So see the point is that you know we are trying to look at families of Meromorphic functions,
we are trying to look at trying to look at normally convergent families of Meromorphic
functions, the reason is because you want to do topology on the space of Meromorphic
functions okay and because that is the kind of you know set up that you need able to prove
the big Picard theorem and the little Picard theorem okay. So you know let me briefly remind
you if you take inspiration from topology alright, what is the topology that you will
put on the space of functions, normally if you have a topological space and you have
you are looking at real valued or complex valued functions then you will restrict yourself
to continuous bounded real valued or complex valued chance that is that is a banach algebra
and it is also a topological space it is complete as a metric space, the metric is induce by
a norm and the norm is essentially convergence in that norm is actually equivalent to uniform
convergence okay.
So the moral of the story is that if you are going to inspiration from topology okay then
trying to do topology on the space of functions is the same as you studying functions under
uniform convergence okay, now this is topological so that means that you are only worried about
continuous functions okay but now suppose you come to complex analysis then you are
worried about holomorphic functions or analytic functions okay and we are worried about something
even worse, we are worried about Meromorphic functions which are actually which have the
additional problem that they can have poles at finitely many at a set of isolated points
okay and of course finitely many if your domain is compact is the whole Riemann sphere on
the extended plane.
So if you are looking at say holomorphic functions on a domain, a domain in the complex plane
or a domain in the extended complex plane that does not matter suppose if you looking
at the analytic functions or holomorphic functions on the domain and you want to do topology
on that set of functions okay. Mind you we have body seen that the set of functions it
is a ring in fact and then if you look at Meromorphic functions it is a okay. It has
algebraic structure but we are now worried about the topology, so you want to do topology
on the set of analytic functions on a domain or you want to do more generally topology
on the set of Meromorphic functions on a domain which means analytic except for poles then
you know if you try to draw inspiration from ology you would just say that this is the
same as studying them under uniform convergence okay because topologically uniform convergence
corresponds to convergence in the space of functions okay but if you come to the case
of analytic functions okay this is not the right thing because you do not get uniform
convergence.
So I was trying to explain to you last time that for example if you take the geometric
series you take the functions that correspond to partial sums of the geometric series okay
then they are of course polynomials and they converge absolutely in the unit disk to the
sum of the geometric series which is one by one minus the variable okay, so the geometric
series is 1 plus Z plus Z square and so on where Z is a variable and you are restricting
Z to be in the unit disk that means you are making mod Z is less than 1 then one plus
Z plus Z square and so on that converges to one by one minus Z that is the high school
formula or geometry series. Now the point is that this convergence is absolute on the
unit disk there is no problem about that but it is not uniform on the whole unit disk okay
that was something that I told you I asked you to check it as an exercise I hope you
have done it, it is very easy to do convergence is not uniform on the whole unit disk is only
uniform on compact subset of the unit disk okay.
So you do not get uniform convergence but you get only normal convergence, so the moral
of the story is that when you want to do topology on a space of holomorphic functions you should
not look at them under uniform convergence you must look at them under normal convergence
so that is the 1st moral 1st lesson to be learned that is what you should keep at the
back of your mind, so that is one thing so you know more generally if you want to extend
this to Meromorphic functions in serve more complicated because now you have poles okay
and the other thing is that of course you know by going from uniform limits to normal
limits things are going to be good okay because normal convergence is just uniform convergence
on compact sets okay it is weaker than uniform convergence as it is okay but it is good enough
for our purposes because as I told you in the last lecture if we have sequence of holomorphic
functions which converge normally to limit function then that limit function is also
holomorphic okay this is something that I explained last time okay.
Essentially it uses Clausius theorem and Morera's theorem and the fact that analyticity is or
holomorphicity is a local property. So therefore there is no harm in relaxing the condition
of uniform convergence with the condition of normal convergence. It means uniform convergence
only on compact sets okay and I also told you philosophically why that is good enough
for complex analysis because the moment you say uniform convergence on compact sets you
get uniform convergence on closed disk because they are also compact and therefore you get
uniform convergence on sufficiently small open disk okay that is good enough for the
analysis for the differentiation theory and then for the integration theory also it is
good because whenever you integrate on the contour, the contour is a compact set therefore
you will get uniform convergence on the contour okay, so that helps in the integration theory,
so for all practical purposes uniform convergence on compact sets that is normal convergence
is good enough okay, so that is what we have to worry about.
