 Alright, so what we did in the last lecture was you know try to explain how the notion of normal family makes sense for a domain which includes the point at infinity okay and the technique as you that we always follow is that you make a 2 piece definition okay. You make one definition for the domain punctured at infinity which will be a domain in the usual plane and then to take care of a neighbourhood of infinity what you do is that you invert the variable z to 1 by w and instead of considering z equal to infinity you have to consider w equal to 0. So you take you consider a neighbourhood of 0 okay which is again it is also a domain in the usual complex plane okay. So therefore you are able to treat the case of normality at infinity okay. So with that we saw that both Montel's theorem and Marty's theorem hold good for domains which are which probably include the which may include the point at infinity okay that is domains in the external complex plane and of course I also told you that it is a matter of terminology that in a family being normally sequentially compact that is sequentially compact with respect to normal convergence that is sometimes abbreviated to simply normal family okay. So now what we are going to do is you know as I told you we have to we have to we have come very close to the proof of the Picard theorem which was the main aim of these all these lectures okay and but before that there is something called there is a theorem which is called as Zalkman's lemma which we have to use okay and see this has got to do with a characterization of non normality okay. So what we so in order to you know motivate this Zalkman's lemma okay what we are going to do is we are going to try to analyse what happens to a normal family at a point okay. So let me make this definition we define normality of a family at a single point to be normality in a small neighbourhood of that point in an open disc containing that point okay. So this is just like definition of analyticity so you see you say a function is analytic if it is differentiable not only at a point but in a small neighbourhood of that point. So in the same way normality is also defined at a point by requiring the property in a small open disc surrounding that point or a non-empty open set containing that point okay. Well of course any open set containing that point is non-empty. So the reason for this definition is that normality is a local property okay to if you have for example a cover of your space which is countable and if you have if your domain can be covered by countably many open sets and on each of these open sets if your family is normal then on the whole domain also the family will be normal because you can use because of countability you can use a diagonalisation argument okay so normality is a local property okay. So the first thing I am going to start with is so I will put the heading as motivation let me use a different colour motivation for Salkman's lemma which is actually a characterization of non-normality. So what we will do is we make the following definition let script F be a family of meromorphic functions defined in a neighbourhood of a point Z not okay of course this point Z not could also be the point at infinity but let us assume that Z not is a point in the plane the case of you know how to treat the case at infinity okay. So suppose this is a family of meromorphic functions defined in a neighbourhood of Z not that means it is defined in an open discontaining Z not okay we say the family script F is normal at Z not if it is normal in a neighbourhood of Z not an open neighbourhood of Z not okay. So well so that is the definition and so the question is suppose a family and of course a family is normal on a domain if it is normal at every point okay further family is normal on a domain if and only if it is normal at each point so I should not say further in fact this is a remark so this is a remark that you can check I mean it just says that the notion of normality is a local property normality is a local property fine so here is the question the question is I now have a family defined in a neighbourhood of a point okay it is a family of meromorphic functions and the question and it is given to me that this family is normal okay and I wanted I want to analyse how the family behaves at that point okay. So the clue to Zalkman's lemma or at least an understanding or motivation for Zalkman's lemma is trying to understand this okay so what we will do is let us so let so I call this as local analysis normal family. So let script F be meromorphic in meromorphic be a family script F be a family of meromorphic functions on mod Z minus Z not less than or equal to rho okay so you see mod Z minus Z not less than or equal to rho is actually a close set it is a compact set okay if that is provided Z not is a point in the usual complex plane if Z not is infinity then you know you have to rewrite this as you must rewrite this as in the you should not use the Euclidean metric but you have to use the spherical metric okay but I am assuming for simplicity that Z not is a point in the plane okay and this is a close and bounded set I am taking a compact set because I can choose such a row certainly it is a family of meromorphic functions at the point Z not so it is defined in a neighbourhood of Z not so it is defined on a small enough disc which contains Z not and if you take a smaller enough radius then the close disc centred at Z not of that radius will also be in the domain where the meromorphic functions are defined where the family is defined so I am taking without loss of generality this close set because the reason is I want a compact set so that I can you know I can get uniform convergence whenever I talk about convergence you know uniform convergence it happens only on compact sets it is normal convergence and whenever I am talk about uniform boundedness it happens only on compact set so for everything I need compact I need a compact set that is the reason why I am doing this okay. So what it means is that the