Okay so this is the continuation of the previous
lecture and we were just discussing Hurwitz’s
theorem which , so let me again just recall
what we are trying to show is the following
if you have a sequence of analytic functions
on a domain in the complex plane and suppose
you assume that the sequence converges normally
to a limit function and you allow this exception
that the limit function can take values, the
value infinity then what can happen is either
the limit function is completely analytic
function or it is completely the constant
function infinity namely the function that
takes every point infinity okay.
So in other words and of course here when
I say I am including functions that take the
value infinity it means that I am not just
taking complex valued function, I am taking
functions with values in the extended complex
plane which is the complex plane along with
the point at infinity okay denoted by this
symbol infinity and of course the convergence
whenever the point at infinity is concerned
the convergence should be taken with respect
to the spherical metric okay which is the
spherical metric on the Riemann sphere transported
via the stereographic projection to the extended
plane okay and you must also remember that
the convergence in the spherical metric is
the same as the convergence in the Euclidean
metric on the complex plane so far as domains
of the complex plane are concerned alright
so in the proof of that result we needed Hurwitz’s
theorem.
You see I wanted to understand the beauty
of that result, the beauty of that result
is following, see when you do a 1st course
in complex analysis and you are not worried
about the value infinity okay you are only
worried about complex options then you know
that if you say a uniform limit of analytic
functions you will get an analytic function
and more generally you need not take uniform
limit but you can take our local uniform limit
which is for example by taking a normal limit,
a normal limit is a locally uniform limit.
So even if you take a locally uniform limit
or a normal limit of analytic functions you
will get an analytic function this is what
you study in the 1st course of complex analysis.
The proof involves just Clausius theorem and
Morera's theorem okay at now what we have
to do is you see we have uhh our aim is to
prove the great Picard theorem, the little
Picard theorem, the Picard theorems and the
problem is that to prove that you will have
to worry about infinity as an isolated singularity
okay and the problem is that you will have
to worry about Meromorphic functions and families
of Meromorphic functions you should do topology
on a space of Meromorphic functions and the
fact is that you allow the value infinity
a Meromorphic function becomes a continuous
map into the extended plane. See if you take
an analytic function and suppose it has a
pole at a point okay then as you approach
the pole the modulus of the function goes
to infinity, so the function goes to infinity.
So the function of course becomes discontinuous
the usual sense but then if you allow the
function to take the value infinity you think
of infinity as actually a value and where
it is the extra point that you have added
get the one-point compactification of the
complex plane. It is the point at infinity
in the extended complex plane, mind you the
extended complex plane is a nice topological
space it is a compact matrix it is a complete
metric space okay. It is because it is just
equivalent topological isomorphic to the Riemann
sphere which have all these properties and
now the point at infinity can be seen on the
Riemann sphere as the North pole.
So you can think of it as a valid point and
now if you take a Meromorphic function and
think of it as taking values not just in complex
plane but also allow it to take the value
infinity then the Meromorphic function becomes
a continuous function because what you will
do is at a pole you will define its value
to be infinity and this definition is continuous
okay, okay it is continuous for the topology
on the extended complex plane or for example
if you want to use a topology and that is
of course the topology induced by the spherical
metric on the extended plane okay. So now
you see by allowing infinity we are value
okay and taking continuous functions which
can take the value infinity you are also allowing
Meromorphic functions, now you see what you
have done is? You have jumped from holomorphic
functions on a domain to all the way to Meromorphic
functions on the domain which means you are
allowing even for functions with poles okay
and then you go one step further and say that
you also allow the constant function infinity.
Maybe the function that assigns every variable
to the value infinity okay it is a constant
function infinity and mind you constant functions
are continuous in any sense of the term okay,
so now since all this has happened now if
you again take normal limit of analytic functions,
a normal limit of holomorphic functions then
you know of course if everything is happening
in the complex plane you are only worried
at complex values and if this is really a
usual convergence okay then you will get the
limit function to be analytic that is nothing
more but the pointers now you are allowing
the extended complex plane, you are allowing
the value infinity and why could it not happened
that sequence of analytic functions convergence
in the limit to Meromorphic functions.
So suddenly a pole some poles can pop up in
the limit that can happen right you do not
expect it to happen by continuity okay but
we have already seen that you can get a function
which is identically infinity example I told
you you take the domain which is the exterior
of the unit circle and you take the function
Z, Z square, Z cube, et cetera so the nth
function is Z power N. Now that sequence of
functions, it is a sequence of analytic functions
mind you in fact entire functions that sequence
converges if you look at it in the usual sense
it will not converge because you take any
value Z with a mod Z greater than 1 Z power
n will diverge because mod Z power N will
go to infinity because mod Z is greater than
1 but if you now include infinity as a value
and think of the extended complex plane, Riemann
sphere in disguise okay.
