Okay so let us continue with what we are trying to do, so you know we are trying to prove
the final aims to prove the Picard theorems okay and the little Picard theorem will be
deduced as corollary of the great Picard theorem okay and but the great Picard… The little
Picard theorem is about the image of an entire function, the great Picard theorem is about
the image of a deleted neighbourhood of an isolated essential singularity okay and what
we are going to now 1st prove is the easiest thing to prove to begin with to get you a
feel of things is the Casorati Weierstrass theorem which says that if you take any neighbourhood
of an isolated essential singularity the image of that neighbourhood of course with the singularity
omitted of course the image of that neighbourhood will always be a dense subset of the complex
plane okay namely its closer will be the whole complex plane okay.
What this means is that it at any complex number you can always find a sequence of points
in any neighbourhood of an isolated essential singularity such that the function values
at those points approaches that complex number okay, so let me write that down let us prove
it, the key to proving the Casorati Weierstrass theorem is the Riemann removable singularity
theorem which we proved last time okay, so let me write that down.
So Casorati Weierstrass theorem, so let Z naught be an isolated essential singularity
of the analytic function f of Z given any complex number W naught we can find a sequence
of points in a neighbourhood of Z naught such that if you take the function values at those
points fZn that tends to W naught okay, so this is the this is the Casorati Weierstrass
theorem okay. You take an isolated essential singularity Z naught of an analytic function
f of Z and then you can always find a sequence of points in the neighbourhood of Z naught
such as the function value at those points tends to the limit W naught okay so in this
way, so what you are what does this mean, it means see W naught can be…f of Z n tends
to W naught means that you can no matter how close you go to W naught confined function
values okay.
So that means that the W naught is in the closure of the set of function values that
what it means, so W naught is the limit point of the image set of f, image set of f is just
the set of values of f okay and what this says is that every complex number is in the
closure of the image set, the set of values that f takes okay and this is this very deep
theorem because what it says is that it tells you that therefore the if you take the image
of a deleted neighbourhood of the isolated essential singularity then the image is going
to be huge a set which is tenses huge okay, the image is going to be dense in the complex
plane.
So that means the image closure is the whole complex plane so it is the image is huge and
this is the this is kind of much weaker when compared to the the great Picard theorem which
says that the images actually the whole complex plane or at most a punctured plane namely
it can at most omit a point okay but you must remember that that omitted point is also in
the closure because it can be approached by the points okay, so so this Casorati Weierstrass
theorem is weaker version of you know the great Picard theorem but it tells you it answers
this question that we have been worried about namely what is the image of an analytic function
okay, so so let us prove this, the prove of this is is going to involve it is just
going to involve Riemann removable singularity theorem, the use of Riemann removable singularity
theorem, so so let me write this also in other words, the image of f is dense in the complex
plane, this is another way of saying it okay.
So so let us go on to the the proof of the theorem, so you know so you know the idea
of the proof is very very simple, what you do in the proof is that you prove by contradiction
okay, so you assume that there is a complex value which is not in the closure of the
image that means there is a complex value which is not in the limit of values in the
image okay that means that the image is bounded away from a certain complex value okay you
assume that and then you show that if this is the case then your function cannot have
an isolated essential… Essential singularity at Z naught but the singularity has to either
be removable or it has to be a pole and this is where we will be using Riemann removable
singularity theorem, so this is the technique of the proof.
The taking of the proof is just by contradiction okay. You you assume that a certain value
is not in the limit okay and then you sure that this will imply that either Z naught
is a removable singularity or it is a pole and both of these are not possible they will
give contradiction because we have assumed that Z naught is an essential singularity
which by definition is something that neither removable singularity nor a pole okay so that
is how the proof works, so let me write down the proof, so prove so let me go to a different
color. Proof by contradiction, assume that there existed W naught complex value
such that such that the conclusion of the theorem does not hold so you assume.
Conclusion of the theorem is that given any complex value W naught okay you can find the
sequence of point Z n such that the function values at Z n approaches W naught, now you
assume that that is not the case assumed that there is at least one W naught or which this
does not happen okay and we will try to get a contradiction, so so what does this mean?
