# Mod-06 Lec-17 Why Normal Convergence, but Not Globally Uniform Convergence,

Alright so we are continuing with our discussion

about studying spaces of Meromorphic functions okay. You have to do topology on spaces of

Meromorphic functions and then you know I was trying to explain last time that you know

in general if you go and draw inspiration from topology okay the usually if you take

the space of functions okay. Then what you do is that you put restrictions like you know

and of course you know you assume that you are worried about functions which are taking

real values or complex values okay. So basically you start with a topological

space if you want in general or you take a metric space and then you look at the functions

on that which are either real valued or complex value okay stop of course you know even if

the set you start with this is just a set, it is not even a topological space. Even then

this set of all functions real valued functions or complex valued functions will form a real

or complex algebra okay that is it will be a ring a commutative ring under point wise

addition and multiplication and it will include the constants which are the real numbers or

the complex numbers as constant functions and it will also have a vector space structure

over the constants. So it will be if you are considering real

valued functions it is a real vector space if you are considering complex valued function

it is a complex vector space and it is in fact it will be real algebra or complex algebra

because it is also ring it is a commutative ring with the ring structure compatible with

the vector space structure, okay the ring multiplication is compatible with the scalar

multiplication okay. So but then this is for any set but then now you know if you make

that set to also to have topological space structure in addition okay then what you can

do isÖ well even before that suppose you still only have a set okay X and you are only

looking at say real valued functions are complex valued functions on the set okay which is

I told you is an algebra okay, what you can do is that if you are looking at bounded functions

okay namely that you are looking at functions whose images in the real line or the complex

line is they are bounded subsets okay. If you are looking at such bounded functions

that forms a subset okay and in fact it forms a sub algebra okay because the sum and product

of bounded functions is again bounded alright and also you get a smaller algebra which consist

of now which consist of only real or complex valued functions but which are bounded okay.

Now the advantage of having this bounded nice is that you can now define a norm on this

vector space structure and make it into normed vector space or normed linear space okay and

the norm is just supremum norm, so what you do is that since you are looking at a function

on a set which is taking real or complex values and since the values it is taking a bounded

you can simply take the modulus of its values and take the supreme of all those values okay

and you get what is called the supreme norm of a function okay. Now this supreme norm will be a finite quantity

okay it will be a finite positive real number and it will be and it will be metric space

structure, so what happens is that if you look at just the set of real valued or complex

valued oceans which are bounded and you add this norm then you get a normed vector space,

it is an norm linear space and it is in fact you know this any norm linear space has a

metric which is just given by basically you know the distance between 2 elements in that

vector space is just the norm of their difference okay that is a metric. Now with respect to this metric it becomes

metric space and then in fact with respect to this metric space it is actually it is

actually a complete metric space okay, it becomes a complete metric space and the completeness

is just because of you know completeness of the real line or the completeness of the complex

plane or which is more generally completeness of the Euclidean spaces RM okay R1 is the

real line, R2 is same as complex plane topological okay. So therefore we get so much if you just

have a set X if you just have a set X let me repeat and you are looking at the collection

of all real valued or complex valued function on X which are bounded that is already a banach

space, it is a complete norm linear space. It complex as a metric space for the metric

induced by the norm okay, now you put the extra condition that the set X is also a topological

space example it may be a metric space alright you put an extra condition now on the set

X is no longer just a set but it is a topological space. Now the advantage that you have these extra

structures of topological space on X tells you that you can further restrict functions

instead of looking arbitrary functions you can look at functions which are tenuous okay.

So now what you can do is you can start looking at art only bounded real valued or complex

valued functions on the set X but you can also insist that you are only going to look

at those among these that are continuous with respect to the topology on X okay and that

is how you get the space of a real valued or complex valued continuous bounded functions

on a topological space X and both of these are banach algebra okay they are commutative

banach algebra over the base field which is either complex numbers or the real numbers

okay and uhh but the point is that the topology on this Banach algebra, this topology has

got to do with actually it is got to do with, it has got to do with point wise convergence

okay. So if you have a sequence of functions which

converges to a limit function in this Banach algebra okay then what it means is that it

not only means if the functions are converging point wise okay it does not only mean point

wise convergence it actually means uniform convergence okay that is the fact, so a sequence

of so let me say it in simple words a sequence ofÖlet X be topological space, let f n be

a sequence of continuous bounded real valued or complex valued functions on this topological

space X then f n converges to f for the topology or the metric space structure on the banach

algebra of functions if and only if f n converges to f point wise with respect to X and the

convergence is also uniform okay, so the moral of the story is that in topology on the space

of functions that you are trying to study has certainly got to do with point wise convergence

and it has in fact got to do with the uniform convergence okay. Now what I want to tell you is that see this

is the motivation from topology but with complex analysis in serve more complicated because

the functions are not just continuous functions, your functions are going to be analytic functions

whenever you consider them and more generally we want to actually worry about families of