Then this Z power N will tent to infinity,
it tends to a point a value in your set and
a sequence of function Z power N at converges
to the constant function infinity at every
point outside the unit circle and low behold
this convergence is even normal, this convergence
is even uniform on compact subsets with respect
to the spherical metric that is amazing thing.
Once you include the value infinity even you
get normal convergence okay but the point
is that it is converging to the constant function
infinity. Now what is the guarantee that something…instead
of converging at all points where infinity
why cannot it just converge at some points
to infinity? Why cannot it converge on to
infinity only at say an isolated set of points?
That means you are going to get a Meromorphic
function okay or why cannot it converge to
infinity on some subsets which is for example
not even isolated. Why cannot such strange
things happen?
Okay so that is the theorem you trying to
prove, the theorem we are trying to prove
is that either what happens is normal what
you expect normally that the limit function
is actually nice complex valued analytic function
or the other extreme happens. Namely the limit
function is always infinity, it is a constant
function infinity you do not get something
in between, you do not get the Meromorphic
functions with poles they will not come in
between okay so you do not get that. So it
does not go wrong in that sense and that is
the theorem we are trying to prove okay.
See we have to be worried about all these
things because you are allowing the value
infinity okay once you allow the value infinity
anything can happen and then you will have
to be very careful and you have to prove things
carefully. See these are all basically ingredients
that you really need but to understand very
well if you want to understand the Picard
theorems, the proof of the Picard theorems
that is the reason why I am doing this pretty
slowly. So now in order to prove that so how
will be prove that what we will do is we will
take sequence of functions on some domain
in the complex plane and assume that the sequence
converges to a function uniformly on compact
sets okay that is normally and of course this
convergence will be…you are allowing the
limit function to take the value infinity.
So the limit will be in functions with values
not in the complex plane but functions with
values in the extended complex plane as of
course the convergence is going to be normal
alright and what you want to show is that
suppose the limit function is not the function
which is identically infinity, you have to
show that the limit function is actually analytic
okay. The limit function is either analytic
which means it is a complex valued analytic
function or it is infinity that is all there
are only these 2 cases there is nothing in
between okay that is what you are trying to
prove.
So what we will do is essentially for that
you know for proving that we need 2 facts
one fact is the important property that the
spherical metric is invariant with respect
to inversion, so the map is Z going to 1 over
Z that defines an automorphism, self-isomorphism
of the extended complex plane okay it is a
homeomorphism in fact isomorphism in the topological
sense and that when you translate it to the
Riemann sphere it simply becomes rotation
by 180 degree around the x axis that is what
I told you last time and of course distances
on the sphere are not going to change if I
rotate the sphere okay that is obvious. So
moral of the story is that this tells you
that the spherical metric is invariant under
the inversion okay that is one fact that we
need.
The other side that we need in the proof is
Hurwitz’s theorem okay which I thing some
of you must have seen in the 1st course in
complex analysis probably some of you have
not seen but anyway I will tell you what it
is. See roughly this is what I was trying
to tell you at the end of the last lecture
and I am now continuing roughly the philosophy
of Hurwitz’s theorem is the following. Suppose
you have a uniform limit of analytic function
and suppose that the limit function is also
a analytic function okay.
Then if you take a 0 of the limit function,
mind you a 0 of the limit function as to be
isolated because the limit function is analytic
and you know zeros of an analytic function
are isolated and you know this is equivalent
to identity theorem if we have seen it in
1st course of complex analysis, so the limit
you take a 0 of the limit function then Hurwitz’s
theorem says that you see since your sequence
of functions converging to a limit function
then the 0 of the limit function also comes
as a cluster point or as a limit point of
zeros of functions which converge to that
limit function.
So that means there is a uhh so you know geometrically
what it says is suppose you have a 0 of the
limit function at a point Z naught then there
is a small neighbourhood around Z naught where
that will be the only 0 this is because it
is zeros of an analytic function are isolated
and the limit function is analytic okay and
then you can use this neighbourhood sufficiently
small so that in that neighbourhood all the
members of your original sequence which converge
to this limit function beyond a certain stage
that means for all sufficiently large indices
subscripts okay. The functions in your sequence
also have zeros and that disk okay and the
number of zeros is also equal to the order
of the 0 of the limit function at that point
and these zeros as you make the disk smaller
and smaller and smaller is 0 actually converge.