family is defined on a bigger open set which contains this close set okay now you see what we want to do is we want to really so you know let me draw diagram so here is Z not alright and here is this there is this disc centred at Z not radius row so this is the disc centred Z not radius row and the point is you give me a family you give me a function small f in the family script f okay and what I want to really do is I want to really analyze this function in that small disc okay I want to analyze this function in this small disc and how do I do it so what I do is you know I have to look closer at the small disc so what I actually do is zoom in okay and how do I zoom in so it is by scaling okay so what I do is that I put I define g for every function f in the family I define this g and what is this g this g is defined with the new variable zeta and this is defined as f of z not plus epsilon times zeta I put this condition I make this definition so what is happening is that this is in this is in the z plane so the variable here is z so this is also the complex plane and then I have another complex plane okay and this is the well this is the zeta plane so the variable here is zeta alright and what I do is I take this I take this unit disc okay I take this unit disc this is this is unit disc mod zeta less than 1 less than 1 okay I take this disc and now I am defining this function g as a function of zeta okay and what is g it is actually you know you have just magnified the behavior of g at the value of g at zeta is the value of f at z not plus epsilon times zeta so what it means is that you know if you start with a and of course you know I am assuming that epsilon is less than rho okay so you know if I take this disc of radius 1 centered at the origin with the variable zeta then if I multiply by epsilon if I take epsilon zeta it will become disc of radius epsilon okay so it will be a smaller disc like this okay and this will have radius epsilon okay and then if I add z not to it I am just translating it to give me a small disc of radius epsilon centered at z not that is what this does okay so you are just multiplying epsilon to zeta then you are adding z not so that is translation by z not okay so basically what you are doing is that when you do this you know you are actually looking at this you are looking at this small disc centered at z not radius epsilon and so you know in principle for what values of zeta is g defined so you see you will see that g is defined for should be rho by epsilon see if mod zeta is less than rho by epsilon okay then mod epsilon zeta will be less than rho and mod z not plus epsilon zeta will be less than therefore also less than rho okay. mod zeta is less than rho by epsilon means that you know mod epsilon zeta is less than rho okay and what does this tell you this tells you that the distance of epsilon zeta from the origin is at most rho and therefore if you add z not to epsilon zeta the distance of z not plus epsilon zeta to z not is at most rho so this is so you see this g is defined on this disc and you see this disc is actually see this disc is larger see because you see rho the original disc centered at z not has radius rho alright now the radius has become rho by epsilon and epsilon being smaller rho by epsilon should be larger if you think of epsilon to be you know epsilon is less than rho so rho by epsilon is greater than 1 okay so what you have done is by this transformation this disc centered at z not radius rho okay if you want to study the function value of the values of the function f there the behavior of the function f there what you have done is you have actually zoomed it by constructing this function g so g is a kind of zooming function okay it is like you have a microscope or a telescope and you have 10 x zoom 20 x zoom you see that also in cameras these days okay you have 5 x zoom and so on so this is some so many x zoom and the zoom is one the zooming factor is 1 by epsilon okay original disc of radius rho has now become zoomed to disc of radius rho by epsilon okay and well the point is that and of course rho by epsilon will be slightly bigger okay you will get so this is epsilon this thing will be rho by epsilon and it is in this bigger disc open disc that the function g is defined and studying this function g in this bigger disc is the same as studying the function f in the original disc okay so you know I will write it like this I will write g of zeta is equal to zoom I will write I will think of it as zooming zoom the function f centered at z0 by a by a magnification factor 1 by epsilon and use the variable zeta okay I will use this notation okay so when I say f zoom f z not 1 by epsilon zeta you are zooming the function f centered at z not so you are zooming your zoom is centered at z not okay and 1 by epsilon is the magnification factor and zeta is the variable of the zoomed function okay now well the so this is how you zoom into the behavior of function at a point okay now I am given that this function I am given that this family script f is normal suppose I am given it is normal at z not suppose it is suppose I am given it is normal in this closed disc centered at z not radius rho which means it is actually normal in an open set which contains that closed disc okay suppose that happens and let us see what it means for us so let me continue like this suppose now that you know script f is normal in mod z minus z not less than or equal to rho okay by that I mean it is normal in a slightly bigger open disc okay then so what does normality mean it means actually normally sequentially compact it is the correct notion of compactness for us when we do complex analysis okay so what it means that you give me any sequence of functions in script f I can always find subsequence which converges normally okay it is normal sequential compactness it is sequential compactness with respect to normal convergence so given any sequence f n in script f we can find a subsequence let me call that as f nk that converges normally in mod z minus z not less