 Yeah, 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 setup that you need to be able to prove the big Picard theorem and the little Picard theorem okay. So now the so you know let me briefly you know 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 functions that is that is a Banach algebra and it is also a topological space it is complete as a metric space the metric is induced 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 take inspiration from topology okay then trying to do topology trying to do topology on the space of functions is the same as do 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 okay 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 or extended plane. So if you are looking at say holomorphic functions on a domain okay a domain in the complex plane or a domain in the external complex plane that does not matter suppose you are 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 already seen that the set of functions it is a ring in fact and then if you look at meromorphic functions it is a field okay it has algebraic structure but we are now worried about the topology okay. 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 topology 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 do it 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 disc to the sum of the geometric series is which is 1 by 1 minus the variable okay. So the geometric series is 1 plus z plus z squared and so on where z is a variable and you are restricting z to be in the unit disc that means you are making mod z less than 1 then 1 plus z plus z squared and so on that converges to 1 by 1 minus z that is a high school formula for geometric series. Now the point is that this convergence is absolute on the unit disc there is no problem about that but it is not uniform on the whole unit disc 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 the convergence is not uniform on the whole unit disc it is only uniform on compact subsets of the unit disc okay. So you do not get convergence but 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 first you know that is the first moral first lesson to be learnt 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 things are 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 you have a sequence of holomorphic functions which converge normally to a limit function then that limit function is also holomorphic okay this is something that I explained last time okay essentially it uses you know Cauchy's theorem and Moreira's theorem okay and the fact that analyticities or holomorphicities local property. So therefore there is no harm in relaxing the condition of uniform convergence to the condition of normal convergence which 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 close disks because they are also compact and therefore you get uniform convergence on sufficiently small open disks 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 a 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 are going to 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 the 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 that is for example that is one of the characterizations of a pole the limit of the function as you approach a pole is going to infinity okay and by that I mean limit goes to the point at infinity okay and of course here you are using the topology on the extended complex plane namely the complex plane with the point at infinity given by the one point compactification which makes it homeomorphic to the three months we have okay. Now there is another pathology and that is the pathology that I was trying to explain towards the end of the last lecture so the pathology was the following you take the exterior of the unit disk mod z greater than 1 that is our 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 is you are looking at these functions powers of z you are looking at 1 which is z power 0 if you want then z and then z squared and z cube and so on okay. Now that is a sequence of functions and the point is that this sequence of functions you can see point wise it will go to infinity because since mod z is greater than 1 mod z to the n to the power of n that is going to go to infinity because it is for a real number greater than 1 you know its higher powers will diverge to infinity so mod z so z power n is so the sequence of functions is going to converge point wise to the function to the constant function infinity namely it is a function that which associates to every point the value infinity okay so you have to worry about this crazy function okay so you have to so this is a pathology that happens that you will have to take care of and the point is that therefore we are forced to introduce a function called 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 external complex plane because after all in the external complex plane infinity is a valid point okay it is a member of that set so you can really think of the function infinity as the constant function with taking the value 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 fn 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 you have this very nice situation it is a very nice pathology you have these z power n which are all holomorphic functions okay 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 compactness 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 a hornest sequence of holomorphic functions you have a hornest sequence of analytic functions which is converging to the function infinity normally that also happens 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 it will happen also for meromorphic functions because you know holomorphic functions are very good melomorphic functions are worse 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 to the function which is infinity okay you should expect the same thing to happen also for meromorphic functions so what I am trying to tell you is if you sum up all this if you want to study topology on the space of meromorphic functions first of all you must study it with respect to normal convergence okay the second thing is you have to introduce this function keep in mind this function which is the function infinity okay and then you have to do 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 me write the let us do the following thing we will worry about matrix on the plane and matrix on the extended plane which are transported from matrix 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 well x, y and so this is my usual complex plane which is the x, y plane and then I have the well you 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 is let me draw this let me draw this thing here which is a Riemann sphere so this is supposed to be 1 this is minus 1 on the x axis and here is my sphere it is 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 third 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 an odd pole okay suppose I start with 2 points z1 and z2 on the complex plane okay see what kind of distances can I define on these points what kind of metric can I define on the complex plane there is a