 Okay, so we have seen in the last lecture the famous theorem of Montel on normality of families and what it says is that if suppose you have a family of meromorphic functions defined on a domain, the domain can be in the extended complex plane. To decide that the family is normal all you have to do is to ensure that all the functions in the family do not take three distinct values in the extended plane okay and you know because you are working with meromorphic functions you have to allow the value infinity because that is the value that a meromorphic function at a pole will take okay and but of course if you are looking at a family of analytic functions okay then the condition is much more simpler you have to just find two complex values which the functions in the family do not take and if that is true then the family is normal okay this is the great theorem of Montel and so you see it is the key to proving the Picard theorems which we will do right. So here is Picard's big or great theorem okay so you know what the theorem is the theorem is that if you take a function which has an isolated essential singularity then the image of any neighbourhood of that singularity is either the whole complex plane or it is the complex plane minus a single value okay that means it means it means that it can at most omit one complex value alright and what is the restatement the restatement is that if it omits more than one value if it omits two values that is something that cannot be that is not possible okay. So what we will do is we will assume that it omits two values and then use the Montel criterion that resulting family of zoom functions is normal okay and examine the limit of the zoom functions and that will give you the proof okay. So let me state it let f of z let f be let f have an isolated let f of z have an isolated at z equal to z not then the image under f any deleted neighbourhood of z not is either all of complex plane or omits at most one value in c. So this is Picard's theorem okay and of course because this is valid for every neighbourhood it means that you know it will take except for one value which it might omit all other complex values it will take infinitely many times because you the same way if you find a point where it will take that value then you can find a smaller neighbourhood you can take a smaller neighbourhood deleted neighbourhood of z not and in that neighbourhood also again it has to take that value and you can go on like this therefore it will take every value except one value infinitely many times okay that is a that is amazing behaviour of function analytic function around a isolated essential singularity okay. So this is the theorem and so what is the proof the proof is by contradiction by contradiction using the zooming process the so you know the zooming process has been we have been using it right from Zalkman's lemma okay and you know what the what is the main idea behind the zooming process the idea behind the zooming process is as you zoom into a normal point then all the zoom functions will converge normally to a constant function and if you zoom to a non-normal point then all the zoom functions will converge to a non-constant meromorphic function okay that is the basically the principle alright. So what we will do is so assume f omits two values in f omits two values in C in some deleted neighbourhood 0 less than mod z minus z not lesser than say rho okay. So you have to show that f takes either takes all values or it will take all values except one so if you want to contradict that you have to assume that f omits at least two values okay. So let us assume that alright so what you do is so here is it so here is my diagram so I have so this is the complex plane this is z plane and I have this point z not and you know I there is this small disc surrounding z not radius rho this disc with radius rho and on this disc f does not take two values two complex values that means there are two distinct complex values which f will never take it may not take many more values also but at least two values it misses okay and what I am going to do I am going to construct a sequence of zoom functions. So what is the sequence of zoom functions what you do is well take any so before that let me write it you know ideologically zoom into the function f at z not itself okay mind you the function in z not is an isolated singular point it is an essential singular point therefore in the deleted neighborhood of z not in this deleted disc this punctured disc okay centered at z not radius rho you throw out the point z not in the punctured disc it is the function is analytic mind you okay and what I am going to do is I am actually going to zoom into z not okay I am going to just zoom into z not and how do I zoom into z not by taking smaller and smaller discs of smaller and smaller radii which are centered at z not and of course I have to exclude z not because z not is a point of singularity of f okay so what you do is zoom in to z not okay and so let me say that is take a sequence epsilon n tending into 0 plus 0 less than epsilon n lesser than rho okay so you take a sequence of smaller and smaller radii okay and let g n of zeta be the zooming of f the same function f centered at z not the scaling factor is 1 by epsilon n and the variable is zeta okay so this means that you are just taking f of z not plus epsilon n times zeta this is the function okay so this is what my g n is so I am using that single function I am constructing a family of functions using the single function f I am constructing a family of functions g n okay and where are these g n is defined so you see the so this is a z plane and then if you look at it correspondingly you have the you have the so this is the complex plane which is z plane then I also have the complex plane which is zeta plane okay and in the zeta