They converge to the 0 of the limit function,
so in other words what it says is that you
know if the limit function has 0 at a point
then all the functions in your sequence beyond
a certain stage should also have zeros in
a neighbourhood of that 0 of the limit function
okay. So a limit function cannot get is 0
just like that okay it cannot happen that
you know all your limit functions never have
any zeros then suddenly you know in the limit
is certainly the limit functions as 0 pops
ups out of the blue out of nowhere that does
not happen okay. So you see intuitively this
is very nice to believe but the problem with
mathematics complex analysis of mathematics
is that you have all these intuitive things
you believe that such thing should not happen
by continuity you believe at such nice things
should always happen but then prove them is
the that is where the meat lies you have to
really sit down and work it out and that is
why all these analysis is being done okay.
So the key to Hurwitz’s theorem is the so-called
argument principle which I will try to recall,
so what is this argument principle, so the
argument principle is as follows, what is
this argument principle? So the situation
is like this you are in some domain, so there
is a domain here and well and inside the domain
there is some there is a simple closed contour
gamma okay and of course the interior of that
on 2 are also lies in the domain okay.
So both gamma and the interior of gamma okay
they lie inside the domain and you have a
function f which is Meromorphic on the domain,
it is Meromorphic on the domain means that
it is analytic with the exception of an isolated
subset where it has poles that is what it
means okay you have a Meromorphic function
and assume that, so you know the function
f has uhh only singularities it has poles
okay and of course it will have zeros also
but you know any way zeros of an analytic
function are isolated you know that, so there
is some subset of the domain which is disjoined
from the subset of poles which were the function
has zeros okay, now what you do is you make
sure that this contour gamma does not pass
through a 0 or a pole, so you assume that
f is not equal to 0 on gamma f has no pole
on gamma okay. Now what is argument principle
if you recall it?
So the argument principle is if you integrate
over gamma the logarithmic integral of f okay
which is by definition and integral over gamma
f dash of Z by f of Z D Z okay because you
know D log is a suggestive notation, the derivative
of log f is one by f times the derivative
of F, so it is f dash by F. So integrating
D log means you are integrating f dash by
f alright, so you calculate this integral
of f dash over f you calculate this integral
over gamma okay and mind you this integral
is very well-defined because you see notice
that the integrant is f dash by f the only
problem is that for the integrant is where
f has zeros because when f has zeros then
f is in the denominator, so f dash by f will
have poles okay but I have not allowed any
zeros of f to lie on gamma.
Mind you when you integrate thing the variable
of integration is varying only on the region
of integration, here the region of integration
is gamma which is the contour and on the contour
f is not going to vanish. So that function
I have written down f dash by f there is going
to be no problem with the denominator and
the numerator there is going to be no problem
because you see if a function has a pole at
a point then its derivative will have a pole
of order 1 more at the point okay because
if a function has a pole at a point Z naught
it locally looks like some g of Z by Z minus
Z naught power N where N is the order of the
pole at Z naught okay and if you take this
g of Z by Z minus Z naught our end and differentiate
it once you will get Z minus Z naught power
n plus 1 in the denominator, so that means
you know if it has a pole at a point then
its derivative will have poles of 1 higher-order
at that point.
So the only way this integrant and get into
trouble because of the numerator is because
of the poles of f but then I have also told
you that none of the poles of f should lie
on gamma, so the integrant has got nothing
uhh has no problems on gamma, so this integral
is well-defined okay and the argument principle
is that this integral is going to give you
2 pi i times the number of zeros minus the
number of poles inside gamma okay that is
the argument principle, so this is equal to
2 pi i times number of zeros minus number
of poles and here of course n zeros is number
of zeros of f inside gamma and n poles is
number of poles of f inside gamma okay.
So what I am saying here is that when you
say number of zeros or number of poles I want
you to realize that you have to count them
with multiplicity that is very very important,
so For example if you have a pole you may
have only 3 poles inside gamma but each pole
may have different orders okay suppose you
had a 3 double poles inside gamma then the
number of poles will become 6 because we have
to count the double pole twice even though
it is one and the same point okay, similarly
0 should be counted with multiplicities okay
for example if you take Z power N at 0 the
multiplicity is N as a 0, if you take one
by Z power N at the origin, the multiplicity
of the pole is N, you should think of it as
n poles the point is geometrically it looks
like only one point at actually algebraically
there are nth of them because there are n
factors alright.
