 Alright, so you see I am continuing with this motivation for Zalkman's lemma, alright and so let me recall what we were doing, so I will let me use a different colour. So basically you are trying to understand you know the behaviour of a normal family at a point, okay that is the understanding that is you know the key to understanding the statement of Zalkman's lemma, okay. How does a normal family of meromorphic functions, how does it behave at a point at a given point? So you know the important outcome of this analysis is that you know you can characterize normality at a point and normality at a point is defined as normality in a neighbourhood of that point, in some neighbourhood open disc surrounding that point, okay and then so you see this means that normality can be defined locally and it is in fact a local property like analyticity, okay. So we start with the family script f, these are meromorphic functions defined in a neighbourhood of the point Z0, I am assuming the point Z0 is in the complex plane it could have been also a point in the extended complex plane namely it could have been the point at infinity, okay all arguments will work but you have to make the right modifications you know how to do that because we know how to deal with the point at infinity and taking infinite limits and things like that, okay. So all right, so well what I told you last time is that you start with this family script f and what you can do is given a function small f in script f, okay you want to study that function in a neighbourhood of the point Z0, okay. So what you do is that you know you take this neighbourhood of this point which is actually it is a disc centred at Z0 radius rho, okay you take this neighbourhood and I am also assuming that the boundary of that disc is in the domain where the functions are defined, okay. So and the reason is why the reason why I am including the boundary is because I want a compact set, okay the disc along with the boundary forms a compact set it is a closed and bounded set, okay and well so suppose I want to study a particular function small f in the family script f at the its behaviour very close to the point Z0 then what I do is I zoom into the point Z0 and try to study the behaviour of the function and how do I do this zooming I do the zooming in this way so I define this zoomed function here, okay g. So given f I zoom f to get a new function g, okay and what is this zooming you are zooming the function f centred at Z0 and the zoom factor is 1 by epsilon where epsilon is supposed to be a positive small positive quantity so that 1 by epsilon is large in fact epsilon is to be taken less than rho, okay and then you define this new function which is for this f you define this function g which is a zoomed function g is just you take f look at its behaviour in a small disc around Z0 and then you zoom the behaviour by a factor 1 by epsilon, okay and you so what you do is that so this is how the zooming is defined the zoom function g of Zeta is f of Z0 plus epsilon Zeta which means that actually what you have done is you are actually translating Z0 to the origin, okay and then what happens is that you know this whole disc centred at Z0 radius rho along with the boundary translates to disc centred at the origin with radius rho by epsilon, okay so 1 by epsilon is the scaling factor and mind you you should think of it like this epsilon is small so 1 by epsilon is large so rho by epsilon is greater than rho so you have actually zoomed, okay and so the behaviour of the zoomed function is being studied here, okay and in this disc and well and I am calling this zoomed function g so this is a very very simple thing and the point is that the difference between f and g is just a you know it is a bilinear transformation it is actually it is translation and it involves a translation it involves a scaling, okay so what you have done is you have taken the variable Zeta you have scaled it by epsilon, okay you have multiplied it by epsilon mind you epsilon is a positive real quantity and then you have translated by Z0, okay so the g and f are of the same type of function, okay g is analytic if g is analytic if and only if f is analytic g is meromorphic if and only if f is meromorphic and whatever if g has a pole at a certain point then f will also have a pole of the same order at the corresponding point, okay so and conversely so g and f are literally the same function except that you have made a change of variable alright but the point is that g is a close up look of f it is you are looking at f very close in a neighbourhood of Z0 that is the whole point, okay now what you do is that this is what you do if you have a single function but what you could have done is you could have done this to a sequence of function so the point is if I knew that this family script f is normal then suppose you assume that the family is normal in this close disc which means actually it is normal in an open set which contains this close disc including the boundary, okay and you know suppose this family is normal you know normality is