plane what happens is that you know if I take this disc centered at the origin radius if I take the radius to be rho by epsilon n okay that is for mod zeta less than rho by epsilon n mod epsilon n zeta will be less than rho and therefore this is the so this is the disc in which g n is defined but the only thing is it is not defined at the origin because at the origin the origin corresponds to z not and f at z not is not defined because we are not a singularity of f okay so let us write that down g n is defined so let me write this here g n of zeta is defined in 0 less than mod zeta less than rho by epsilon n and of course you know the point about this whole business is that since epsilon n tends to 0 plus rho by epsilon n tends to infinity and therefore g n zeta is going to be the you can talk about convergence or normal convergence of g n zeta on the whole punctured plane okay except that there is a whole plane except the origin because eventually any compact subset of the of the punctured plane other than which does not contain the origin is going to be is going to be contained in the domain of g n zeta for n sufficiently large okay so let me write this down note that any compact subset of C not containing the origin the origin is going to be in the domain of definition of g n zeta for n sufficiently large okay so now I now I wanted to just watch see after all you know the values of g n in this punctured disk centered at the origin radius rho by epsilon n correspond to exactly the values of f in the punctured disk centered at z not the values of f inside this whole disk punctured disk centered at z not radius rho they are exactly the values of g n in the punctured disk centered at the origin radius rho by epsilon n because this is just a scaling and a translation okay now you see now you know f but what is what have we assumed about f we have assumed that f omits two complex values okay f f is an analytic function and it omits two at least two complex values in this the disk punctured disk centered at z not radius rho therefore each of the g n's will also omit those two values in their domains okay and of course each of the g n's are also analytic functions because they differ from f only by bilinear transformation consisting of a scaling and a translation okay so but now you know we are in good shape because what you have done is we have we have we have been able to get a family of we have been able to get a family of functions g n which are analytic okay and which omit two values now immediately Montel's great theorem will tell you that they had it has to be normal and you will get a convergent subsequence and then you have to examine the limit okay and the limit will give you a contradiction alright. So basically the contradiction will be that the limit function at the origin will tell is examining the limit function at the origin will tell you that the origin has to be either a pole or a removable singularity for f which is not true I mean analysis of the limit function will tell you that the you take this limit function this limit function will also be defined on the punctured plane okay because all the original g n's are all defined outside 0 okay so if you analyze the limit function see the limit function is like zooming into f at z0 infinitely many times okay so the behavior of the limit function at the origin which is an isolated singularity will reflect upon the behavior of f at z0 and by analysis we will show that if you analyze the limit function there are only two possibilities z0 should either be a removable singularity or it has to be a pole and both of these are contradictions because I assume z0 for f to be a essential singularity okay and that is how the proof goes so it becomes as simple as that so let me write this down note that g n is normal family as it also does not assume the values that f omits okay so what does it mean if it is a normal family it means that you know and it is a normal family and mind you for g n because these are zoom functions whose domains are becoming bigger and bigger and bigger okay you can think of them as a normal family you know in with a limit function in the punctured plane alright so what you must understand is that if you take for example if you take the punctured unit disk okay if you take the punctured unit disk then for n sufficiently large all the g n's are going to be defined their domains are going to become bigger and bigger and they are going to contain the punctured unit disk so you can say that the if you want to take the if you want to talk about the domain of the g n's okay you can assume that for n sufficiently large the domain contains unit disk if you want or for that matter any finite disk okay with of course origin omitted okay and when you take the limit function that is because g n is a normal is normal if you take the limit function will be defined on the whole punctured plane because it will make sense because you are covering every point in the in the plane because for every point in the plane if you take epsilon n sufficiently small rho by epsilon n becomes sufficiently large and g n's beyond a certain stage will be defined at that point and therefore the limit of all the if you take a convergent subsequence of g n's as the limit will also be defined at that point okay. So let me write this thus g n k thus there exists g n k a subsequence that converges normally to g of zeta if you want on c minus the origin okay so this happens because what is the meaning of normal family normal means that normally sequentially compact that means give me any sequence that is an normally convergent subsequence. So if I when g n itself is a normal family it is already a sequence so it will have a normally convergent subsequence so you have a subsequence g n k which converges normally to g on c minus 0 okay but the point is