So when you write number of zeros minus number
of you have to count multiplicities. It is
not just the number of points which are zeros
minus the number of points which are poles
that is not correct okay. Now well this is
called the argument principle and you know
the point is that if your function is actually
analytic there are going to be no poles and
what you are going to just get is 2 pi i times
number of zeros okay and why is this called
the argument principle, the reason why it
is called the argument principle is that if
you take this number on the right side and
divided by i will get 2 pi times an integer
okay and this 2 pi times an integer as you
know it is actually an angle 2 pi n is an
angle, so 2 pi itself is one full rotation,
so actually this 2 pi times is integer is
actually the it is an angle and what is that
angle? That is the change in the argument
of f as you go around right that is the change
in the argument that you get as you go around
and in fact there is a change in argument
of f as you go around and the way to see it
is like this you know.
If you wright log f as Ln mod f plus i times
argument of F, suppose you write it like this
okay then you know you can see that if I put
a D to this then I will get D log f is D Ln
mod f plus i d arg f, this is what I will
get and now if I integrate this over gamma
okay if I integrate this over gamma, what
you will get is? So I will get integral over
gamma d log f is equal to integral over gamma
d Ln mod f plus i times integral over gamma
d arg f, now you see what is this? This will
be 0 see d Ln mod f it will be an exact differential,
it will be 0 and basically you see it is a
derivative of Ln of mod f and Ln of mod f
is a nice real valued function okay.
So it is the exact uhh so the 1st integral
is exact and for an exact integral by fundamental
theorem of calculus the integral will be final
value minus the initial value of the anti-derivative.
The anti-derivative the 1st integral is Ln
mod F, so it is Ln mod f final value minus
Ln mod f initial value but this is a close
look, so final value will be the same as initial
value okay therefore it will be 0, so 1st
integral is just 0 and what is the 2nd integral?
The 2nd integral is the change in the argument
of F, it is a change in the argument of the
function f of Z as Z goes one around gamma
okay and that is what this 2 pi i times number
of zeros minus number of poles okay and now
you see that you know there is this i here
there is this i here alright, if you get rid
of this i you now see the reason why it is
called the argument principle, what it actually
says is that integral over gamma d arg f is
actually 2 pi times number of zeros minus
number of poles.
So what it actually says is that if you integrate
the logarithmic derivative what you are going
to get is just the change in the argument
of f and what is that change in the argument
of F? It is 2 pi times an integer and so the
change in the argument of f is mind you it
is multiples of 2 pi, it is a multiple of
2 pi and what is that multiple? That multiple
is actually number of zeros minus number of
poles, and that is what the argument principle
says and one needs this for Hurwitz’s theorem
and how does one get the proof of Hurwitz’s
theorem on this? It is very simple you just
you in use uniform continuity. Let me go to
the proof of Hurwitz’s theorem I am just
giving you a short sketch details can be filled
in later.
So here is the proof of Hurwitz’s theorem
so what is the situation in Hurwitz’s theorem?
You are given that f n converges to f normally,
f in the domain D, this is in a domain D in
the complex plane and f n and FR analytic
okay f has zero Z naught in d of multiplicity
otherwise order N okay this is given and what
is Hurwitz’s theorem, Hurwitz’s theorem
is that all the f n also will have n zeros
okay, they will have n zeros counted with
multiplicities in a small neighbourhood of
Z naught and these as n tends to infinity
the zeros will come close and closer and closer
and in the limit they will all coalesce to
this 0 Z naught that is Hurwitz’s theorem,
so Hurwitz’s theorem says that the 0 of
a limit just does not pop-up like that, it
pops up as the limit of zeros of the original
function which gave that limit okay that is
what Hurwitz’s theorem says.
So what do we do? We prove this just by using
the argument principle is pretty easy, so
see you have Z naught here alright and then
what you do is you choose a sufficiently small
disk centred at Z naught radius delta okay,
so this say radius delta mod Z minus Z naught
less than delta is contained in D you take
a sufficiently small disk and in fact you
also make sure that I will also put mod Z
minus Z naught less than or equal to delta
is D which means that I am also including
the boundary, the reason is I want compactness
by including the boundary I am making it a
closed disk, closed disk is compact because
it is closed end bounded and why do I need
that because I can now use normal convergence
because normal convergence means that whenever
you have a compact subset it is uniform convergence.