the correct notion of compactness that we require normality actually means normally sequentially compact it means that given any sequence you can find a subsequence which converges normally, okay so converges normally means converges uniformly on compact subsets so assuming that the family script f is normal and I want to study the behaviour of the family at a given point Z0, okay so if I take a sequence fn in this family then you because of normality you can find a subsequence fnk that will converge normally on this disc in fact it will converge uniformly on that disc because that is a close disc, okay normal convergence means it is uniform on compact subsets and this disc centred at Z0 radius rho is you know it is a compact subset and again at the back of your mind you should remember that you know this argument will work even if Z0 is a point at infinity the only thing is that if Z0 is a point at infinity you will have to think of this you should write this disc you know if you want with the Euclidean metric if you want with the spherical metric or you must invert the variable and look at neighbourhood of 0, okay so you can deal with the case when Z0 is the point at infinity also, okay but in any case given this sequence fn I have this subsequence which converges normally so let me call the limit function as f and then we have already seen this whenever you take a normal limit of meromorphic functions the limit is either identically infinity or it is meromorphic, okay and now look at what happens to the zoomed functions so if fnk converges to f normally then the zoomed functions for the sequence will converge to the zoomed function of the limit, okay so this is obvious, okay because the difference is only the difference is just a change of variable given by a bilinear transformation consisting of a scaling and a translation, okay so this gnk will converge normally to g, okay and all this is happening mind you because your zoom factor is 1 by epsilon it is all happening here it is happening in this open disc centred at the origin radius rho by epsilon which is greater than rho, okay and well this is where it is happening now you see the point is that whenever a family of meromorphic functions converges then the family of spherical derivatives will also converge, okay now this is just saying that you know the taking the spherical derivative will preserve the normality the convergence and the normal convergence, okay so fnk converges to f normally so what will happen is that you will get this which is that the spherical derivative of fnk converges normally to the spherical derivative of f, okay and we already know that because fnk converges normally to f g and k converges normally to g, alright and therefore the spherical derivatives of the g and k hash they will converge normally to g hash, okay and but the point is that you see the original family that the family script f you have assumed is normal so this is this family script f is normal and Marty's theorem tells you that you know that a family is normal if and only if the spherical derivatives are normally uniformly bounded, okay and since we are already looking at a compact set, okay there is a uniform bound for all the derivatives the spherical derivatives of the functions in the family mind you have to use spherical derivatives because functions in the family are not analytic functions they are meromorphic functions so you cannot talk about an usual derivative at a pole, okay but then you can talk about spherical derivative at a pole. So by Marty's theorem the Marty's theorem mind you tells you that normal sequential compactness is equivalent to normal uniform boundedness of the spherical derivatives, okay and it is a generalization of the Montel's theorem for analytic functions which says that you know the normal sequential compactness is equivalent to uniform boundedness normal uniform boundedness of the original family, okay and but the point is when you go to Marty's theorem you go to from the uniform boundedness of the normal uniform boundedness of the given family you switch to the normal uniform boundedness of the family of spherical derivatives, okay that is a quantum jump that you make from Montel's theorem for analytic functions to when you go to Marty's theorem which is for meromorphic functions. So there is this uniform bound M which works for all functions in the family script F so it will work also for these f n k's it is a bound for not the functions but it is bound for the derivatives, okay spherical derivatives. So all the spherical derivatives of the f n k's are bounded by M and then well what happens is therefore now if you take the limit so there is a chain rule that is working here, there is a chain rule the spherical derivative of the zoomed functions g and k hash is epsilon times the spherical derivative of the f n k, the spherical derivative of the g n k's is epsilon times the spherical derivative of the f n k's because this so you know the idea is that you are zooming by a factor 1 by epsilon, okay you are going closer to the point by you