, okay, except that, the whole plane
except the origin. Because eventually any compact subset of the punctured plane other
than which does not contain the origin is going to be, is going to be contained in the
domain of gn zeta for n sufficiently large, okay.
So, so let me write this down, note that any compact subset
of c not containing the origin, the origin is going to be in the domain of definition
of gn zeta for n sufficiently large, okay. So now, now i want you to just watch. See
after all, you know the, the values of gn in this punctured disc centred at the origin
o and radius rho by epsilon n correspondence to exactly the values of f in the punctured
disc centred at z0, the values of f inside this whole disc, punctured disc centred at
z0 radius rho, they are exactly the values of gn, in the punctured this centred at the
origin, radius row by epsilon n, because this is just a scaling and data solution, okay.
Now you see now you know f, but what is, what have we assumed about f, we assumed that
f omits 2 complex values, f is analytic function and it omits at least 2 complex values in
this disc, punctured disc centred at z0 radius rho. Therefore each of the gns will also omit
those 2 values in their domains, okay. And of course each of the gns are also analytic
functions because they differ from f only by bilinear transformation consisting of scaling
under and incarceration. But now you know we are in good shape because what we have
done you know, we have been able to get a family of, we have been able to get a family
of functions gns which are analytic and which omit 2 values.
Now immediately montel’s great theorem will tell you that there has to be a normal and
you get a convergence of sequence and then you have to examine the limit, okay. And the
limit will give you contradiction, all right. So basically the contradiction will be set
the limit function at the origin will, examining the limit function at the origin will tell
you that the origin has to be either pole or removable singularity for f which is not
true. I mean analysis of limit function will tell you that the, you take this limit function,
this limit function will also be defined on the punctured plane, okay.
Because all the original gns are all defined outside 0, okay, so if you analyse the limit
function. See the limit function is like zooming into f at z0 infinitely many times, okay.
So behaviour of the limit function at the origin which is an isolated singularity will
reflect upon the behaviour of f at z0. And by analysis we will show that if you analyse
the limit function, there are only 2 possibilities z0 should either be removable singularity
or it has to be a pole and both of these contradictions because i have assumed z0 at f to be a essential
singularity, okay. And that is how the proof goes, so it becomes as simple as that.
So let me write this down note that gn is a normal family as it also does not assume
the values that f omits, okay. So, so what does it mean, it is a normal form means, it
means that you know, and it is a normal family and mind you for gn because it is a zoomed
function, whose domains are becoming bigger and bigger and bigger, you can think of them
as a normal family you know in with a limit in the punctured plane, all right. So
what you must understand is that if you take, for example if you take the punctured unit
disc, okay, to take the punctured unit disc then for n sufficiently large, all the gns
are going to be defined, their domains are going to become bigger and bigger and they
are going to contain the punctured unit disc, okay.
So you can see that if you want to take the, if you want to talk about the domain of the
gns, okay, you can assume that for n, p on, for n sufficiently large the domain contains
unit disc if you want. Or for that matter, any finite disc with of course the origin
omitted, okay. And when you take the limit function that is because gn is a normal, is
normal, if you take the limit function, the limit function will be defined on the whole
punctured plane because it will make sense, because you are covering every point in the
plane literally. Because for every point in the plane if you take epsilon n sufficiently
small, rho by epsilon n becomes sufficiently large and gns beyond a certain state will
be defined with that point.
And therefore the limit of all the, if you take convergence of sequence of gns, the limit
will also be defined with that point, okay. So let me write this, thus gn k, thus there
exists gn k case of sequence that converges normally as g on, g of zeta if you want on
c minus the origin, okay. So this happens, because what is the meaning of normal family?
Normal means that, normally sequentially compact, that is give me any sequence, there is a normal
convergence subsequence. So when gn itself is a normal family, it is already a sequence,
so it will have a normally convergence of sequence. So you have a sub sequence gn k
which converges normally to g on c minus origin, okay.
But the point is that it is not, it is not, what is important is that it is g lives in
a neighbourhood of 0, okay and now, now try to understand each gn is analytic on, on the
punctured disc, on the punctured disc centred at the origin, okay. Therefore this limit
function g is a normal limit of analytic function. We have already seen such a normal limit can
have only 2 possibilities, either the, either the normal limit can completely be analytics
or it can be identically infinity, these are the only 2 possibilities. So let us, so let
us write that down. Thus g is identically infinity or g is analytic in c minus0, okay.