Now the other important thing that I want to tell you is that you see at least if you
are working with Meromorphic functions you know the Meromorphic functions have poles
okay, so at a pole a function is going to behave in a bad way in the sense that the
modulus of the function is going to blow up to infinity okay, so for example that is one
of the characterisations of a pole, the limit of the function as you approach a poll is
going to infinity and by that time limit goes to the point at infinity okay and of course
here you are using the topology on extended complex plane namely the complex plane the
point at infinity given by the one-point compactification which makes it holomorphic to the Riemann
sphere okay.
Now is another pathology and that is the pathology that I was trying to explain what is the and
of the last lecture, so the pathology was the following, you take the exterior of the
unit disk or Z greater than 1 that is a variable, we are on the Z plane the complex plane and
you are taking the exterior of the unit disk mod Z is greater than 1 and what you are doing
as you are looking at these functions powers of Z you are looking at one which is Z power
zeros if you want then as Z square and Z cube and so on okay. Now that is a sequence of
functions and a point is that this sequence of functions you can see point wise it will
go to infinity because sense mod Z is greater than 1 mod Z to the n to the power of n is
going to go to infinity because it is for a real number greater than 1 you know its
higher powers will diverse to infinity, so mod Zůso Z power n so this sequence of
function is going to converge point wise to the function with the constant function at
infinity.
Namely it is a function which associate every point the value at infinity, so you have to
worry about this crazy function okay so this is a pathology that happens that you have
to take care of and the point is that therefore we are forced to introduce a function call
infinity okay and this function infinity is what it is just the constant function infinity,
namely it is the function which maps every value to infinity that is what it is and then
if you think of that as a function I mean it is a function of course theoretically if
you want it is a function from your domain to the extended complex plane because after
all in the extended complex plane infinity is a valid point okay it is a member of that
site, so you can really think of the function infinity as he constant function taking the
value at infinity provided you extend your values to not just complex values but also
the extended plane you include the value at infinity that is one thing.
So in that sense you can say that this sequence of functions f n of Z is equal to Z power
n that converges to infinity you can say that and when I say that converges to infinity
I mean that it converges point wise in the exterior of the unit circle to the function
which is infinity okay. So you have this is very nice situation, it is a very nice pathology
you have these Z power n which are all holomorphic functions in fact they are entire functions
they are just polynomials and they converge to the function infinity in the exterior of
the unit disk the convergence is again a normal convergence it is uniform on compact sets
okay. It is still a normal convergence is not just a point wise convergence but it is
in fact even a normal convergence in a way I will explain to you.
So what is the moral of the story? The moral of the story is you have sequence of holomorphic
functions you have sequence of analytic functions which is converging to the function infinity
normally that also happens. You see this is the extreme case that happens and this also
has to be taken care of in our arguments okay and mind you if this is happening for holomorphic
functions will happen also for Meromorphic functions because you know holomorphic functions
are very good Meromorphic functions are worst because they have poles, so even for a family
of holomorphic functions even for a family of analytic functions if you can get normal
convergence the functions which is infinity okay you should expect the same thing happen
also for Meromorphic functions.
So what I am trying to tell you is that if you sum up all this if you want to study topology
on the space of Meromorphic functions 1st of all you must study with respect to normal
convergence okay the 2nd thing is you have to introduce this function keeping mind this
function which is the function infinity okay and then you have to justify this business
of trying to make sense of normal convergence okay and so let me begin by trying to you
know explain at least in this particular case where this normal convergence comes from okay,
so let us do the following thing will worry about metrics on the plane and metrics on
the extended plane which are transported from metrics on the Riemann sphere okay.
So here is what I am going to do so let me draw a diagram, I have this so this is my
usual complex plane xy, so this is my usual complex plane which is the xy plane and then
I have the of course you know we are going to compare everything with the Riemann sphere
using the stereographic projection, so what I am going to do let me draw this thing here
which is Riemann sphere, so this is supposed to be 1 this is minus 1 on the x axis and
here is my sphere its cross-section on the plane is the unit circle, so it is going to
be so this is my Riemann sphere as it is and then of course I have this 3rd axis which
I will not call as Z I will call it as u because Z is supposed to stand for x plus i y okay
Z is supposed to be x plus i y and well and here is the North pole okay suppose I start
with 2 point Z 1 and Z 2 on the complex plane okay.