are zooming into the point by a factor 1 by epsilon, okay but then the spherical derivative becomes smaller. This spherical derivative gets multiplied by the factor epsilon mind you epsilon is a small quantity so multiplying by epsilon makes the quantity smaller, okay whereas 1 by epsilon is a zooming factor so as you zoom closer your spherical derivative is going to become smaller, okay that is what is happening, okay. So for the zoomed function if the original functions have this bound M then the zoomed functions have the bound epsilon M and then because the zoomed functions g the spherical derivatives of the zoomed functions converge normally to the spherical derivative of the limit function which is the zoomed limit function what it will tell you is that the limit function also if you take its spherical derivative is bounded by this epsilon times M, okay. Now the point is that you know all this happens in this domain which is the disk center the origin radius rho by epsilon, right and mind you must remember that as I make a smaller and smaller and smaller I am zooming closer and closer to the point z0 but then the zoomed functions are in larger and larger disk center at the origin because rho by epsilon becomes larger and larger as epsilon becomes smaller, okay. So now the point is that you repeat this game, okay by changing the zoom factor, okay. So what you do is instead of taking single epsilon you take a sequence of epsilon which go to 0, okay a sequence of which go to 0 of course from the right namely that you take sequence of positive epsilon that go to 0, okay. So if you do if you repeat the game with that what you do is that now you know the nth function g n, okay well I think I have not written it correctly last time the nth function g n is the zooming of the nth function f n by the factor 1 by epsilon n center at z0 with the variable zeta so that should have been an f n here, alright. So g n is the zooming of f n, alright but the only thing is that now the zooming factor is also dependent on n, okay. f n is zoomed by a factor 1 by epsilon n and I am calling it as g n, okay just what we did some time ago was keeping all these epsilon n's the same, the same epsilon but now I am making the epsilon n's smaller as n becomes larger, okay well so if I do this what happens is mind you if I make the epsilon n smaller then the domains on which the zoom functions work these domains become bigger, okay so you know so I get these domains, okay mod zeta lesser than well I think there was a row there was a row there was a row there, mod zeta less than row by epsilon n and these are increasingly larger disks as epsilon n goes to 0 these disks cover the whole plane, okay and in fact as n that means as n tends to infinity you are getting a you will cover the whole plane in particular you will cover any compact subset of the plane. So what you can say is that these zoom functions g n you know you can talk about normal convergence on the whole plane, okay. So actually what happens is that the anyway as before the f nk's will converge to f, okay and therefore the f nk hash will converge to f hash where hash denotes the spherical derivative and then the corresponding zoom functions g nk hash will converge to g hash all these convergences are all normal, okay but the only thing is that this when you come to the zoom functions this normal convergence is on all of c it is on all on the whole complex plane because now you have covered the whole complex plane you have covered every compact subset of the complex plane by a sufficiently large disk centered at the origin where that means all the g n's all the g nk's beyond a certain k they are all defined on any compact subset beyond a certain stage, okay. So fine so you have this and the point is that you know the g nk's if you take the spherical derivatives they are bounded by epsilon nk times the bounds for the original functions which is m, okay and therefore this limiting argument will tell you that the limit function g hash the limit function g if you take its spherical derivative it has to be 0 and therefore you get a constant function on all of c, okay. So you see the point is therefore you know as you go closer and closer and closer to a point and look at the zoom functions, okay then the zoom functions they tend to become constant that is what it means, okay the zoom functions become constant. So you know to sum up all this all I am saying is that you take a normal family you take a point where the family is normal and then what happens is that as you take any sequence of in that family you can always find a convergence of sequence if you study it at that point, okay then the zoom functions will you know they will tend to a constant function that is the whole point, okay and Zalkman's lemma is all about you know being able to find zoom functions which converge to a non-constant meromorphic function, okay. So something opposite to this happens, okay so what I want to do first is that I need to you know first of all tell you that I can repeat this argument with not just one point