What it means is that it means that W naught cannot be approached by image points okay
the another way of saying this is that you are trying to say that there is a neighbourhood
of W naught okay which has nothing to do with the image okay which does not intersect the
image.
So that means there is an Epsilon greater than 0 such that this open disk centred at
W naught radius Epsilon is disjoint from the image of f okay that is the that is what it
means okay so let me write that down so thus that exists Epsilon greater than 0 such that
mod W minus W, W naught less than Epsilon does not meet the image of f, image of f Im
f is just the set of values of f and set of values of f includes all values at f takes
in a also in the deleted neighbourhood of the isolated singularity Z naught which
we have assumed is essential, okay so so what does this mean, so what do I do now?
Now you see so let us rewrite that thus for all Z not equal to Z naught of course okay,
we have mod f of Z minus W naught is greater than equal to Epsilon this is what it means
okay. So so let me again tell you what I wrote down before what I wrote down before is that
the disk centred at W naught radius Epsilon does not meet the image okay, it means that
there is no point in the image whose distance from W naught is less than Epsilon, so it
means that if you take any point in the image the distance from that point to W naught is
at least Epsilon greater than Epsilon that is what I have written down, so when you write
mod f when I write mod f Z minus W naught is greater than not equal to Epsilon I am
actually saying that the distance between f of Z which is the point to the image of
f the value of f at Z and W naught is at least Epsilon.
Now so you know the advantage with having this kind of thing is that see whenever you
have a function which is bounded away from 0 okay, what is the advantage of having
a function on bounded away from 0? The advantage of having a function bounded away from 0 is
that you can invert it okay it is reciprocal make sense okay, so here f Z minus W naught
mind you is also function f of Z is a function, f Z minus W naught is just a function f of
Z added to the constant minus W naught and adding this does not change the analyticity
of the function except at the point Z naught where which of course we are not going to
worry about okay.
Mind you when I write f of Z I am of course assuming Z is in the domain of analysis the
of f it is implicit and of course Z is not Z naught because at Z naught is not a point
where f is analytic to begin with I have assume that Z naught is an essential singularity
it is a singular point okay, so so the point is that since I have mod f minus W naught
is greater than Epsilon 1 by f minus W naught makes sense as function okay, so and what
it tells you is that that function in a deleted neighbourhood of Z naught is bounded by 1
by Epsilon okay that is what it says, so now look at g of Z defined to be 1 by f of Z minus
W naught okay in in a deleted neighbourhood of Z naught. Now look at this function, now
you see this function mod g Z is greater than or equal to is less than or equal to 1 by
Epsilon you have that that is just inverting mod f Z minus W naught greater than equal
to Epsilon.
So what you have is now you have 2 things see g of Z makes sense as an analytic function
okay it is because it is a reciprocal of an analytic function and it is defined where
the denominator does not vanish, so f Z minus W naught never vanishes because if it vanishes
then it is modulus will be 0 and vice a versa but the modulus is always bounded away from
0 it is always greater than equal to Epsilon, so f Z minus W naught never vanishes okay
and mind you wherever f is analytic, f Z minus W naught is also analytic okay because it
is just f of Z added to the constant minus W naught adding a constant does not change
the analyticity of a function because or constant function is also analytic and sum of analytic
functions is analytic okay, so f Z minus W naught is also analytic in a deleted neighbourhood
of Z naught okay and it is non-zero.
So it is reciprocal 1 by f Z minus W naught is as well analytic in a deleted neighbourhood
of Z naught okay, so what I have now is I have this function g Z it is analytic in the
deleted neighbourhood of Z naught okay Z naught is a singular point but look at the last inequality,
this function is bounded in a neighbourhood of Z naught. Now Riemann removable singularity
theorem will tell you that Z naught has to be a removable singularity for g okay, so
since so let me write that down, since g of Z is analytic in a deleted neighbourhood of
Z naught, Z naught is also a singular point of g of g and further is a removable singularity
by Riemann’s theorem on removable singularities since it is bounded okay, so here is where
we are using the Riemann’s removable singularity theorem okay.