Meromorphic functions okay and you know Meromorphic functions are slightly more complicated because

they are poles and at the poles they go to infinity because by definition at a pole the

function goes to infinity. So it means that you have to do something the complication

you have to worry about is that you cannot just consider real or complex valued functions

okay for example if you are looking at Meromorphic functions you have to take functions with

values in the extended complex plane. You have to include the value at infinity because

that is the value you will assign to the function at a pole okay that is 1 point okay then the

other thing is about. So you know the modification as you come from

the topological point of view which is the motivation of our situation which is we want

to study Meromorphic functions, the 1st modification is you have to include the value infinity

okay, so you cannot just look at complex valued functions so you have to look at actions which

values in the Riemannís sphere essentially being thought of as the extended complex plane,

so you should not take functions values in C we should take function values in C union

infinity okay. Now and thankfully C union infinity is not

very bad in fact it is very good because it is in fact a metric space it is a complete

compact metric space okay and that is because of the stereographic projection which identifies

it with the Riemann sphere okay that is one part of the story, the other part of the story

is that I told you that from the topological point of view topology on the space of function

has got to do with point wise convergence and in fact it has got to do with uniform

convergence but you see again when you come to complex analysis when you are studying

holomorphic functions or analytic functions you do not get uniform convergence in general

what you get is only uniform convergence on compact sets you do not get uniform convergence

just like that on whatever domain you are working with okay. You only get normal convergence which is point

wise convergence which is uniform only when restricted to compact subset, so you see therefore

what I am trying to tell you is that say when you when we take inspiration or when we try

to think of an analogy from topology and move it to complex analysis okay. In topology what

you are thinking of is just space of continuous functions, bounded continuous functions okay

and they are real valued or complex valued okay and the topology has got to do with exactly

uniform convergence but when you come to complex analysis you are worried about space of analytic

functions or holomorphic functions or you are more generally worried about space of

Meromorphic functions okay. Then the complication is that 1st of all is

that you have to worry with values in the extended plane you have to add the value a

point at infinity to get a value at infinity the other thing is that you will also have

to worry about is uniform convergence is less because uniform convergence you do not get,

so I will tell you why I will give you a very very simple example. The simplest example

of that is for example gotten by looking at the you know the geometric series which is

you know the fundamental series that we all have been studying from high school and if

you really if you go back to your 1st course in complex analysis you will see that almost

everything that you have proved about power series started with an argument that has got

to do with you know simple properties of geometric series okay. So let me say that so look at the geometric

series, let us look at this geometric series Sigma Z power n, n equal to 0 to infinity

okay this is simply one plus Z plus Z square and so on this is a familiar series and you

know you have all studies this, this is actually we write this as one by 1 minus Z if mod Z

less than 1 okay. This is something that we have all seen from probably high school okay.

Now but what is going on here, so the 1st thing is that one by one minus Z the function

on the right is an analytic function in the domain mod Z less than 1 which is the unit

disk okay and whatever we have written on the left side which is the geometric series

is actually the Taylor representation. It is just the Taylor expansion over the function

centred at the origin, so it is actually the MacLaurin expansion okay. So this is just MacLaurin expansion, so all

you are saying is that one by one minus Z has a MacLaurin expansion given by the geometric

series okay. Now the fact is that if you were to go little back to your 1st course in complex

analysis you would have noticed that you know the big deal is that this convergence isÖthe

point is that this convergence is absolute okay it is uniform on compact subsets okay.