usual metric which is d of z1, z2 so I will put d sub e for the Euclidean metric and you know what that is it is simply modulus of z1 minus z2 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 this line segment okay joining z1 and z2. Then the other thing that you can do is you can take these 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 p2 is this point here on the sphere which is the unique point of intersection of the line joining n and z2 on the sphere okay and similarly p1 is unique point on the Riemann sphere which is the intersection of the surface of the sphere with the line joining n and z1 okay and now you see p1 and p2 lie on the sphere now what I can do is that I can measure the distance between p1 and p2 okay now that distance I am measuring in 3 space okay because now everything once you draw the Riemann sphere you are actually in R3 and your R2 which is the xy plane corresponds to the complex plane alright. So what you could do is you could define this you could define the following new distance also d let me call this as dc for chordal distance so dc is the chordal distance from z1 and z2 it is just length of the chord from p1 to p2 where p1 the stereographic projection of p1 is z1 and the stereographic projection of p2 is z2 okay so this is another distance that I can define it makes see what this distance does is that actually it is the metric in R3 after all the chord joining p1 p2 is exactly the line segment from p1 to p2 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 okay and so it is a metric space you know whenever you have metric on a space and you restrict to a subspace 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 through the stereographic projection you are transporting that metric to the complex plane because after all the stereographic projection is a bijection between the extended complex plane and the Riemann sphere okay. The moment you have a bijection of a set with a metric space you can transport the metric on the metric space to the set so I have just transported the basically what I have done is I have simply transported the metric on R3 restricted to the Riemann sphere I have simply transported it to the plane that is what this DC is so this DC also will D sub C that is also a metric you can check that and that also makes the complex plane into a metric space okay and but the big deal is that you know all these metrics are all equivalent okay namely the topology is that they induce on the complex plane they 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 say 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 that 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 distance to the point at infinity I cannot do it with the Euclidean metric because first 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 add this extra point I have to add this extra topology the topology of the one point compactification but then I 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 homeomorphic to the extended plane okay and therefore I am led to look at the metrics on the Riemann sphere and so this is the Caudal metric okay so D sub C is the Caudal metric and then here is a third metric which is the spherical metric. So D sub S of Z1, Z2 this is the length of the arc of the minor arc from P12, P2 along this length of the minor arc from P1 to P2 along the great circle through P1 and P2. So you see this is a spherical distance what is a spherical distance is a spherical distance actually I am trying to measure distance on the Riemann sphere on the surface of the sphere so it is a curved distance okay and I am trying to measure the shortest curved distance and you know you can imagine this if you basically what one is doing is that one is doing a kind of some kind of Riemann 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 on the surface passing on the surface okay and then you can measure the lengths of each of these 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 geodes if you give me 2 points on the sphere on the surface of the sphere then the geodesic is exactly the following you for those 2 points you get a big circle a great 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 this 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 sub s it is a geodesic for the sphere for any 2 points on the sphere okay. So the great circles of the geodesics okay the minor arcs of the great circles of the geodesics for points on the sphere so what is happening is that now I have all these 3 matrix the beautiful thing is that these 3 matrix give you matrix not only on the complex plane the point is they give you matrix on the extended plane okay see what I have drawn here is for z1 and z2 imagining z1 z2 as points on the complex plane but I can very well make z1 or z2 to be the point at infinity now when I say I make the point z1 or z2 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 a north 4 so basically what I am doing is I am 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 this these distances help you to give a matrix on the extended plane and the topology induced by all these matrix is 1 and the same all these matrix are equivalent okay so this is the fact that you need to check from you know this is very easy in fact to check ideologically let me tell you how to do it how will you check the 2 matrix are equivalent for a topological space it is very simple what you do is that you show that you take a point 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 that is true okay for all the 3 matrix so therefore all these 3 matrix will give you 1 and the same topological space structure on the extended complex plane which is the same as the 1 point compactification and that will be exactly homeomorphic 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 functions f n converges normally to infinity to the 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 second and the third 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 some somewhere here may be in a may be use a different colour all the metrics below are equivalent on C alright and the last 2 metrics are equivalent on the extended plane the last 2 or the latter 2 so let me make some space let me get rid of this the latter 2 namely these 2 are equivalent on the extended plane C union infinity okay and of course so I rubbed off 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 is 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 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 to the function which is infinity is something that is hard to as it is if you do not analyse it is very hard to digest okay so it is very important that you understand what is happening here and you must understand that therefore even if you are looking at normal convergence of you know honest holomorphic analytic functions you can still end up with the function at the function which is infinity okay so you see what is happening see so here is so again let me so again let me draw another diagram so you see here is the here is your complex plane as it is in R2 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 it