See what kind of the distances can I define on these points, what kind of metric and I
define on the complex plane that is the usual metric which is D of Z 1, Z 2 so I will put
d sub e for Euclidean metric and you know what that is? It is simply modulus of Z 1
minus Z 2 it is just the distance between these 2 points okay. This is the good old
metric that we use always in Euclidean space right. It is actually the length of the line
segment okay joining Z 1 and Z 2. Then the other thing you can do is you can take the
images of these points on the Riemann sphere because of the stereographic projection and
you can measure the distance between those 2 points and call that as the distance between
these 2 points, so what you can do is so here is the stereographic projection, so p 2 is
this point here on the sphere which is the unique point of intersection of the line joining
n and Z 2 on the sphere okay and similarly p 1 is unique point on the Riemann sphere
which is the intersection of the surface of the sphere with the line joining n and Z 1.
Now you see p 1 and p 2 lie on the sphere, now what I can do is that I can measure the
distance between p 1 and p 2 okay. Now that distance I measuring in 3 space because now
everything once you draw the Riemann sphere you are actually in R 3 and your R 2 which
is xy plane corresponds to the complex plane alright, so what you can do is you could define
the following new distance also d let me call this as d c is the chordal distance from Z
1 and Z 2 it is just length of the cord from p 1 to p 2 where p 1 the
stereographic projection of p 1 is Z 1 and the estimated projection of p 2 is Z 2 okay
so this is another distance that I can define it makesůsee what this distance does is that
actually it is the metric in R3 after all the cord joining p 1 p 2 is exactly the line
segment p 1 to p 2 in 3 space and I am just taking the length of that line segment, so
it is actually the metric in R3 it is metric in R3 and so it is a metric space you know
whenever you have a metric on space and you restrict to a substance then the subspace
also becomes automatically a metric space.
So this distance will make the Riemann sphere into a metric subspace of R3 okay and what
we are doing is that to the stereographic projection you are transporting that metric
to the complex plane because after all this geographic projection is the bisection between
the extended complex plane and the Riemann sphere okay. The moment you have bisection
of a set with a metric space it can transport the metric on the metric space to the set,
so I will just transported theů basically what I have done is I have simply transported
the metric on R3 restricted to the Riemann sphere. I have simply unsupported it to the
plane that is what this d c is.
So this d c also will also d subsea that is also a metric you can check that that also
makes the complex plane into a metric space okay and
but the big deal is that all these metrics are all equivalent okay namely the topologies
that they induce on the complex plane there are all the same that is the whole point okay
and that is very important because what it tells you is, it tells you the following thing
if I want to study convergence of functions you know as long as you are worried about
continuous functions I can use any of these metrics and the point is so let me tell that
in advance, why I am worried about these extra metrics is because I can also define the distance
of a point on the complex plane to the point at infinity okay because that the point at
infinity will correspond to a finite point namely the North pole on the Riemann sphere
and distance to that is something that I can measure okay, so that is the advantage.
The advantage is you see I want to be able to measure the distance the point at infinity
R I cannot do it with the Euclidean metric because 1st of all the point at infinity is
not in my set okay it is an extra point I have added for compactification and once I
had these extra points I have to add this extra topology, the topology of the one-point
compactification but then that is not enough I have to even make it a metric space and
where will I get the metric structure? The only way is I will have to get this metric
structure from the Riemann sphere which is what is holomorphic to the extended plane
okay and therefore I am lead to look at the metrics on the on the Riemann sphere, so this
is the chordal metric okay, so d subsea is the chordal metric and then here is the 3rd
metric which is the spherical metric so d sub s of Z 1, Z 2 this is the length of the
arc of the minor R from p 1 to p 2 along is a length of the minor arc from p 1 to p 2
along the great circle through p 1 and p 2.