So what it means is that so it means that g can be redefined at Z naught so that you
get function which is analytic at Z naught as well okay and the fact that you can define
g redefine g at Z naught okay should tell you that f has to have at Z naught either
a removable singularity or a pole, it cannot have an essential singularity that is the
conclusion of all this and that is the contradiction to the hypothesis that Z naught is actually
an essential singularity and that is how we get the proof of the theorem of the Casorati
Weierstrass theorem okay, so let me write that down. Thus limit Z tends to Z naught
g of Z exist call it g of Z naught and then g extends to an analytic function at Z naught
okay, so this is what removable singularity means you can take the you can take the so
so this is probably the right time for me to recall the Riemann removable singularity
theorem, what does it say?
It gives you an equivalence of 4 statements, the first statement is that the point in concern
the isolated singularity concern is is removable okay which is equivalent to saying by definition
that the function can be extended to an analytic function at that point. The 2nd condition
is slightly weaker that you can extend the function continuously to that point okay namely
the condition is that the limit of the function as you approach that point exist okay as a
finite complex number and then the 3rd condition was the condition equivalent condition that
involves the Laurent expansion and that condition was that if you write out the Laurent expansion
about removable singularity you actually get Taylor expansion namely there are no negative
powers there is no principle part, there is no singular apart okay.
And then the 4th condition which was the most amazing was the condition that the function
is bounded in a neighbourhood of the singularity okay bounded of course bounded means bounded
in modulus okay and I told you that that is the weakest of all the 4 conditions and that
boundedness in the neighbourhood of a singularity can only happen if the singularity is a removable
singularity that is essentially Riemann’s removable singularity theorem.
So so you know so because of that the limit Z tends to Z naught g of Z exist let us call
it as g of Z naught so g extends to analytic function at Z naught, so here I am using several
equivalent version of the Riemann removable singularity theorem you must realize that.
Now the question is it all depends on what is the value of g of Z naught is, the
point is that if g of Z naught is 0 okay then f has a pole at Z naught okay and if g of
Z naught is not 0 then f has a removable singularity at Z naught okay and then thus we have manifest
a contradiction to what we have assumed okay so let me write that down.
Now if g of Z naught is 0 we see that limit Z tends to Z naught f of Z has to be infinity
this has to happen because you know the limit as I said tends to Z naught g of Z is
0 so the limit is Z tends to Z naught 1 by f Z minus W naught to 0 that means the denominator
has to become unbounded, so that means and if f Z minus W naught has to become unbounded
in modulus then f has… because W naught is just a constant f has to become unbounded
in modulus and that is a condition for a pole okay, so so if g of Z naught is 0 then limit
Z tends to Z naught f of Z is infinity and Z naught is a pole, is a pole of f and you
see Z naught is a pole of f which is not possible so that is ruled out so you have ruled out
the case that g of Z naught is 0 the only other case is when g of Z naught is not zero
okay.
If g of Z naught is not equal to 0 then limit as Z tends to Z naught f of Z is actually
W naught plus 1 by g of Z naught will get this okay which means again by Riemann’s
removable singularity theorem that f has a removable singularity at Z naught okay
which is again not possible, so so in both cases you get a contradiction and we are done
okay so that that is the that brings you do the end of the proof okay so so you see we
are applying the you must see that we have applied the Riemann’s removable singularity
theorem twice we have applied it once to g and then we have applied it in one of the
cases to f itself okay, fine. So this so this theorem is a very nice theorem and so what
it tells you is that you take neighbourhood of an isolated essential singularity and take
its image you are going to more or less fill up the whole complex plane.
You are going to get the images that is dense and we have to move towards the proof the
the great Picard theorem okay. Now what I am going to do next is going to deal with
the point at infinity okay, so you see you see in this proof itself for example when
I wrote down limit Z instead to Z naught f of Z is infinity okay I am using the point
at infinity, so you would have seen the point at infinity as the extra point that is added
to get 1 point compactification of the complex plane and you would also have seen it as Riemann’s
sphere in the 1st course and but anyway I want to recall these things because you see
there is very important from now on to be able to think of infinity both in the domain
of definition of the function as well as in the range of values of the function.