is very hard to draw that and in a 3 dimensional diagram so let me do the following thing let me use another colour you know I am 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 a xy plane okay and of course you know the third axis is I am calling it as u now you see so let me change colour 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 see this these red lines that I have drawn they are supposed to extend outside the unit circle to 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 it is a neighbourhood of infinity okay and what is its image under the stereographic position it is exactly the upper hemisphere okay it is the whole upper hemisphere which is which you can see clearly is a neighbourhood of the north pole okay that is what that is the region you are looking at alright. Now you see take a compact subset of mod z greater than 1 take a compact subset it 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 has 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 outer circle centered at the origin sufficiently small inner radius greater than 1 and sufficiently large outer radius greater than of course the inner radius okay so you see if I take some compact set here so here is some compact set K compact set in in in the complex plane then you see this K of course lies inside suitable annulus so it is going to it is going to look like this you know I am going to get something like this I am going to get I am going to get this annulus here so I am going to get this annular region mind you this is the annular region on the complex plane consisting of the region between these two circles and I am also including the boundaries to make it compact okay so it is a close 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 annular 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 annular is 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 under the stereographic projection I am going to get this I am going to get something like this here this is what I will get I will get a curved annular region centered at the north pole alright and now you see 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 for each point z z0 f n of z0 converges to infinity as n tends to infinity I cannot say that I can say that in a topological sense okay 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 if you look at the distance the spherical distance between a point z in I take the point z in k my compact z 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 I am what I have written there is actually infinity of z and I am saying the 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 n greater than or equal to capital N irrespective of 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 large n large enough n such a large enough n exists the point is that this large enough n does not have anything to do with this z it will work for all z in the compact set that is the uniformness that is the uniformness of the convergence on the compact set so n this n is so there exists this n and this is independent of z and you see this is this fact is true this fact is very you 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 well mod z power n is going to lie in this annulus okay and you know if you see if I this is 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 take its image in the on the Riemann sphere what I am going to get is a 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 for this statement so moral of the story is now you know you are able to justify that this sequence z power n converges normally to the function infinity okay on the exterior of the unit disc in the in the extended plane okay and the point is that you are using spherical metric okay that is the advantage you are using spherical metric because it allows you to give you to measure the point distance even to the point at infinity from a finite point in the complex plane which you cannot do with the Euclide metric and the way all this is done it will also it will also extend the usual definition of normal convergence see suppose you have a sequence of analytic functions which converges to a honest analytic function itself okay a finite valued function a function that does not take values infinity then what is the usual convergence that we talk about the usual convergence that we talk about when you for example when you do first course in complex analysis the usual convergence that you talk about is with respect to the 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 a honest sequence of holomorphic functions on a domain analytic functions on a domain suppose it is converging normally again to an analytic function on that 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 metric so you do not lose anything. So what I am saying is that this definition of a 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 you 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 they are this as far as a subset of the complex plane is concerned convergence under this spherical metric is the same as convergence under the Euclidean metric normal convergence under the spherical metric is the same as normal convergence under the Euclidean metric because they are equivalent okay. So this is one 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 that I mean include we need to worry about that is include the function the function infinity that takes the value infinity at every point of your domain then we need to define we may define normal convergence as follows. Let fn be a sequence of holomorphic functions or analytic holomorphic is the same as analytic functions on a domain t in the complex plane we say fn converges converges to f on d normally okay. If the spherical distance between fn 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 fn converges to f on the domain okay. If the spherical distance okay that converges to 0 okay mind you this spherical distance is a function of z so z is varying on the domain. So I want you to understand this z is varying on the domain this quantity here that is this quantity here is also a function of z it measures for each z it measures the spherical distance between fn of z and f of z okay and what I want is that function of z should go to 0 the constant function 0 uniformly in z on compact subsets of z of the domain. So I want it to go to 0 normally on the domain 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 fn 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 take complex values you have to also allow functions take the value infinity but then notice if you take the value infinity if you allow functions take the value infinity then you can include meromorphic functions because you can define the value of the meromorphic function 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 certain symmetry rotational symmetry that is there about for the spherical metric that I will explain in the next lecture but the point is so the important observation is that this same definition works with holomorphic replace by meromorphic and that is all that we need to do all the analysis we want. So I will stop here.