So inside this closed disk and in particular
on the boundary of the closed disk which is
a nice circle centred at Z naught of radius
delta mind you that is a compact set that
is also closed end bounded. Even on the circle
I have uniform convergence because it is a
compact subset of D okay that is the reason
why I am including a circle and now you see
you can of course choose this disk in such
a way that Z naught is the only 0 of f that
is because you know f is analytic function,
zeros of an analytic function are isolated
which means that if you give me 0 of an analytic
function I can find a sufficiently small disk
surrounding that 0 where there is no other
0 okay, so you make that also. So let me write
that Z naught is the only 0 of f in the above
disk and of course am not uhh removing that
there is any other 0 even on the boundary
of that disk okay mind you alright then what
does your…
Now what does your now let us write out the
argument principle integral over mod Z minus
Z naught is equal to delta, D log f is going
to be 2 pi times in this is what I will get
I think I will also get an this okay 2 pi
i n this is what I will get okay. If I write
the change in argument and I have to remove
i if I am just writing the logarithmic integral
then I will get 2 pi i okay. Actually it is
2 pi i times number of zeros minus number
of poles mind you the limit function f is
analytic, so it has no poles okay so it has
only zeros and the only 0 it has is at the
center which is Z naught that is because I
have made sure that there are no other zeros
nearby, the zeros of an analytic function
are isolated and what is the order of that
0 at Z naught, it is an okay that is what
I have used here that is one part of the story.
Now look at the following thing, f n if on
the other hand on this side suppose I write
integral mod Z minus Z naught is equal to
delta and if I write d log f n suppose I write
this, what will I get? I will get number of
zeros minus number of poles of f n times 2
pi i where I count zeros and poles of f n
inside that open disk with multiplicity okay
that is what I am going to get and essentially
what will happen is you see, so this is going
to be some you know let me use better notation
okay so let me put this as capital N sub n
small n okay times 2 pi i and again you know
I will not have any poles because the f n
are all anyway analytic functions there are
no poles. Normally I should write number of
zeros minus number of poles times 2 pi i but
there are no poles here okay.
So I get 2 pi i times N n and now notice this
is the big deal if I take the limit as n tends
to infinity of this integral okay, now you
see this is the point f n converges to f normally
on D and the set on which I am doing that
I am worried about is the set where I am doing
this integral it is this circle because you
see I am worried about this integrant, the
integrant the variable of the integration
varies over the region of integration, the
region of integration is the circle, so I
am worried about this circle but this circle
is compact it is a compact subset of D and
on this circle therefore there is uniform
convergence.
Now you know because of uniform convergence
I can interchange limit and integral okay,
now this is one of the important properties
of uniform convergence. Uniform convergence
allows you to do things like changing limit
and substitution which is continuity and then
limit and differentiation which is differentiability
term by term and it also allows you to change
limit and integration which is the same as
saying that you can integrate term by term
okay. So because of uniform convergence I
can push the limit inside okay and when I
push the limit inside what will I get? I will
get the thing on the right I will get simply
integral mod Z minus Z naught is equal to
delta limit n tends to infinity d log f n
and that is equal to actually this limit n
tends to infinity d log f n is just d log
f okay, so what I get is, look at the moral
of the story, the moral of the story is that
limit as small n tends to infinity capital
N 
sub small n is capital N this is what I will
get finally this is just by applying argument
principle but you see what does it say?
Capital N sub small n is a sequence of integers
okay and a sequence of integers is tending
to a constant means that the sequence becomes
constant beyond a certain stage okay. See
what does this mean, this means that for n
sufficiently large N n is the same as capital
N sub small n is the same as capital N, see
sequence means it should come closer and closer
but when these are integers closer means they
are the same literary okay. If you say one
integer is within an Epsilon of another integer
they have to be the same okay if Epsilon is
less than 1 alright, so N n is equal to N
for n sufficiently large, what does that mean?
It means this f n they have what is this capital
N sub small n? It is the number of zeros of
f n inside that this and what have you got?
You have gotten that beyond a certain stage
all the f n have exactly n zeros and these
n zeros and that n is the same as multiplicity
of 0 of the limit function f at Z naught that
is what it says and now mind you whatever
I have done here will work if I make delta
is smaller. After all if I make delta smaller
the right side is not going to change because
I am always going to get only the order of
0 of f at Z naught, so therefore this whole
argument will work if I make delta smaller
and smaller and smaller and smaller, so that
means what? All the zeros of f n beyond a
certain stage you see they are all you know
they are going to go and cluster they are
going to go closer and closer and closer and
there are going to cluster and they are going
to finally coalesce into 0 at Z naught and
that is what Hurwitz’s theorem.