than rho okay I can put less than or equal to in fact mod z minus z not less than or equal to rho is actually compact and therefore it will converge even uniformly there okay so it converges normally so what happens is that you will get this subsequence f nk of z it will go to some function f z okay and f is a normal limit of metamorphic functions so you know what is going to happen you already seen this a normal limit of metamorphic functions is either identically infinity or it is also a metamorphic function it could even be analytic okay so then of course then as we have seen either f is identically infinity or f is metamorphic this is something that we have seen alright but the point I want to do I want to emphasize is that is the following issue you see you have these functions f nk they are converging to f alright and what is happening is that now you know mind you I have the zoomed functions okay I have the zoomed functions by zooming factor epsilon less than rho so look at the zoomed functions so what happens is that I take g nk which is actually the zooming of so g nk of zeta is a zooming of f nk with respect centered at z not with the magnification factor 1 by epsilon and I am using the variable zeta okay and similarly I get the limit I get a I get zoomed function for the limit function also so I get g also is equal to zoom the zoomed function of f centered at z not magnification factor 1 by epsilon my variable is zeta okay so this is g of zeta so I have that and of course you know if f nk converges to f then the zoomed functions of the f nk namely the g nk will converge to the zoom function of f which is g and this is just because the zooming is just scaling followed by a translation okay so after all what is the zooming the zooming is you multiply zeta by epsilon and then you add z not to it okay that is just a factor in fact it is actually a bilinear transformation it is a linear fractional transformation it is a mobius transformation okay so it will preserve all properties convergence everything okay so you get this now you see so let me write this this is normal normally so here also I will get so this will imply that this will also converge normally okay and the point I want to make is that you see I have assumed that the original the original family is normal okay and we have seen namely Marty's theorem that normality is equivalent to the spherical derivatives being normally bounded okay so if you recall you know Marty's theorem says that a family of meromorphic functions on a domain the domain could even be a domain in the external complex plane it could contain the point at infinity such a family is normal if and only if the spherical derivatives of those functions the family of spherical derivatives is normally uniformly bounded so it should be on every compact subset of the domain the family of spherical derivatives should have a uniform bound okay so by Marty's theorem you know that the family script f itself has a has if you take the spherical derivatives then that has a uniform bound okay so let me write that down by Marty's theorem script f has a uniform bound so script f hash so when I put script f upper hash it means the family of spherical derivatives of the functions in f okay this has a uniform bound in mod z minus z not less than or equal to rho I have this okay because mod z minus z not less than or equal to rho is a compact set alright and on a compact set I will have a uniform bound for spherical derivatives and that is equivalent to normality okay this is something that I have this is Marty's theorem so what it means that the if you take the if you take any if you take any function f in so let me not use f let me use h because I have already used f for the limit if you take a h in f then h hash this is spherical derivative of h is just an m for all h okay and same m so you have this you have uniform bound alright and this bound applies to all members of spherical derivatives of all members of the family f script f so it up so it applies also to the f n case okay so in particular what you will have is you know you will have that f n k hash the spherical derivatives you see these things are going to be bounded by m is going to happen because all these f n k's are anyway in the family okay but you know we have also seen this earlier if a family of function monomorphic functions converges normally to a limit function then the spherical derivatives will also converge okay the taking the spherical derivative will respect convergence okay so you also have from you also have from here you also have from here if you want so I need to go like this that f n k if I take the spherical derivatives that will go to f hash okay you have this okay and of course you know if I by the same token if I do it to the g n k's the spherical derivatives of g n k's will go to the spherical derivative of g okay mind you g n k is just f n k translated and scaled so when you translate and scale a function its nature does not change so if you take a meromorphic function and translate and scale it you will again get a meromorphic function okay after all translating and scaling is not going to change the nature of singularities okay and if you translate and scale an analytic function you will again get an analytic function okay and so on and so forth so the g n k's are also meromorphic functions and mind you you have to remember that the I am considering the f n k's in mod z minus z naught less than or equal to rho but I am actually considering the g n k's in mod zeta less than rho by epsilon okay that is the that is the disk that is the zoomed disk centered at the origin where the zoomed functions are being looked at okay so I will let me write this here on mod zeta less than rho by epsilon okay this is there epsilon is less than rho right so this is a quantity greater than 1 fine so I have this and I also have the same thing for the for these guys I will also have g n k ash going to g ash alright and of course it is again normal this is also normal okay and the point is that but there