Now let us look at both of these cases. Suppose, g is identically infinity, okay, suppose g
is identically infinity, so what does it mean, what this will mean, see, think of it heuristically,
g is f zoomed you know infinitely at z0 and if g is identically infinity, what you are
actually saying is that f is infinity in the neighbourhood of, f is going to be infinity
in the neighbourhood of z0, right, because the values of g are just limits of values
of gns and the values of gns are just values of f in smaller and smaller neighbourhood.
So if g is identically infinity, okay, that means that the value, the gns are getting
larger and larger in modulus, okay.
And that means that the values of f are getting closer and closer to infinity as you approach
z0. But that means z0 is a pole, but that is not possible because z0 is an essential
singularity, so this is not possible, so you ruled out this case, okay. So let me write
this down. This means that g of zeta is infinity for all zeta in c minus infinity. I will have
to make use of the fact that you know gn converges to g, okay, g is not just, it is not simply
limit of gns. Of course it is point wise limit of gns but it is not, it is more than that,
it is not just a point wise limit, it is just a normal limit.
So the convergence is uniform complexion since, okay. So you know gn of zeta converges to
g of zeta uniformly on mod zeta is equal to say r for any r greater than 0. Because you
see mod theta equal to r is a circle in the case of plane centred at the origin, radius
r and that is certainly a compact circuit, it is closed and bounded. And g is a normal
limit, therefore the convergence gn to g should be uniform on any compact subset, so it has
to be uniform on mod theta equal to r, okay. But, but of course what is, but what is g
zeta? G zeta is infinity, if g is identically infinity. So what this and and and what does
uniform convergence mean, it means that all the gns, they will come to within an epsilon
of g zeta, okay, if you take n sufficiently large irrespective of zeta, right, therefore
that means, okay.
But of course you want to come to an epsilon of infinity so you will have to be careful
and you have to use a spherical metric. So let me write this down, thus given epsilon
greater than 0, the spherical distance between gn zeta and g of the zeta which is actually
infinity can be made less than epsilon for n sufficiently large. And for all
theta with mod theta equal to r, okay but now, but what is gn zeta? You see gn zeta,
our definition is just f of z0 plus epsilon n theta, this is what it is. Okay.
And as you know, you see even if mod zeta is r, if your, your epsilon ns are becoming
smaller, so you are covering smaller and smaller disc, you are covering smaller and smaller
circles centred at z0, okay. And therefore what you are saying is that the function values
of f on smaller and smaller circles centred at z0 are getting close to infinity, okay.
And and and that is enough to tell you that f, the limit of f let as z tends to z0 is
actually infinity, which means that z0 has to be a pole. But that is a contradiction
to our assumption that z0 is actually an isolated essential singularity, okay.
So let me write this down, this means that the spherical distance between f of z and
infinity can be made less than epsilon uniformly on compact subsets
of mod z minus z0 lesser than epsilon n for n sufficiently large, okay. So, you know because,
what you must understand is that this, this r is, this capital r is at our disposal. You
can make this capital r as small as you want, close to 0, you can make it as large as you
want, okay. So you can, you can literally cover all the , you can cover all the
circles centred at z0 of fixed radius, okay, below a certain positive value.
So in some sense you are, therefore you are able to cover complete deleted neighbourhood
of z0, okay, that is the whole point. So, well, so , so this implies z0 is a pole
of f, which is a contradiction. Because you will assume z0 to be an essential singularity,
okay. So does, you know you, the limit, the zoomed limit function g cannot be identically
infinity, okay. Therefore what is the other possibility, it has to be only be an analytic
function in the, with, in the punctured plane, punctured r, okay. Thus g is analytic in the
punctured plane, okay. And what does that mean, it means of course the origin is a singularity,
origin is an isolated singularity for g, okay. And now you can ask what kind of singularity
it is.
But you know the point is that again you should not try to study the singularity of g at the
origin, if not do that. Because after all g at the origin is going to reflect f at z0,
okay. So, mind you the gns are all zoomings of f at z0 and their limit is g, okay. So
g is some kind of infinite zooming of f at z0. G at the origin is infinite zooming of
f at z0, okay. Therefore you should not study g at the origin but you must make use of the
fact that g is analytic in the outside the origin. So if you again take this mod theta
equal to r, it would take circles centred at the origin, this is the plane, radius r.
Mind you again that is a complex set and g being analytic, if continuous, then on the
compact that it will be bounded, okay. And now this bound will apply to f in the neighbourhood
of z0, okay. And we will tell you that f is bounded in the neighbourhood of z0 but then
riemann’s removable singularity will tell you that z0 has to be removable singularity
and again that is a contradiction. And that is the proof of the theorem, okay, proof of
the theorem is so simple. So let us go to the other case, again there exists an m greater
than 0 such that again on mod z, mod zeta equal to r greater than 0, there exists an
m greater than 0 such that mod g is less than or equal to m, okay.