So you want to have a situation where you can talk of a function variable, an independent
variable going to infinity and its value at for example the value of function at infinity
you want to say that and you also want a function to take the value infinity okay you want to
include infinity into your set of values of the independent variable and the set of values
of the independent variable, so you have to deal with infinity carefully and usually in
the 1st course probably this is sometimes not covered very thoroughly, so I want to
just revise these things so that you are comfortable about thinking about limits at infinity and
infinite limits okay, so that is what I am going to do next.
So and I need that because because of the following reason, see I want to be able to
think of infinity as one of the values of a function okay and I also want to be able
to think of infinity as singularity okay see For example if you take an entire function
okay then the point at infinity is of course approachable by infinity is of course approachable
by any by any curve on the complex plane which is not bounded okay so you can always approach
the point at infinity and then the question is whether the function is analytic at infinity
or it is not analytic at infinity, so you want to think of infinity as a singular point
okay and then the question is what kind of singularity is it because you know we have
already classified singularity as either removable or pole or essential, so the question is I
want be able to think of infinity the point at infinity as a singularity and question
or study when they singularity is either removable singularity or a pole or essential singularity,
so the point at infinity is very very important.
So so let me go to that so I will take another color the point at infinity infinite limits,
infinite values so this is what I am going to I am going to tell you about okay,
so well so the idea is as follows, so what we do is let us so 1st of all let me you know
the approach to everything is since we are doing calculus approach is always through
limits, so let me recall what finite limit is okay as we as an independent variable approaches
finite value, so you know see limit so when I write limit Z tends to Z naught okay when
I write this what does it mean? For Z naught in C for let Z naught be a complex number,
okay.
What does limit Z tends to Z naught mean, see it means that you going closer and closer
to the point Z naught okay so basically what it means is that and what does it mean to
say that you are going closer and closer to Z naught, if you think of Z as a moving point
a variable point then you are saying that the distance Z to Z naught is becoming smaller
and smaller okay so the limit Z tends to Z naught can be interpreted as limit mod Z minus
Z naught tends to 0 okay that is how you can interpret it okay so this means so let me
write that here means mod Z minus Z naught tends to 0 okay means we let so let me write
that we are letting mod Z minus Z naught tends to 0 and mod Z minus Z naught mind you is
the distance between Z and Z naught and what it means is that, if you go to definitions
what is the business of letting something go to 0.
In analysis trying to let something go to 0 is the same as making it as small as possible,
so that is where your Epsilon comes in, so usually we use Epsilon for the values of the
function and use delta for the values of the variables, so let me use delta so the point
is that you are putting you are choosing delta as small as you want and you are letting mod
Z minus Z naught less than delta okay so let me write that down which in turn means
means let delta be small consider mod Z minus Z naught less than delta and let delta tends
to 0 this is what it means you are making something small means you are actually taking
values of that which are getting closer and closer to 0 okay and you know therefore you
know of course all this conveys very clearly what is happening topologically mod Z minus
Z naught less than delta is actually a disk, it is an open disk centred at Z naught radius
delta.
It means that you are going infinitesimally small very very small neighbourhood of Z naught
and the smaller the delta is the smaller the neighbour is, so you are basically looking
at you are just concentrating attention at a very small neighbourhood of Z naught that
is what it means okay and now you know in the same way… so now this this can be
used to also define what limit Z tends to infinity means okay so, so limit Z tends to
infinity, what we make of this? What you make of limit as Z tends to infinity, okay so now
this is the point where you will have to have a little bit of imagination okay, so the idea
is that 1st of all you should be able to think of infinity as a point okay as a concrete
point and the 2nd thing is that once you think of it as a concrete point in a space then
you can think of limit Z going to infinity just as you thought of limit Z going to Z
naught you know the limit as Z going to Z naught meant that you were going to a small
neighbourhood of Z naught.
Now if you can think of infinity as a point in the same way limit Z tends to infinity
means at your going to a small neighbourhood of infinity okay, so all you need is the way
of thinking of infinity is a point on a space where you can think of a small neighbourhood
around that point okay and the key to this is as you would have seen in the 1st course
in complex analysis, the key to this is the so-called Riemann’s stereographic projection
okay, so let me explain that, so let me write this down as go to a small neighbourhood neighbourhood
of infinity, now what does that mean? What does that mean? So the key is the Riemann’s
stereographic projection that is the key.