is an inequality coming now see what is g n k g n k of zeta is by definition f n k of z naught plus epsilon zeta this is a zoomed function so what is g n k ash of zeta what is it what is it it is see it is by definition it is supposed to be 2 times the modulus of the derivative of g n k with respect to zeta divided by 1 plus modulus of g n k of zeta the whole square this is the definition of spherical derivative okay and but then this is what is the g n k dash zeta in the numerator it is the derivative of g n k with respect to zeta it is the derivative with respect to zeta okay and mind you this formula for spherical derivative also works when zeta is a pole okay we have seen that you know by continuity if the pole is of higher order than the spherical derivative at the pole is 0 if the pole is a simple pole then spherical derivative the pole is actually 2 divided by modulus of the residue at the pole is something that we have seen already okay so there is no problem if zeta is a pole okay this formula works but then you know if I differentiate g n k with respect to zeta it is the same as differentiating f n k and then you know by the chain rule differentiating the argument of the f n k with respect to zeta and that will bring me a multiple of epsilon so what I will get is I will get epsilon times f n k ash of z this is what I will get okay not z in fact I have to plug in the right substitution scaled variables z naught plus this is what I will get okay and but what is but the f n k's are all bounded by what the f n k's are all bounded by m alright so this is bounded by epsilon m okay so the model of the story is that because you because of your scaling okay the see what you must understand is that you zoomed the small disc center at z naught radius rho to a larger disc okay you scaled and you got the zoom function but what has happened is that the bound for the spherical derivative has become smaller because it is got multiplied by this the inverse of the zooming factor the zooming factor is 1 by epsilon where epsilon is very small okay then your your bound the bound for the zoom function becomes much more smaller okay now that is what it says now this is true for every n k but you know g n k hash converges normally to g hash and each of the g n k's is bounded by epsilon m therefore you know g hash will also be bounded by epsilon m so these two put together will tell you that g hash will also be bounded by epsilon m okay so this will happen alright so that is because of a normal convergence okay so the so the model of the story is that so this is what is happening so so what have we shown so far you take a point z naught a point where a family is normal okay and then you take any sequence in the family you will get a convergent subsequence normally convergent subsequence you look at those those functions in the subsequence okay and zoom them then the zoom functions their spherical derivatives become very small and in fact the zoom functions if you take the if you take the limit of the zoom functions the limit function will be a meromorphic function whose spherical derivative is very small that is what that is what it says okay and now you know so this is the first step of the argument now what I am going to do is I am going to introduce a level of a level of complication by what I am by doing the following thing what you do is instead of considering a single epsilon you consider a sequence of epsilon going to 0 imagine you are considering I am using the same epsilon here for all the functions okay but suppose for g n I use an epsilon n okay and such that the epsilon n's are going to go to 0 okay that means I am doing ultra zooming I am I am zooming into smaller and smaller and smaller and smaller neighbourhoods is that not okay if I do that then what will happen is that you can you can guess what is going to happen this g hash will be 0 because see g hash will the thing on the left side g n hash g n k hash that is going to be bounded by epsilon n k m okay and if I take limit as n k tends to 0 epsilon n k is going to go to 0 so epsilon n k m is going to go to 0 because m is anyway finite quantity therefore g hash is going to go to 0 and what does it mean if g hash goes to 0 it means that g is constant a spherical derivative of function cannot be 0 unless it is a constant it can be even the constant function that is uniformly infinity mind you for the constant function which is uniformly infinity also the spherical derivative is 0 okay in line with the fact that the philosophy that whenever you have a constant function the derivative of any type should be 0 okay. So the moral of the story is this is the moral of the story the moral of the story is that if you if you zoom in a family of you know convergent normally convergent family of functions at a point okay then the limit function is going to be constant that is the whole point okay. So let me add that level of complication but now there is something very nice what is happening is that this so let me put this in a different color here whatever is happening is happening happens in mod zeta lesser than epsilon I am sorry rho by I think it was rho by epsilon okay this was the disc where everything is happening okay but now you know if I make these epsilon go to 0 okay by choosing epsilon n which go to 0 something nice is going to happen as epsilon n goes to go to 0 rho by epsilon n go to infinity. So your discs are becoming bigger and bigger and bigger and the beautiful thing is that any compact subset of the complex plane the zeta plane will become contained in a sufficiently large disc and therefore you get a family of functions for which you can talk about normal convergence on the whole plane okay. So this G hash will be defined on the whole plane and you will get a constant function okay that is the whole point okay. So let me write this down now let us consider a sequence of epsilons say epsilon n going to 0 so epsilon going to 0 plus so they are all positive quantities and you know let me say even I mean say even decreasing okay