So you see this is a spherical distance, what is a spherical distance? This is spherical
distance actually am trying to measure distance on the Riemann sphere on the surface of the
sphere, so it is the curve distance okay and I am trying to measure the shortest curve
distance and you know you can imagine thisůbasically what one is doing is that one is doing a kind
of some kind of Riemann in geometry, what is happening is that you have a surface okay
you imagine some nice smooth surface you have 2 points okay then you can try to connect
those 2 points by many arcs by many arcs on the surface passing on the surface okay and
then you can measure the lens of each of the arcs and you can take the smallest length
okay and the arc of smallest length is called a geodesic okay.
So now what is happening is that if you take the sphere it is quite easy to see that the
if you give me 2 points on this sphere, on the surface of the sphere then the geodesic
is exactly the following for those 2 points you get a big circle, a great circle a circle
of largest radius on the surface of the sphere passing through those 2 points and you take
the minor arc okay any 2 points of a circle will split the circle into 2 arcs and you
take the minor arc the one of smaller length and take the length of that, that is exactly
the spherical distance okay and that is what I am denoting as d of s it is a geodesic for
the sphere for any 2 points on the sphere, so the great circle are the geodesic minor
arcs of the great circle are the geodesic for points on the sphere.
So what is happening is that now I have all these 3 metrics, the beautiful thing is that
these 3 metrics give you metrics not only on the complex plane the point is they give
you metrics on the extended plane okay see what I have drawn here is for Z 1 and Z 2
imagining Z 1 and Z 2 as points on the complex plane but I can very well make Z 1 or Z 2
to be the point at infinity. Now when I say I make the point Z 1 or Z 2 to be the point
at infinity I cannot see it on the complex plane okay but I can see its image on the
Riemann sphere it is the North pole, so basically what I am doing is I simply taking 2 points
on the Riemann sphere and I am measuring their distance, the distance between them either
the chordal distance or I am measuring the spherical distance, so the moral of the story
is that these distances help you to give a metric on the extended plane and the topology
induced by all these metrics is one and the same, all these metrics are equivalent okay.
So this is the fact that you need to check from you know this is very easy fact to check
are logically let me tell you how to do it how will check the 2 metrics are equivalent
for a topological space it is very simple, what you do is that you show that you take
an for each point of the topological space you take small open balls with respect to
one metric and show that it contains a small open ball with respect to the other metric
and do this for both metrics symmetrically and then you are done okay, so you can see
pictorially you can see that it is true okay if for all the 3 metrics, so therefore all
these 3 metrics will give you one and the same topological space structure on the extended
complex plane which is the same as the one-point compactification and that will be exactly
holomorphic to the Riemann sphere by the stereographic projection okay and the advantage of doing
all this, why do all this? You can ask me why do all this?
The advantage of doing all this is that now I can say, now I can make sense of the following
statement. A sequence of f function f n converges normally to infinity with a function infinity
it makes sense now because I can say I can say convergence with respect to this metric
one of these metrics namely the 2nd and 3rd one which are also defined for the point at
infinity okay and that is the reason why we need to use that okay, so let me write this
down, all the so let me write somewhere here may be use a different color. All the metrics
below are equivalent on C alright and the last 2 metrics are equivalent on the extended
plane. The latter to so let me make some space let me get rid of this, the latter two namely
these 2 are equivalent on the extended plane C union infinity okay and of course I rubbed
of Z equal to x plus i y so let me write it here okay and now here comes the here
comes our the advantage of this.
We can now say that f n of Z equal to Z power N, n greater than or equal to 1 converges
normally to infinity on mod Z greater than 1 you can say this you can make this statement
it makes sense okay and and why is that correct? You have to do a little bit of you know
why I am spending so much time on this is because you see this normal convergence
the function which is infinity is something that is hard toůas it is if you do not analyse
it is very hard to digester, so it is very important that you understand what is happening
here and you must understand that therefore even if you are looking at the normal convergence
of holomorphic analytic functions can still end up with a function which is infinity okay.
So you see what is happening?
So again let me draw another diagram, so you see here is the here is your complex plane
as it is in R 2 and this is the origin and of course I have, so let me draw the Riemann
sphere here okay so this is the situation and you see of course mod Z greater than 1
is this is the region of the plane exterior of the unit circle okay, so basically so this
is mod Z greater than 1 yes it is very hard to draw that in three-dimensional diagram,
so let me do the following thing let me use another color you know I just looking at so
it is all these, so it is this exterior of this unit circle on the complex plane which
is thought of as xy plane okay and of course you know the 3rd axis is I am calling it as
u.