So let me recall what that this, so basically so the idea is you should have seen this,
so what you do is the following. So let me draw a diagram so here is well this is the
three-dimensional space, so this is my so let me draw it like this, so this is my
usual x y plane, this is the origin okay and this is the x axis and this is the y axis
okay or rather this is if you want to be right-handed then that is the y-axis as the negative y-axis
right, so so if you want the 0.1 is here 0.1 on the y-axis is here, so this point is 1,
0 this point is 0, 1. So if you think of this x y plane
as usual complex plane then you have 2 coordinates x and y and of course 1, 0 is the 0.1 complex
number 1, 0, 1 is the complex number I okay the point is I want to put in a 3rd axis which
are normally in three-dimensional you would call the Z axis but you do not want to use
Z because Z is already supposed to be x plus I y.
So you use let us see with something else for it, you use u if you want okay some books
use u so let me also use it, so what we do is now you take the three-dimensional space
now the snore xyz but it is xyu there is a positive u axis then what you do is that you
draw this you draw this circle I mean you draw this sphere centred at the origin and
radius 1 okay so so let me rub these coordinates off because it will make things easier for
me to draw, so you know draw this, so I have this circle I have this circle, this is the
unit circle on the complex plane and then I have this 0.1 with u coordinate 1 okay and
x and y coordinates 0 okay, so it will be the North pole of a sphere of this sphere
centred at the origin and radius 1, so what I am going to get is am going to get something,
so I am going to get something like this, so here is my sphere.
So this is this sphere is, this is the sphere that is called the Riemann’s sphere, it
is the sphere centred at the origin radius one unit and this point here which has coordinates
0, 0, 1 or x y and u is called the North pole so I will use the word I will put the symbol
N okay and of course even if you project it all the way down you are going to get the
point with ordinate 0, 0, minus 1 which is the South pole so-called South pole okay you
can think of the Earth as a sphere and you have the North pole and the South pole it
is just like that and so what is stereographic projection? So what it does is that so let
me call this sphere let me give you a name for the sphere, so let me call this as I will
put this S like a dollar symbol and put S 2 and I will put a subscript R, so this is
the standard topological notation.
The 2 on top S is supposed to denote as sphere okay the 2 on top this supposed to denote
the dimensions okay it is the surface of the sphere okay and mind you I am not taking the
solids sphere I am only taking the surface of the sphere which is surface okay, so it
is two-dimensional and by dimension I mean real dimension, so it is real two-dimensional
okay and the subscript R is to remind you that this is being done in real space
okay, this is being done in real space, real 3 space so the ambience space here is R 3
, so the ambience space here is R 3 in 3 space the only thing is that I am treating the x
y plane as the complex plane and instead of the usual Z axis I am calling it the U axis
because Z is already reserved for x plus i y of now what you do is well this is a stereographic
projection there is very very simple projection what it does is that it goes from the sphere
I will call this sphere the Riemann’s sphere okay.
So it is called the Riemann’s sphere okay it goes from the Riemann’s sphere minus
the North pole to the complex plane to the complex plane okay and what is a map?
So what you do is you take any point on the sphere okay and mind you you are taking a
point on the surface of the sphere okay and you are not taking the point at infinity I
mean you not taking the point N you are not taking the point N, so as I just said inadvertently
the point and will be the missing point at infinity okay so that will be the analogy
so we will see that, so you take any point P here on this sphere other than the North
pole and then what you do is you joined this thing the straight line passing through N
and in P okay that straight line will go down and hit the plane at some point and that point
will give you complex number because for me any point on the plane is a complex number.
I have thought of the x y plane as a complex plane and that is the complex number to which
I am going to send P to okay and you can see clearly it is a project map okay, so projection
map and that is why this called the stereographic projection okay. So so basically what I do
is that I take this line from N passing through P and then it will go and hit the let
me call this as phi of P, so this is the map phi which sends P to phi of P and phi of P
is a complex number, phi of P is a complex number and this is a stereographic projection,
so you have if you want to think of it as a projection it is like this okay.