And, but after all, since gn converges to g normally, in fact uniformly on mod zeta
equal to r, because, again because it is a compact set we have fn f is bounded
in a deleted neighbourhood of z0, this again implies by the riemann’s removable singularity
theorem
that z0 is removable singularity of f, again a contradiction. And you know that finishes
the proof. There are only 2 choices for g and both choices lead to contradictions. So
that is the of the famous big picard’s theorem. And we can, as a corollary we can deduce the
little picard’s theorem.
What is the little picard’s theorem? It tells you that the image of the complex plane
under entire function is again the whole plane or the plane minus the point. And now what
is the proof, the proof is very simple, taken entire function, of course we should take
a nonconstant entire function, okay because if you take a constant entire function, the
image of a constant function is always only one point. So you must be careful, i must
have been carefully saying that statement. If you take a nonconstant entire function,
then you know the image of the complex plane should be either the whole complex plane or
complex plane minus a point, it can omit only one value at most.
What is the proof, very simple, take the, take the entire function and look at the point
at infinity, okay. The point at infinity becomes the isolated singularity because the complex
plane is a deleted neighbourhood of infinity in the external complex plane, okay. So your
function f, your entire function f has infinity is an isolated singularity, okay. Now for
an isolated singularity what are the possibilities? It can be removable, it can be pole or it
can be essential. If it is removable, it means that, if infinity is a removable singularity,
it means that f is bounded at infinity and that means that by louisville’s theorem,
f has to be a constant.
So if we take f to be a nonconstant entire function, okay, infinity cannot be removable
singularity, all right. So it can only be a pole or essential singularity. If infinity
is a pole, then you have already seen that f has to be polynomial, okay. And you know
a polynomial will take all values because of the fundamental theorem of algebra. So
if f is a, f has infinity as a pole, it is a polynomial, the image is a complex plane
under f is the whole complex plane, using the fundamental theorem of algebra. So the
only thing is infinity is an essential singularity.
If infinity is an essential singularity for f, apply the great picard’s theorem, okay.
F can, any neighbourhood of infinity has to be mapped by f into the whole complex plane
or the complex plane minus the point. At the complex plane itself is a deleted neighbourhood
of infinity, so f has to map the whole complex plane or the complex plane minus the point,
that is it. Okay, so i will just write this down.
So corollary, picard’s little theorem, is f is a nonconstant entire function, then
f c is equal to c or c minus z0, okay, for some z0 in c. So proof is infinity
is an isolated singular, is an isolated singular point
for f in c union infinity as c is a deleted neighbourhood of infinity where f is analytics,
okay. Thus infinity is either removable pole or essential. If infinity is it removable,
f is bounded at infinity, so by louisville f is constant, a contradiction. Because i
am assuming f is a nonconstant fashion, okay, nonconstant entire function. If infinity is
a pole, we have seen earlier that f has to be polynomial, be a nonconstant polynomial,
which assumes all complex values with the fundamental theorem of algebra, okay.
So the only other case is if infinity is a pole, i mean if infinity is an essential singularity,
infinity is an essential singularity, plus the big picard’s theorem, by the big picard’s
theorem or the great picard’s theorem, f of c is c or c minus point z0, okay. So that
finishes the proof of the little picard’s theorem. And therefore you see you are
able to prove the picard’s theorem very easily. And the key to all this is, this this
really great theorem of montel, it says, this is the criterion for normality of a family,
okay.
And it is a very very simple criteria, in the sense that if you know is family functions,
if it is a family of meromorphic functions, if you know that it omits 3 values, okay,
in the extended plane, then you what is normal. If it is a family of analytics functions,
that means it omits 2 complex values, then you know again it is normal. And the advantage
of normality is that it is a kind of compactness. Namely it is normal sequential compactness
which allows you to extract from any sequence is of sequence which converges normally, that
is which converges uniformly at compact subsets, okay.
So that finishes the proof of the picard theorem which was the main aim of this course, okay.
What i would like to next do is to tell you that, to tell you how, how powerful zaltzmann’s
lemma is in several other contexts, okay. Mind you that this reasonably simplified
proof of the great picard theorem was possible because of the montel’s theorem on common
normality, okay. And that in turn was proved by zaltzmann lemma, okay.
So these are all actually all the simplifications are because of zaltzmann lemma, that is the
most important thing. But the zaltzmann is, so similarly in the proofs of various other
theorems and various other theories of complex functions, zaltzmann lemma provides us with
easier proofs of some very deep results and also provides us with new results. And i will
try to outline those results in the coming lectures, okay. So i will stop here.