that it is not it is not what is important is that it is g lives in a neighbourhood of 0 okay and now try to understand each g n is analytic on the punctured disk on a punctured disk centered at the origin okay therefore this limit function g is a normal limit of analytic functions we have already seen such a normal limit can have only two possibilities either the normal limit can completely be analytic or it can be identically infinity these are the only two possibilities. So let us write that down thus g is identically infinity or g is analytic in c minus 0 okay. Now let us look at both of these cases suppose g is identically infinity okay suppose g is identically infinity so what does it mean what this will mean see think of it you know heuristically g is f zoomed you know infinitely at z not and if g is identically infinity what you are actually saying is that f is infinity in a neighbourhood of f is going to infinity in a neighbourhood of z not right because the values of g are just limits of values of g n's and the values of g n's are just values of f in smaller and smaller neighbourhoods. So if g is identically infinity okay that means that the g n's are getting larger and larger in modulus okay and that means that the values of f are getting closer and closer to infinity as you approach z not but that means z not is a pole but that is not possible because z not is an essential similarity so this is not possible so you have ruled out this case okay. So let me write this down this means that g of zeta is infinity for all zeta in c minus infinity I will have to make use of the fact that you know g n converges to g normally okay g is not just the it is not simply limit of g n's of course it is point wise limit of g n's but it is not it is more than that it is not just the point wise limit it is a normal limit so the convergence is uniform on compact subsets okay. So you know g n of zeta converges to g of zeta uniformly on mod zeta is equal to say r for any r greater than 0 because you see mod zeta equal to r is a circle in the zeta plane centered at the origin radius r and that is certainly a compact set it is closed and bound and g n g is a normal limit therefore the convergence g n to g should be uniform on any compact subset so it has to be uniform on mod zeta equal to r okay but of course what is but what is g zeta g zeta is infinity if g is identically infinity so what this and and what does uniform convergence mean it means that you know all the g n's they will come to within an epsilon of g zeta okay if you take n sufficiently large irrespective of zeta right that is what it means okay but of course you want to come to an epsilon of infinity so you will have to be careful and you will have to use a spherical metric okay so let me write this down thus given epsilon greater than 0 the spherical distance between g n of zeta and g of zeta which is actually infinity can be made less than epsilon for n sufficiently large and for all zeta with mod zeta equal to r okay but now but what is g n zeta you see but g n zeta our definition is just f of z0 plus epsilon n zeta this is what it is okay and as you know and you see even if mod zeta is r okay if you are your epsilon n's are becoming smaller so you are covering smaller and smaller disc you are covering smaller and smaller circles centered at z0 okay and therefore what you are saying is that the function values of f on smaller and smaller circle centered at z0 are getting close to infinity okay and that is enough to tell you that f the limit of f as z tends to z0 is actually infinity which means that z0 has to be a pole but that is a contradiction to our assumption that z0 is actually an isolated essential singularity okay so let me write this down this means that this spherical distance between f of z and infinity can be made less than epsilon uniformly on compact subsets of mod z minus z0 less than epsilon n for n sufficiently large okay. So you know because what you must understand is that this r is this capital r is at our disposal you can make this capital r as small as you want close to 0 you can make it as large as you want okay so you can literally cover all the circles centered at z0 of a fixed radius below a certain positive value so in some sense you are therefore you are able to cover a complete deleted neighborhood of z0 okay that is the whole point alright. So well so this implies z0 is a pole of f which is a contradiction because we assumed z0 to be an essential singularity okay so thus you know f the limit the zoomed limit function g cannot be identically infinity okay therefore what is the other possibility it has to only be an analytic function in the punctured plane punctured at the origin okay thus g is analytic in the punctured plane okay and what does that mean it means that it means of course origin is a singularity origin is an isolated singularity for g okay and now you can ask what kind of singularity it is but you know the point is that again you should not try to study the singularity of g at the origin you should not do that because after all g at the origin is going to reflect f at z0 okay so because mind you the g n's are all zoomings of f at z0 and their limit is g okay so g is some kind of infinite zooming of f at z0 g at the origin is infinite zooming of f at z0 okay therefore you should not study g at origin but you must make use of the fact that g is analytic in the outside the origin so if you again take this mod zeta equal to r if you take a circle centered at the origin the zeta plane radius r mind you again that is a compact set and g being analytic is continuous there on a compact set it will be bounded okay now this bound will apply to f in a neighbourhood of z0 okay and we will tell you that f is bounded in a neighbourhood of z0 but then Rayman's removable singularities will tell you that