It says that the 0 of a normal limit of analytic
functions that 0 does not just pop-up just
like that, it comes zeros of the original
sequence of functions beyond a certain stage
okay that is what Hurwitz’s theorem. So
now we will have to use this to give the proof
of a fact that if a sequence of analytic functions
does not converge to an analytic function
then it has 2 completely converge to infinity,
the constant function infinity. So now let
us go ahead with and to prove whatever we
were trying to prove.
So we go back to the old problem that you
are worried about, the statement that we 1st
gave, so let f n be a sequence of analytic
functions on D which is in the complex plane
in domain. So f n is in our notation get is
in H of D, H of D stands for the analytic
function or holomorphic function on D and
mind you this H of D is contained in m of
D, m of D is a set of Meromorphic function
on D that is further contained in the set
of all continuous functions from D to C union
infinity okay and mind you here are when I
am writing it like this I am treating the
Meromorphic functions also as functions which
can take the value infinity and the Meromorphic
function the value at a pole is defined to
be infinity mind you okay and the sequence
f n is in here and you assume that f n converges
to an f point wise okay.
So f n converges to f normally with respect
to the spherical metric okay and mind you
I have to allow the spherical metric because
I have the point at infinity also as a value
alright and mind you this is sitting inside
the set of all maps, not necessarily continuous
from D to C union infinity okay and this script
C is continuous maps, this is continuous maps
alright and mind you let me again insists
this is convergence with respect to this spherical
metric okay point wise convergence with respect
to the spherical metric. Why I need the spherical
metric is because the limit function at some
point can be infinity then I will have to
measure distance with respect to infinity
and I can do that only with the spherical
metric okay.
Now I have already proved a lemma last time
it is something that you or you know that
whenever you have a normal limit of continuous
functions it is continuous, so there is quite
clear because a normal limit is a locally
uniform limit and therefore and you know uniform
limit of continuous functions is continuous
and the function which is locally continuous
is also continuous because continuity is a
local property, so this f this limit function
f is certainly continuous okay f is continuous,
so f is actually here. f is continuous map
from D to C union infinity and what is the
claim? This is the big theorem that you trying
to prove, the claim is if f is not analytic
then f is identically infinity that is our
theorem, that is what we are trying to prove
okay.
So we have to show that if f is not analytic
or let me write uhh more logical way if f
is not identically infinity then it is analytic
this is what we have to show okay so let me
find out you have situation where…so the
limit function f either lies it is either
the function here which is the constant function
infinity or it is here itself it is in H of
D itself. It cannot go into this it cannot
become Meromorphic that is it cannot become
honestly Meromorphic okay so you know it is
like saying that what is the meaning of saying
it cannot be honestly Meromorphic it means
that suddenly a pole cannot pop-up okay in
the limit you have sequence of analytic functions
it is converging to limit function.
A pole simply cannot pop-up out of the blue
of course the original sequence does not have
poles because they are analytic functions
okay by continuity you should expect this
also to happen but again you know this is
the point is has to be proved and you can
see it is like there is already the flavour
of Hurwitz’s theorem which says that when
you have limit of analytic functions 0 of
the limit just does not just pop up out of
the blue it comes from a limit of zeros of
the original functions. So that is what we
trying to prove, so assume that f is not identically
infinity then show that f is analytic okay
and what does it mean? It means that if f
does not is not going to be the identically
the function infinity, it cannot assume infinity
even at a single point mind you. It has to
assume only complex values point there is
a strong thing there. If it assumes value
infinity at a point and if that point is isolated,
point where it is assumed the value infinity
it means it is a pole okay but that does not
happen that is what it says, alright. That
is what we have to prove.
So how does one see that well you see you
take the subset D sub D infinity this is a
set of all Z in the domain where f of Z is
infinity okay and this is a proper subset
of your domain it is the proper subset of
the domain because you have assume that f
is not identically infinity. If f is identically
infinity then it is the same as saying D infinity
is D okay but when you say f is not identically
infinity it means that the limit function
f has some point where it has some finite
complex value, a value which is different
from infinity in the extended complex plane
okay. Now mind you this is the closed set
actually okay because it is the inverse image
of infinity under f which is a continuous
map I already told you that f is continuous
and infinity uhh single point is always closed
okay in one-point compactification, so if
you take the inverse image of infinity that
will be D infinity, so D infinity is just
f inverse infinity and that is closed inside
the it is a closed subset just by continuity
of f alright. So I will stop here.