These 2 come because of the Weierstrass m test okay with the Weierstrass m test mind

you is a tool that will help you to decide whether a series converges uniformly and the

way it works is that it will always give you absolute convergence because what the Weierstrass

m test does is actually it tests the absolute series that is the series gotten by putting

mod to the terms of the original series and it test that series for convergence and you

know absolute convergence implies convergence it is stronger than convergence. So that is what we use to okay that this series

converges absolutely and uniformly on any closed subset of the disk but the point is

that if you take the whole disk the convergence is not uniform okay it is this is something

that you can work out okay. You assume that you can do it in 2 ways either you can directly

write out estimates or you can For example assume it converges uniformly on the whole

disk and you will get glaring contradiction okay stop so the point about so this is the

point I am trying to tell you, see look at this fact that this series does not converges

uniformly on the unit disk, so you see what look at what we are getting let me put F1

of Z equal to 1, so let me let me write this here about converges absolutely and uniformly. So I am using absly for absolutely and ufly

for uniformly as abbreviation okay on compact subsets of mod Z less than 1 and since any

compact subset of mod Z less than 1 the unit disk is always contained in a closed disk

centred at the origin you can actually verify this condition only for closed disk centred

at the origin contained inside the unit disk okay and that is how it is done when you do

the Weierstrass m test you take the radius of that closed disk as theÖand its powers

as the Weierstrass Constance the m n that I used in the m test okay, so the point is

so let me write this in red because this is very very important, so let me write here

but not uniformly okay. So this is something that I want you to check

if you have not objected please check it, please check it because it is a lesson that

you know it is not the way you always wanted to be. It is not uniform but notÖso when

I say but not uniformly I mean it is not uniform on the whole disk okay you get uniformness

only on compact subsets of the disk, so what does it mean? Suppose I write the sequence

of functions which are given by the partial sums of the series, so I write f 1 of Z is

1, f 2 of Z is 1 plus Z then I write f 3 of Z to be 1 plus Z plus Z square and so on then

you see what we are saying is that f n tends to f okay because by definition that is what

definition of conversion of a series means it means its convergence of the partial sums

and f n is the nth partial okay, so f n converges to f and mind you how is this convergence,

this convergence is only normal. It is absolute of course on the whole disk

it is absolute on the whole unit disk it is absolute but it is uniform only on compact

sets that is it is normal but it is not uniform on the whole disk. So therefore you see you

now have a problem in adapting in trying to think off or in trying to compare with the

situation in topology. See in topology if a sequence of function converges to a function

f if you wanted in the space of functions then it should be point wise convergence and

uniform convergence okay but that is not happening here that is not happening here. What is happening

here is that you are getting point wise convergence no doubt f n convergence to f point wise is

always there the problem is that it is not uniform, it is not uniform on the whole domain

which is a unit disk, it is uniform only on compact subsets you are getting only normal

convergence you see that is a technical point. So you know so what I am trying to say is

the following thing say if you take the domain D to be mod Z less than 1 the unit disk okay,

the set of all Z such that mod Z less than 1 okay and then you know I can do this I can

take L D which is the set of all maps from D to C, this is a set of all set theoretic

maps D to C and this is an algebra this is C algebra okay one would take B of D to be

the set of all among all the functions you take the bounded functions okay such that

you take the set of all f such that f is bounded okay and then inside this you take the set

of all continuous functions and you know that you know this is an inclusion of uhh these

are all inclusion of banach algebra okay. Now here you see I am looking at this last

set below this is the set of all continuous bounded complex valued function on the desk

alright. Now you see what happens is that if you watch I have all these functions f

1 of Z is 1, f 2 of Z is 1 plus Z, f is 3 of Z is 1 plus Z plus Z square I mean these

are just the partial sums of the geometric series, now you see and you know f n converges

to f which is one by one minus Z okay each of it is of course bounded because if you

uhh you can use a triangle inequality for example f 1 is of course constant so it is

bounded and f 2 is 1 plus Z and mod f 2 is mod of 1 plus Z which is less than or equal

to 1 plus mod Z by the triangle inequality that is less than or equal to 2. Similarly f 3 is equal to 3 okay you see that

all the f n are bounded suddenly, bounded continuous functions and the limit function

f which is one by one minus Z which is the limit of the geometric series that function

is also continuous, holomorphicÖf does not even belong here f is not even here okay,

so f is not bounded because values of f can become very large if Z tends to 1 or example

on the real axis if I approach from the left okay, so in any case what I want you to understand

is that you see if you have f n tending to f of course you do not have it in this space

okay but you have f n tending to f in it is rather funny you have this space H of D okay

you have this space H of D, this H of D is the set of all holomorphic functions on D,

analytic functions on D okay and the point is that there are analytic functions like

one by one minus Z which are not bounded okay that is because they have a singularity on

the boundary of D okay. So what is happening is that this this convergence

is happening here in the space of holomorphic functions and you see that you know it certainly