z0 has to be a removable singularity and again there is a contradiction and that is the proof of the theorem okay the proof of the theorem becomes so simple okay so let us go to the other case again there exists an m greater than 0 such that again on mod zeta equal to r greater than 0 there exists an m greater than 0 such that mod g is less than or equal to m okay and but after all since g n converges to g normally in fact uniformly on mod zeta equal to r because again because it is a compact set we have fn f is bounded in a deleted neighbourhood of z0 this again implies by the Rayman's removable singularity theorem that you know z0 is a removable singularity of f again a contradiction and you know that finishes the proof there are only two choices for g and both choices lead to contradictions okay so that is the proof of the famous big Picard theorem and we can as a corollary we can deduce the little Picard theorem what is the little Picard theorem it tells you that the image of the complex plane under an entire function is again the whole plane or the plane minus a point and now what is the proof the proof is very simple take an entire function okay of course you should take a non constant entire function okay because we take a constant entire function the image of a constant function is always only one point so you must be careful I must have been careful in saying that statement if you take a non constant entire function then you know the image of the complex plane should be either the whole complex plane or complex plane minus a point you can omit only one value at most what is the proof very simple take the take an entire function okay and look at the point at infinity okay the point infinity becomes a isolated singularity because the complex plane is a deleted neighborhood of infinity in the extended complex plane okay so your function f your entire function f has infinity as an isolated singularity okay now for an isolated singularity what are the possibilities it can be removable it can be a pole or it can be essential if it is removable it means that if infinity is a removable singularity it means that f is bounded at infinity and that means that by Leuel's theorem f has to be a constant so if I take f to be a non constant entire function okay infinity cannot be a removable singularity alright so it can only be a pole or an essential singularity if infinity is a pole then we have already seen that f has to be a polynomial okay and you know a polynomial will take all values because of the fundamental theorem of algebra so if f has infinity as a pole it is a polynomial the image of the complex plane and f is the whole complex plane you are using the fundamental theorem of algebra so the only thing is infinity is an essential singularity if infinity is an essential singularity for f apply the great Picard theorem okay f can f has to any neighborhood of infinity has to be mapped by f into the whole complex plane or the complex plane minus a point but the complex plane itself is a deleted neighborhood of infinity so f has to map the whole complex plane onto the whole complex plane or the complex plane minus a point that is it okay so I will just write this down so corollary Picard's little theorem if f is a non constant entire function then f of c is equal to c or c minus z0 okay for some z0 in c so proof is infinity is an isolated point singular point for f in c union infinity as c is a deleted neighborhood of infinity where f is analytic okay thus infinity is either removable pole or essential if infinity is removable f is bounded at infinity so by Liouville f is constant a contradiction because I am assuming f is a non constant function okay non constant a non constant entire function if infinity is a pole we have seen earlier that f has to be a polynomial be a non constant polynomial which assumes all complex values by the fundamental theorem of algebra so the only other case is when infinity is a pole I mean if infinity is an essential singularity if infinity is an essential singularity apply the big Picard theorem by the big Picard theorem or the great Picard theorem f of c is c or c minus a point so that finishes the proof of the little Picard theorem and therefore you see you are able to prove the Picard theorems very easily okay and the key to all this is this really great theorem of Montel which says which is a criterion for normality of a family okay and it is a very very simple criterion in the sense that you know if you know a family functions if it is a family of meromorphic functions if you know that it omits 3 values okay in the extended plane then you know it is normal if it is a family of analytic functions and it omits 2 complex values then you know again it is normal okay and the advantage of normality is that it is a kind of compactness namely it is normal sequential compactness which allows you to extract from any sequence a subsequence which converges normally that is which converges uniformly and compact subsets okay so that finishes the proof of the Picard theorems which was which were the main aim of this course okay. What I would like to next do is to tell you how powerful Zalkman's lemma is in several other contexts okay mind you that this reasonably simplified proof of the great Picard theorem was possible because of the Montel's theorem on normality okay and that in turn was proved by using Zalkman's lemma okay so these are all actually all the simplifications are because of Zalkman's lemma that is the most important thing okay so but the Zalkman's so similarly in the proofs of various other theorems and various other theories about complex functions Zalkman's lemma provides us with easier proofs of some very deep results okay and also provides us with new results and I will try to outline those results in the coming lectures okay so I will stop here.