Z0 but I can even take the sequence of points tending to Z0, okay. So now what we do is now take a sequence Zn going to Z0 in say the same disk mod Z minus Z0 is not equal to rho if you want, okay well for that matter if you take any Zn sequence Zn going to Z0 the sequence beyond a certain stage will lie in that disk, okay but I am assuming this whole sequence lies in that closed disk center at Z0 radius rho, okay and what you do is now you zoom at each Zn, okay see all this time we were zooming only at Z0 but now what you do is you zoom at each Zn so you know they so you define Gn of Zeta to be the zooming of you zoom the function Fn but now centered at Zn and you know the zooming factor is 1 by Epsilon n and the new variable is Zeta. So this means that Gn of Zeta is well it is Fn of Zn plus well Epsilon n times Zeta so this is the thing but then I have to worry about one has to worry about where this will what about the domains of the Gn's, okay you will have to worry about that but the point is that essentially as n tends to infinity you are coming closer and closer to Z0 so the domains of the Gn's is going to again cover the whole complex plane, okay. So one can write out the details for that so you know the picture is like this the picture is that you know so I have this Z0 and you know there is so there is Zn and then I have this so inside Z0 there is this big disk with radius rho where I am concentrating my attention and of course even the boundary is included though I am putting a dotted line well and what I am doing is that Zn I am taking you know I am taking a smaller disk radius Epsilon n, okay and well and then you know so Zn is Zn's are tending to Z0 so well Zn plus 1 is closer to Z0 if you want and there is it and then I am taking a much smaller disk and the radius of that disk is Epsilon n plus 1, okay. So you know I am coming I am taking smaller and smaller disks about points which are going closer and closer and closer to Z0, okay and the point is that on this I have this zoomed function, okay and then if I go to Zn plus 1 I have another zoom function it is Gn plus 1 of Zeta and this is Fn plus 1 centered at Zn plus 1, yeah so it is Epsilon n plus 1 Zeta so this is just assuming of Fn plus 1 centered at Zn plus 1 with the factor 1 by Epsilon n plus 1 and the variable Zeta, okay. So I have these zoom functions, okay and well of course you know I will have to assume that the Epsilon n's are chosen in such a way that the disk centered at Zn radius Epsilon n is within my big disk centered at Z0 with radius rho, okay but that will happen eventually even if you did not assume it beyond a certain stage it has to happen, okay but we assume therefore without loss of generality that this happens that the diagram is like we have already shown it so you know for example well you know so if you take so I am assuming that if you take Z1 then I am taking the Epsilon 1 that I am taking is such that the disk centered at Z1 radius Epsilon 1 lies inside this big disk centered at Z0 radius rho, okay and the point is that if this does not work I can replace the Epsilon i's by some Epsilon i's which I can calculate, okay. So the condition I want is that you know the distance of Z0 to any point of the disk should not be greater than rho, okay it should be less than rho that is the condition I want. So well let us assume that the picture is like this, all right you do we do the same calculations as before you see fn the you know you have this fn case they converge normally to f they converge normally to f on normally this is normally and well and what happens is that you will therefore get that the spherical derivatives will converge to f hash normally and you will also get that the zoom functions will also converge normally to g and so will their spherical derivatives, okay. So you will get all this as before mind you the only thing now is that you also change the center of the zooming, okay the centers of the zoomings are all different now, okay but it does not that does not change the calculation for the spherical derivative, okay because spherical derivative for the zoom functions with respect to Zeta and as far as Zeta as concerned the Zn's are all constant, okay. So well again what you will get is that you will get that again you will get that you know the original functions f and k their spherical derivatives are bounded by m and this will tell you that the zoom functions their spherical derivatives are going to be bounded by epsilon the corresponding scaling factor times m I mean not scaling factor the corresponding epsilon times m and if you take a limit as n tends to infinity you will get that the limit zoomed function which is the zoomed function of the limit function that will have spherical derivative 0, okay that is because the epsilon nk tends to 0 and m is a fixed constant positive constant and the moment you and as before the moment you know that the spherical derivative of meromorphic function is 0 it has to be constant. So this will tell you that g is constant on the complex plane mind you this constant could have