it is point wise convergence that is for sure it is point wise convergence okay but it is

not uniform okay it is only uniform on compact sets, so you see why I am saying all this

is that I want you to I want you to see that when you go to complex analysis things are

very different I mean in normal topology you look at bounded functions and then you look

at among those bounded functions you look at continuous functions okay but then look

at what is happening in the context of complex analysis even in the case of a very simple

domain which is a unit disk it is a bounded domain. Even in that case you have a sequence of functions

which are very much in the space but they converges to a point which is outside okay

and the convergence is of course point wise but it is not uniform is only uniform on compact

subsets okay, so this should tell you that you know that at something has to be done

if you want to do topology on the space of holomorphic function something more serious

has to be done okay and of course you know let me tell you that if you if you take the

Meromorphic functions on D, if you take the space of Meromorphic function on D then things

are you know slightly more complicated I cannot think of them as functions from D to C okay.

I can think of them as only as actions from D minus the poles of that Meromorphic functions

to C and if I want to think of them as functions on D I will have to include the point at infinity. So what I will have to do is I have to take

this set, set of all f from D to C union infinity I will take this C union the point at infinity,

I will take this guy I will take this and let me call this as something else star or

L star of D this is a bigger set than this, so this is subset of this and this is here

and this is here okay so the moral of the story is that you have to worry about this

value at infinity and you will have to worry about the fact that life if you are looking

at analytic functions you are not going to get uniform convergence, you are going to

get uniform convergence only on compact sets okay and the simplest example namely geometric

series tells you that okay. Of course there are situations where you can get uniform convergence

on an unbounded set that can happen but they are special cases okay. For example you know let me give you one example

take the zeta function, take Zeta of Z be you know this Sigma 1 pi n power Z, n equal

to 1 to infinity okay and this is by definition well this is Sigma n equal to 1 to infinity

n power minus Z and the way you define n power minus Z is you define it as E power minus

Z n okay using properties of logarithms and this n is actually the real logarithm okay

it is real logarithm and the fact is that this function here is Riemannís zeta function

and of course you know it is very very famous it is the most famous function introduced

of course by Riemann 157 years ago okay and it is the most mysterious function in all

of mathematics and the Riemann hypothesis which is a conjecture of both that function

is one of the most famous unsolved problems and the effect of solving that is like you

would have solved thousands of theorems in number theory okay. So it is a very very deep function, very mysterious

function at the fact is that this converges uniformly for any closed right half plane

to the right of real part of Z equal to 1, so let me write this converges uniformly and

absolutely 4 mod Z greater than equal to 1 plus Epsilon, Epsilon positive uhh so I should

not say mod Z I mean real part of Z, so this is you know this region is something like

this, so you have so this is 1, 1 plus Epsilon is somewhere the right of that and in fact

if you take this shaded region along with the boundary, this is a shaded region where

you get uniform convergence okay it is a half plane, it is the half plane along with that

vertical line which passes through x equal to 1 plus Epsilon okay and that is a close

set mind you because the boundary is included it is a close set and this is an unbounded

close set, it is not a compact set but inspired of this being unbounded you get actually uniform

convergence okay and in fact therefore you will get convergence on you call right half

plane, so what you will get is that it converges on the real part of Z greater than 1 okay. So, order of the story is that there are special

cases where you can get uniform convergence on a huge set even on an unbounded set okay

but that is not but that will happen only in special cases, you cannot expect it to

happen always alright, so there is an issue the issues that whenever you are studying

spaces of analytic functions or more generally Meromorphic functions you can expect only

normal convergence okay that is one issue. Then see now what I want to say is that in

any case, so to sum up what I want to say is that we will have to worry about when we

are worrying about analytic functions or Meromorphic functions you have to worry about 2 things,

one thing is that is not uniform convergence it is normal convergence is what you will

get. So whenever so you must keep this in mind

as a philosophy, whenever you are worrying about sequences of functions converging in

theorems in complex analysis, mind you this is equivalent to doing topology on the space

of such actions and the topology is being done essentially by looking at point wise

convergence which is normal okay this is one fact that you should always remember and the

other thing that you will have to remember is that you have to worry about the value

infinity when you are worried about Meromorphic actions okay and the way you deal with infinity

is by always appealing the stereographic projection and comparing infinity the North pole and

comparing neighbourhood of infinity with the neighbourhood of the North pole via the stenographic