Now you see, so let me change color again so I have this let me draw this also so that
this is u and I have the North pole here okay, now you see these red lines that I have
drawn they are supposed to extend outside unit circle the whole exterior of the unit
circle and what they are going to give me? They are going to give me this domain mod
Z greater than 1 if you consider it as a domain in the extended plane it is a neighbourhood
of infinity okay and what is its image under the stereographic projection it is exactly
the upper hemisphere okay it is the whole upper hemisphere which you can see clearly
is neighbourhood of the North pole okay that is the reason you are looking at alright.
Now you see take a compact subset of mod Z greater than 1 take a compact subset is a
close and bounded subset of mod Z greater than 1 okay and so you know if you are looking
at a compact subset on the plane okay then any compact subset in mod Z greater than 1
on the plane to be has to lie within a sufficiently well-chosen annulus okay it should
lie within an annular region consisting of an inner circle and an ouster circle centred
at the origin sufficiently small inner radius greater than 1 and sufficiently large radius
greater than of course the inner radius. So you see if I take some compact set here, so
here is some compact set, k compact set in the complex plane then you see this is k of
course lies inside suitable annulus, so it is going to look like this you know I am going
to get something like this, I am going to get this annulus here so I am going to
get this annular region mind you this is annular region on the complex plane consisting of
the region between these 2 circles and am also including the boundaries go make it compact
okay.
So it is a closed and bounded set it is compact and this is a compact set and the point I
want to make is that instead of just considering any compact set k in the complex plane which
is lying in this domain mod Z greater than 1 it is enough to just consider such annuli
which lie in the exterior of that circle okay the exterior of the unit circle and you see
if you watch carefully how is this annulus going to be given by well this annulus is
going to be given by mod Z less than small r I mean less than capital R less than or
equal to capital R less than or equal to small r which is greater than 1 this is how it is
going to be where small r is the radius of the inner circle capital R is the radius
of the outer circle okay.
This is what this annulus is going to be given by and well what is its image going to be
on the Riemann sphere stereographic projection I am going to get this, see I am going to
get something like this here this is what I will get, I will get a curved annular region
centred at the North pole alright and now you see that
now you can see why the f n converges to the function infinity normally okay because our
definition of convergence is in the following sense okay. So the definition of convergence
will be of course point wise convergence okay but then point wise convergence if you try
to write it in the metric it will create a problem when you put the point at infinity
okay, so I cannot say that or each point Z naught f n of Z naught converges to infinity
as n tends to infinity I cannot say that. I can say that in a topological sense but
I cannot say that in a metric sense but if I use Euclidean metric but then if I use a
spherical metric I can say that okay.
So the moral of the story is that if you look at the distance the spherical distance between
a point Z I take the point Z in K my compact set okay. If I look at the spherical distance
between f n of Z which is in this case Z power n okay and the point at infinity okay and
here you see what I mean here is infinity of Z. See infinity of Z I am thinking of the
function which is constant function which gives the value infinity to every point so
what I have written there is actually infinity of Z and I am saying that it is spherical
distance between f n of Z and infinity of Z that can be made uniformly less than Epsilon
irrespective of Z if I choose n sufficiently large, so I can make this less than Epsilon
okay for an a grade than or equal to capital n irrespective ofů
So given if I start with an Epsilon greater than 0 okay for Z in K I can make the spherical
distance between f n of Z which is Z power n and infinity I can make it less than this
Epsilon for n whenever small n is greater than or equal to a large enough n such a large
enough n exist the point is that this large enough n does not have anything to do with
the Z. It will work for all Z in the compact set that is the uniformness that is uniformness
of the convergence on the compact set, so n this n is so there exit this n and this
is independent of Z and you see this fact is true this fact is very. See suppose I give
you an Epsilon okay what is the spherical distance between f n of Z and infinity it
is actually the spherical distance between Z power n and infinity alright and Z power
n is going to lie where, it is going to lie in mod Z power n is going to lie in this annulus
okay and you know if you see if Iů
The inner radius of this annulus is r power n small r power n the outer radius is capital
R power n and you know if I and since R is greater than 1 if I increase n R power
n is small R power n itself is going to shoot up okay, so moral of the story is that this
region if I have its image in the on the Riemann sphere, what I am going to get is sufficiently
small annular region surrounding the North pole and clearly the spherical distance can
be made less than Epsilon for any point in that region okay, so that is you know that
is pictorial justification or the statement.