anyway sequence going to 0 eventually is going to be decreasing sequence right strictly decreasing sequence you can think of it as smaller and smaller neighbourhoods okay. Then what happens is that we get for you know so here is a funny thing you define g n of zeta to be you zoom f so you take f you write z0 plus epsilon n zeta okay this is actually the zooming of the function f centred at z0 scaling factor is not is now 1 by epsilon n and the variable I am using is zeta okay and mind you this 1 by epsilon n is going to go to infinity because epsilon n is going to 0 plus 1 by epsilon n is going to infinity that means I am actually I am doing ultra zooming you know at the point z0 and I am looking at the zoom functions all right now see this is this is defined as I told you this is defined in mod zeta less than rho by epsilon n okay and the point is that this goes to infinity this goes to plus infinity as n tends to infinity because epsilon n goes to 0 rho is a fixed quantity all right and note that what will happen is that g n k hash will go to g hash as before okay this will be normal but the beautiful thing is this will be normally on C on the whole of C that is the beautiful point earlier this g n k hash going to g hash was on only on this on this bounded domain it was on mod zeta less than rho by epsilon it was normal on the convergence g n k hash going to g hash was it was normal convergence on this bounded domain but now since these epsilon are becoming smaller these bounded the g n s the g n s are being defined on bigger and bigger domains that is an increasing sequence of disks open disk centered at the origin that will eventually cover the whole plane so it will eventually cover any compact subset of the plane and therefore on any compact subset of the plane I can say that this sequence is going to converge normally because I will have to consider the sequence only beyond a certain stage okay see whenever you are looking at convergence the first finitely no matter how large the first finitely many terms do not matter okay. So the moral of the story is that if you buy the advantage of taking smaller and smaller epsilon is that you are zooming into the point you are zooming into the behavior of the functions at that point but then you get a limit function which is defined on the whole plane so you get a meromorphic function on the plane so this g it will be a meromorphic function on the whole plane g will be a meromorphic function on the whole plane alright and the nice thing is that as I was telling you the bound for the spherical derivatives of g n will be what it will be epsilon nk times m so you will get g nk hash is bounded by epsilon nk times m okay because you know earlier we got the bound as epsilon m for g nk okay but now the epsilon is epsilon nk in this case so I will get this okay and this goes to g hash and as k tends to infinity and well this goes to 0 as k tends to infinity. So the moral of the story is g hash is 0 and this implies that g is a constant and when I say g is a constant mind you g could be the constant function is identically infinity that is also allowed okay so you know how do you argue that g is a constant mind you g becomes g anyway the limit function g is meromorphic on the whole plane okay so it has only poles on the plane okay and outside the poles it is an analytic function okay but outside the poles the spherical derivative is 0 means ordinary derivative is 0 okay and the ordinary derivative is 0 the function is constant so except for these isolated set of points which are poles everywhere else the function is constant and those points cannot be there you cannot have even a single pole because if you have a pole as the points approach the pole the modulus of the function values goes to infinity how can it be constant in a deleted neighborhood of a pole so that force of the function has to be identically constant that is how you get g is identically constant g is only a meromorphic function okay so this is the important thing the important thing is that if you take a normal family so what have we proved you take a point where a family is normal okay and you take any sequence in the normal family you can find a normally convergent subsequence if you take those sequence of functions and do zoom in at the point you are able to construct see they will the zoom functions will converge to a constant function that is what it says okay if you look at a normally convergent family of meromorphic functions zoomed at a point you will get only a constant function that is what you are saying okay this is the behavior of a normal family at a point okay and what I want to do next is introduce another level of complication okay and this next level of complication is to vary Z0 also okay so instead of considering a point Z0 all this time I had fixed Z0 but now what you do is you take a sequence of points Zn which goes to Z0 okay and at each Zn you zoom to epsilon n for the function fn and do this process okay and you will still get the same result okay and beautiful thing is that whatever you get there okay that behavior is good enough to characterize normal families okay and the point about Zalkman's lemma is the negativity of that statement okay so roughly Zalkman's lemma is a lemma that is actually a theorem which will explain when family is not normal it will give you necessary and sufficient conditions for a family to be not normal and guess what the condition will be the condition will be that you can do all this but the limit function that you get G it will be non-constant that will be the only difference you will get a limit function which will be a non-constant meromorphic function so its spherical derivative will not be 0 okay and that is a characterization of non-normality okay that is the see this is the motivation for Zalkman's lemma okay so I will in my next lecture I will tell you how to instead of considering Z0 consider a sequence Zn which goes to Z0 and have the same argument okay and continue with Zalkman's lemma alright so I will stop here