projection, so these are the techniques that we should use okay. Now what I will tell you is that at least

this is again something that you should have seen in 1st course in complex analysis but

let me recall. Suppose you have a domain in the complex plane suppose you have a sequence

of functions analytic functions defined on the domain okay, so these are honest analytic

functions they are holomorphic functions. Suppose the sequence converges to a function

normally on the domain then the limit function is actually analytic, it is holomorphic okay,

so it is not that it will go out of the space of analytic functions okay, so I thing you

should have seen the proof of that but in any case let me recall it. So let so the 1st thing is you take a limit

of analytic function you should get an analytic function okay at least that you must have

otherwise you cannot go ahead but the point is we haveÖthe reason why it works out well

is because we have agreed that whatever we do it will be point wise convergence which

is normal and this normal convergence is the technical thing with which we can do a lot

of things because actually it is a local version of uniform convergence. It is stronger than

a local version of uniform convergence okay, see when you say there is a normal convergence

you are saying it is normal the convergence is uniform in compact sets okay. Now there are 2 types of compact sets that

you can think of which are important for complex analysis, one kind of compact set is a canny

point, take a small disk around that point okay and then take its closure that is a compact

set but the beautiful thing is it is a compact set which contains an open set around that

point, so if you have a normal convergence then you will have certainly a uniform convergence

and therefore on sufficiently small neighbourhood of every point that is because given any point

all you have to do is you have to choose a sufficiently small neighbourhood whose closure

is also in your set okay. Basically you will have to choose a sufficiently small circle

surrounded by that points such that the circle and the interior of the circle is in your

set which will happen because you are only working with open sets and points inside open

sets. So therefore what you will get is you will

get uniform convergence on sufficiently small disk, so you must remember normal convergence

does not give you uniform convergence on the whole set but locally it gives you convergence,

uniform convergence it gives you uniform convergence on small small disk okay, sufficiently small

disk that is one fact then the other fact is what is another kind of compact sets that

you are always interested in you see as you know one of the most important techniques

in the theory of complex analysis is integration, integration theory and what do you integrate

on you integrate on contours and what are contours? They are of course compact sets

because they are curves basically there are just continuous images of a closed and bounded

interval compact interval on the real line, so they are compact. So the advantage of that is that the other

important type of compact set that you are always interested in are contours and then

uniform convergence on the contour will tell you that integrating the limit function on

the contour is the same as integrating each of the functions on the contour and then taking

the limit, so you can interchange the limit and the integral that is what compactness,

uniform convergence for contours tells you, so you see thatÖ therefore moral of the story

is that while normal convergence is not uniform convergence on the whole set, it still gives

you everything that you need locally for the differentiation theory. It gives you everything that it gives you

normalÖ It gives you actually uniform convergence on sufficiently small disk surrounding every

point and further integration theory it does give you uniform convergence on every contour

therefore everything goes through nicely okay, so what this tells you is you really do not

need uniform convergence on a whole open set and the example of the geometric series tells

you that that is what is expected and that is what you will get you will not get anything

better than that okay, so that is good enough okay. So let me write this down let f n from D to

C be sequence of analytic functions, analytic or holomorphic functions on a domain D inside

the complex plane. Suppose f n converges to f normally in D then f is analytic, so the

moral of the story is that you know under normal convergence, normal convergence preserve

analyticity. When a normal limit of analytic function is analytic that is all I am saying

okay and so let me tell you what is the way you prove this, the way you prove this is

1st of all because each f n is continuous okay f will become continuous and why is that

true because continuity is a local property show that f is continuous on the domain it

is enough to show that f is continuous on sufficiently small disk surrounded at every

point but if I take a sufficiently small disk okay such that its closure itself is contained

in the domain then I will in fact get uniform convergence and you know uniform limit of

continuous function is continuous therefore locally I will get continuity but continuity

is a local property if it is true locally it is true globally. So I get continuity okay. Same argument applies

to analyticity, how do I check a function is analytic if it is locally analytic it is

analytic because analyticity is also a local property, so what I have to do is that I have

to take any point in the domain I have to show that in a small disk surrounding that

point the function is analytic, how do I do that? It is very very simple, if I take a

small disk surrounding that point of course I have uniform convergence okay because you

have taken a sufficiently small disk, the closure of that disk is a compact subset of

the domain okay. Now how do I get analyticity, it is very simple I simply use Moreiraís