very well been the constant value infinity that is allowed, okay because we are in the context of meromorphic functions but the whole point is now you have got this important characterization you take a point Z0 where a family is normal, okay then what happens is that you take any sequence of point Zn going to Z0 and you take a sequence of radii which are going to 0 the epsilon n's, okay then given any sequence in the family you can find a subsequence such that if you take the subsequence and zoom it with respect to this sequence the zoom functions converge uniformly on compact subsets of C that is normally unsee to a constant function this is the behaviour of a normal family at a point, okay a normal family in a neighbourhood of a point, okay and the amazing thing is that the whole point is that this characterises normal families, okay this is by this you can actually characterise normal families, okay so let me write this down and the fact that you can characterise normal families like this is actually the philosophy behind Zalkman's lemma and the way it works is it tells you when this is contradicted. So Zalkman's lemma will tell you that such a thing will not happen if the family is not normal, okay so let me write this so proposition is well let script F be normal at Z0 which means that it is a normal family in an open disk containing Z0 and of course all my families of functions are Meromorphic, okay so let me write here normal family of Meromorphic functions at Z0, okay given any sequence Zn tending to Z0 and any sequence of radii epsilon n tending to 0 plus, okay for any sequence fn in the family script F, okay we can find a subsequence fnk such that the zoom sequence of fnk with respect to Zn and epsilon n that zoom sequence converges to a constant function normally on all of the complex plane, okay. Gnk of Zeta is equal to the zooming of fnk with respect to centre Zn Zk radius 1 by I mean scaling factor 1 by epsilon k new variable Zeta which is by definition you just take fnk of Zk plus epsilon k times Zeta this is the zooming, okay now such that if this is this zoomed family then Gnk converges to a constant normally on the whole complex plane. So this is the characterization of this is how normal families behave, okay and the big deal about so this is normality at a point, okay this is normality at a point Z0 which means which by definition is normality in a small open disc containing Z0 that is what it means, okay and if you want to cover this to normality on a whole domain then you know you have to say a family is normal on a domain if it is normal at each point of the domain, okay you can define it point wise. So now let me state Zalkman's lemma which will tell you and you will appreciate what it is all about but you know with all this background that I have given you will see that Zalkman's lemma is something natural to expect, okay. So here is Zalkman's lemma and it is actually well it is called Zalkman's lemma but it is actually theorem. So what is a lemma? The lemma is about normality it is about non-normality, okay see I will tell you in very short words what we just saw is that if you have a normal family in the zoom functions go to a constant, okay Zalkman's lemma tells you that if you have a family which is not normal you can get hold of a zoomed family of functions which will go to a non-constant meromorphic function that is all that is the whole point, okay. So let me write that of course you know you should always keep saying this in as little words as you can so that you grab the main idea but then of course when you write statements you will have to be you know you have to bring in lot of notation you have to worry about lot of notation and then of course proofs are even more slightly complicated but the point is that you should always be able to zoom out you know and say things in just a few words because that is how you will map it in your memory and remember it and so that you do not lose track of the idea, okay. So here is a lemma suppose script F is a family of meromorphic functions on a domain D that is not that is not normal. So here is a non-normal family then what happens is that there exists a sequence Zn converging to Z0 in D and epsilon n going to 0 plus with and further a sequence fn in the family script F such that the zoomed functions gn of zeta is equal to the zooming of the fn's centered at the Zn's scaling factor 1 by epsilon n new variable zeta namely fn of zeta plus epsilon n zeta converges normally on C to a non-constant to a non-constant that is the whole point to a non-constant meromorphic function g on the whole complex plane such that okay well the fact that you know it is a non-constant meromorphic function so you do not expect the derivative to be spherical derivative to be 0. So the spherical derivative at the origin will be 1 and all the spherical derivatives is bounded by 1 okay so this is Zolkman's limit. So it tells you see it tells you that the behaviour is exactly opposite to what you saw for a normal family okay so we will try to prove this in the next lecture.