Now now the beautiful thing is that so the beautiful thing is that this map phi is actually
a Bijective map you can very will see that if P changes then phi P will change so it
is an map okay and conversely give me any point on the complex plane it is so the form
phi P for a unique point on this on the Riemann sphere because I can get that point by simply
joining that point to the North pole that will hit the sphere at a certain point and
that will be the point which will be mapped to the given point under phi okay so it is
very clear that this map is Bijective okay it is very clear that this map is Bijective
and in fact you can try it out as an exercise this map is actually a homeomorphism there
is a topological isomorphism.
See the complex plane as a topology and this topology is also a topology that it is a same
as a topology that the plane inherits as a subspace of three-dimensional space okay you
take the x y plane take the complex plane x y plane and treated as plane in 3 space
and you take the natural topology on R 3 you restrict that topology to the subset that
is the same as the topology on the complex plane okay, so the topology on the complex
plane is same as the topology that it inherits as a subspace from 3 space and this sphere
is also living in 3 space, so it is also subset of the 3 space so it also has inherits a topology
okay and the fact is that this map from this sphere to the plane is in fact continues Bijective
map which is open and therefore you know its inverse is also continuous so it is homeomorphism
it is a topological isomorphism there is a map which is continuous whose inverse is also
continuous okay and of course the inverse being defined because the map is Bijective
okay.
So the beautiful thing is that phi is actually homeomorphism okay so the fact that phi is
a homeomorphism so let me write that down but before that let me tell you something
the fact that phi is a homeomorphism tells you that therefore you can think of the whole
complex plane as a punctured sphere okay see S2 minus N is a punctured sphere, it is the
sphere minus the North pole and what this homeomorphism tells you, what does a homeomorphism
tells you? It tells you that the 2 spaces are topologically the same, same means up
to isomorphism, so when you say phi is an isomorphism, topological isomorphism namely
homeomorphism you are actually saying that you get this is just another way of saying
that the complex explain can be thought of topologically as a punctured sphere that is
the significance of this statement okay.
So let me write that down, so here let me take another color phi is check so this check
is something that is more or less obvious but you should do is phi is a homeomorphism
okay I am certain many of you would have done this in the 1st course in complex analysis
but it is not very difficult to do if you have not done it, So phi is a homeomorphism
which tells you that the complex plane can be thought of as a punctured sphere okay,
now the punctured sphere this is only 1 point namely the North pole and now you know you
have you wanted a point at infinity you wanted to attach to the complex plane a point at
infinity but the point is where do you attach it?
I mean you cannot see it okay but then if you look at this picture you think of the
complex plane like a punctured sphere now what is the extra point that you will have
to add to make it the whole sphere and mind you when you make it the whole sphere only
then it becomes compact okay if you remove a point from a sphere it loses compactness
okay because it will not be closed since we are in Euclidean space a subset is compact
if you know only if it is closed and bounded, so this sphere this of course any subset of
the sphere is of course bounded but the problem is that unless it is closed it is not compact,
so the only way to make it compact is to hand that missing point that in this case it is
the North pole okay.
So here comes the nice upshot of this what you do is you think of the complex plane plus
the point at infinity you denote the point at infinity as with the symbol of infinity
and think of it as an extra point that you add to the set of complex numbers and then
what the topology you give, you give the topology which makes the natural extension of this
homeomorphism phi into a homeomorphism from this sphere to the complex plane which is
a complex plane with that extra point at infinity added okay, so define phi from the Riemann
sphere to C union infinity by sending N to the point at infinity okay.
So the North pole maps to the point at infinity and once you do this what we have done is
you have given a Bijection between this sphere and C union infinity mind you C union infinity
will be called the extended complex explain, it is the complex plane plus the point at
infinity and therefore the complex plane plus the point at infinity is now nicely identified
as a sphere okay and the advantage now is that you have therefore the point at infinity
being thought of as the North pole on the sphere and it is a point on the topological
space you can do topology in a neighbourhood of the North pole and think of it as doing
as working in a neighbourhood of infinity okay that is how you think of infinity the
point at infinity, so so I will stop here.