theorem. What I do is that I take a point, I take a sufficiently small disk surrounding

that point open disk okay. If I take the closure of the disk then the

convergence is actually uniform because it is a compact subset, so the moral of the story

is that to check that the function is analytic inside that disk I just have toÖ I know already

it is continuous so I have to only check that the integral over any closed path is 0, the

integral over any simple closed contour is 0 is what I have to check but if I want to,

so I try to integrate the function around a simple closed contour inside that disk but

then integrating the function is the same as integrating the sequence, each member of

the sequence of functions and then taking the limit, I can change the integration and

the limit okay, so but if I integrate each of those functions in the original sequence

around a simple closed contour I am going to get 0 because they are analytic that is

because of Clausius theorem. So what I get is that the integral over this

limit function over any simple closed contour in a sufficiently small disk surrounding a

point is always 0 and Moreiraís theorem will tell me therefore that it is analytic and

therefore it is locally analytic and therefore it is globally analytic that is it okay, so

this is how you do it with sequence of analytic functions but the problem is that when you

start worrying about Meromorphic functions things becomes complicated, so you know so

let me look at the following example, so let me take another example and see the points

about these examples is that you should know what kind of things happens, so here is another

example. What you do is you take the domain D to be the exterior of the unit disk okay

you just take the set of all mod Z greater than 1. Mind you this is an deleted neighbourhood

of the point at infinity you know the exterior of a disk is always a deleted neighbourhood

of the point at infinity. If you add the point at infinity this becomes

actually a neighbourhood of infinity in the extended plane okay and this is the deleted

neighbourhood of infinity and what you do is you put f n of Z you put it as Z power

n. Take this sequence of functions okay, now what will happen is you can see if you fixÖmind

you if you take any value of Z greater than one with modulus greater than 1 that is it

is basically a point lying outside the unit circle okay in the complex plane okay and

if I plug it in this sequence f n I will get higher powers of that point and that is going

to go to infinity in modulus as n tends to infinity because mod Z is greater than 1,

mod Z power n is also greater than 1 and mod Z power n will tend to infinity as n tends

to infinity. So what happens is that at every point this sequence convergence to what? The value it converges to is the point at

infinity okay it converges to the point at infinity alright. Therefore the limit function

is what? The limit function is a function which maps the whole exterior of the unit

disk the point at infinity, it is a constant function with the value infinity okay that

is what you get in the limit. Now this is the kind of thing that this is the kind of

pathology that happens and we have to take care of this okay. So mind you in this example

all these f n are in fact holomorphic functions f n is a sequence of holomorphic functions,

f n is in fact a sequence of holomorphic functions and f n converges to the function which is

constantly equal to the value infinity at every point and of course you know if you

check very carefully is convergence is even uniform on compact subsets okay, so here is

something that you do not expect that is happening. A nice sequence of functions even holomorphic

functions, even analytic functions on a domain can actually converge uniformly to the function

which is constantly equal to infinity at all points okay and that is no function by any

of our standards okay. It is certainly not Meromorphic function there is no set on which

it is analytic, it is basically a constant function, it is the function that maps the

whole domain onto the point at infinity, so let me write that down so f n converges to

infinity and what is this infinity? Where infinity of Z is the point at infinity for

all Z in D okay and this infinity this belongs to the set of all functions from D to the

extended plane okay. In fact it is that unique function which maps

the whole domain to the point at infinity a, so this is one standard pathology, so you

see what is happening even if sequence of holomorphic functions even a sequence of good

analytic function is going uniformly to infinity alright this is happening, so the moral of

the story is that when this happens for good holomorphic functions it will also happen

for Meromorphic questions. So the moral of the story is that whenever

you study sequence of Meromorphic functions okay and particularly we will be interested

only in normal convergence we will have to define what is meant by normal convergence

of sequence of Meromorphic functions that has to be done carefully because we will have

to worry at the value about the value at infinity and we will also have to worry about the point

at infinity if your domain also includes the point at infinity you have in the domain also

if you have the point at infinity and in the range also if you have the point at infinity

then it is a double complication you have to worry about it okay but in any case we

are going to be worried about sequences of Meromorphic actions which are converging normally

and the 1st pathology you should expect and this is the only pathology that you have to

really worry about is that that sequence may converge uniformly to the function which is

infinity everywhere, so you will have to do this you have to add this function artificially

okay as a function in your list of functions okay and deal with it, so this is something

that one needs to worry about okay, so let me stop.