So the moral of the story is now you know you are able to justify that this sequence
Z power n converges normally the function infinity okay on the exterior of the unit
disk in the extended plane okay and the point is that you are using the spherical metrics
okay that is the advantage you are using the spherical metrics because it allows you to
give you to measure the distance even to the point at infinity from a finite point in the
complex plane which you cannot do with the Euclidean method and the way all this is done
it will also extend the usual definition of normal convergence. Suppose you have a sequence
of analytic function which converges to a analytic function itself okay a finite valued
function.
A function that does not take values infinity then what is the usual convergence that we
talked about, the usual convergence that we talk about when you for example when you do
a 1st course in complex analysis the usual convergence that you talking about is with
respect to Euclidean metric okay you have to only worry about the Euclidean metric,
nobody is worried about the point at infinity to begin with okay. Now if you take a usual
sequence of holomorphic functions on a domain, analytic functions on a domain suppose it
is converging normally again to an analytic function on the domain okay. Then suppose
this convergence is in the usual sense I am saying this convergence is also correct with
respect to the spherical metric.
The reason is because the spherical metric when you restrict it to the usual complex
plane it is equivalent to the usual Euclidean metrics, so you do not lose anything. So what
I am saying is that this definition of sequence of functions converging to another function
normally on a domain that works irrespective of whether you are using the Euclidean metric
or whether you are using the spherical metric but the point is it helps you when infinity
values are taken it helps you because when infinity values are taken you cannot use Euclidean
metric you can use only the spherical metric okay. So the normal convergence under the
spherical metric is just an extension of the normal convergence under the Euclidean metric
as far as subset of the complex plane is concerned convergence under this spherical metric is
same as convergence under the Euclidean metric.
Normal convergence under the spherical metric is same as normal convergence under the Euclidean
metric because they are equivalent okay, so so this is 1 point that you need to understand,
so you know so let me say this so let me write this specifically we need to therefore worry
about by the time it include we need to worry about that this include the function infinity
that takes the value infinity at every point of your domain. Then we may define normal
convergence as follows; let f n be a sequence of holomorphic functions or analytic holomorphic
is same as analytic functions on a domain D in the complex plane. We say f n converges
to f on D normally okay. If the spherical distance between f n of Z and f of Z goes
to 0 normally which means uniformly on compact sets
on D where we allow f to be the function infinity this is an exceptional case okay, so you define
normal convergence in the following way.
So what is the normal convergence you have sequence of functions on a domain in the complex
plane you say f n converges to f on the domain okay. If the spherical distance okay that
converges to 0 okay and you this spherical distance is a function of Z, so Z is varying
on the domain, so I wanted to understand this Z is varying on the domain. This quantity
here is also a function of Z it measures for each Z it measures the spherical distance
between f n of Z and f of Z okay and what I want is that a function of Z should go to
0 the constant function 0 uniformly in Z on compact subset of set of the domain, so I
wanted to go to 0 normally on the that is my definition and now the beautiful thing
is you know we need this definition because if you take the domain as I told you if you
take the domain to be mod Z greater than 1 and you take f n of Z to be Z power n such
a definition is necessary.
So what it tells you is that now you have to also worry you are not worried only about
functions which takes complex values you have also allow functions in the value infinity
but then notice if you take the value infinity if you allow function take the value infinity
then you can include Meromorphic functions because you can define the value or the Meromorphic
functions at a pole to be infinity and the beautiful thing is the very same definition
this very same definition works absolutely well if you change holomorphic by Meromorphic.
So that is because of a circle and cemetery rotational symmetry that is the about for
this spherical metric that I will explain in the next lecture at the point is so the
important observation is that this same definition works with holomorphic replaced by Meromorphic
and that is all that we need to do all the analysis we want okay. So I will stop here.