 Alright so what we need to do now is worry about giving the statements of and of course the proofs of you know Montel's theorem and Marty's theorem in the case when the domain includes the point at infinity okay. So if you recall see all this time we have been doing we have been worrying about domains in the complex plane but since the last lecture we also wanted to include the infinity as a point in the domain that means you are looking at a domain in the external complex plane alright. See the problem with including infinity is that you know compact neighborhood of infinity makes sense only in the extended plane okay and because it corresponds to compact neighborhood of the north pole on the Riemann sphere the extended plane being identified with the Riemann sphere okay. So but if you take a compact neighborhood of infinity and delete infinity okay then what you will get is you will get an unbounded domain in the usual complex plane. So basically it is an unbounded domain and along with of course the boundary so on an unbounded domain we generally do not expect uniform convergence okay. We expect uniform convergence only on bounded domains especially on compact subsets okay and of course you know compact implies closed and bounded so since you are in a nuclear space it is the same as closed and bounded. So you do not expect uniform convergence on an unbounded domain okay that is the rule you will get it only on compact subsets. So if you take a compact neighborhood of infinity okay then you will see that it is too much in general it is too much to expect uniform convergence on because it is a it is unbounded as far as the if you look at it from the complex plane point of view any compact neighborhood of infinity for that matter any neighborhood of infinity will be an unbounded set in the usual complex plane okay it will be bounded only with respect to the external complex plane okay. So that is the reason why we defined what is meant by normal convergence what is what is meant by normal uniform convergence of a sequence or normal convergence of a sequence of functions defined on a domain which contains the point at infinity okay. So the way we did it was if you remember if D is a domain in the external complex plane of course if the and of course if infinity is a point of the domain then what you do is that you consider two things you first of all remove infinity and you get D minus infinity that becomes a domain in the usual plane and for such a domain you know what uniform convergence on compact subsets that is normal convergence means so you make that definition and then to deal with the to deal with you know the point at infinity to deal with normal convergence at infinity what you do is that you invert the variable. So what you do is you take a neighborhood of infinity you take a neighborhood of Z equal to infinity and treat it as a neighborhood of W equal to 0 by putting Z equal to 1 by W okay and then you say now this neighborhood of 0 is anyway it is a it is anyway a neighborhood in the complex plane so it makes sense to talk about normal convergence okay. So you define sequence of functions to converge normally on a domain in the external complex plane containing the point at infinity if it individually it converges on D minus infinity and the sequence with the arc the variable change from Z to 1 by W converges again converges normally in a neighborhood of W equal to 0 which corresponds to a neighborhood of Z equal to infinity okay. And then you know with this modification we found that you know the limit function to which the if the original functions are already continuous for example which is the case when you have analytic functions or meromorphic functions then the limit function is also continuous and the limit function is of course unique and continuous because it is a two piece definition there is a definition for the domain minus the point at infinity and there is another definition for a neighborhood of infinity okay. So in principle you could have got two different functions but because of continuity you will get a unique function okay and therefore what we did was we were able to extend these important results namely that you know if you have a sequence of analytic functions on a domain in the external complex plane if that sequence converges normally on the domain then the limit function is either analytic or it is identically equal to infinity and we also proved the same thing for meromorphic functions if you have a sequence of meromorphic functions on a domain in the extended plane and if that sequence converges normally then the limit function is either meromorphic or it is identically equal to infinity. Now what we need to do is we need to worry about Montel's theorem and Marty's theorem which is a meromorphic version of Montel's theorem to you know in the case when the domain of definition of the functions or family of functions is domain in the extended complex plane okay. So for that we will have to define what is meant by normally sequentially compact for a family of functions defined on a domain in the extended plane and that involves a little bit of subtlety but anyway as we will see everything works out well so let me write this down. So suppose D in the extended complex plane is a domain of course non-empty it is an open connected set and let us assume that infinity is a point of T okay so this means that if you remove infinity from D what you will get on the complex plane is an unbounded domain okay and what we want to do is suppose you have a family of functions defined on this domain I would like to say what it means for the family to be normally sequentially compact okay because after all Montel's theorem the usual Montel's theorem and Marty's theorem are just you know the correct generalizations of the Arzela-Ascoli theorem okay which say that you know so Montel's theorem says that for analytic functions you know normally sequentially compact okay that is sequential compactness with respect to normal convergence that is the same as uniform boundedness on compact subsets that is normal uniform boundedness for the family okay and Marty's theorem extends this from analytic functions to meromorphic functions okay. So one can always be very naive and say that well my definition of a family f being sequentially compact with respect to normal convergence is just that it should be you know sequentially compact with respect to normal convergence when infinity is removed okay so on the domain punctured at infinity and to deal with the point at infinity you say that the same family you take the same family and change the variable to from z to 1 by w and you say for this new family defined in a neighbourhood of 0 again you should have you know sequentially compact with respect to normal convergence and that is the this is the definition that you will make in line with what we have been doing and if you will see that it is the correct definition to make okay. So let me write this down we say that a family script f of functions on D with values in c union infinity that is the external complex plane is normally sequentially compact i.e. sequentially compact with respect to normal convergence if number 1 so f is normally sequentially compact the domain D punctured at infinity and now I will have to take care of the point at infinity and for that what I will do is that f let me put this is f sub 1 by w this is the set of all f of 1 by w where f belongs to script f so I put the subscript 1 by w to tell you that I have changed the I have inverted the argument the independent variable in the function okay is normally sequentially compact on a neighbourhood of w equal to 0 because you know a neighbourhood of w equal to 0 will correspond to a neighbourhood of z equal to infinity because w is 1 by z all right. So this is the so this is so I should put this heading as normal sequential compactness domain in the external complex plane so that is the heading right. So well this seems to be the right definition to make whenever you want to deal with the domain which contains a point at infinity you do deal with it in 2 pieces one is you throw the point at infinity okay you throw it away namely you puncture the domain at infinity so you get a deleted neighbourhood of infinity and you give a definition for that and that is easy to give because a deleted neighbourhood of infinity is also a domain in the usual plane okay and the other thing that you do is that to consider a neighbourhood of infinity you consider the neighbourhood of 0 by inverting the variable okay so but now so there are a couple of things that I want to tell you with respect to the notation terminology and literature and also there is another subtlety that I want to point to you about. So the first thing is that you know in the literature this normally this normal sequential compactness is actually abbreviated to normal okay so this is a very very important thing see people just use the word normal this is in fact Montel's terminology that instead of every time saying normally sequentially compact you use the word normal and this is considered as this is being thought of as a property of the family so when you say a family is normal it means it is not normally sequentially compact okay so wherever normally sequentially compact comes you know then you can just replace it with the word normal okay and the whole point is that a family being normal is the correct notion of compactness of a family okay that is the whole point okay. So you know if you are working with general topology and if you are working with continuous functions say real valued or complex valued functions and you are working on a you know compact metric space or for that matter you are even working with continuous functions with taking values in from one compact I mean it takes values in another compact metric space okay then usually compactness because you are in the context of metric spaces compactness is the same as sequential compactness okay so and there what happens is that you when you say sequential compactness the idea is that you are able to say that every sequence admits a convergence of sequence okay and this convergence is with respect to what it is with respect to for example if you are working with real or complex valued functions it is with respect to the supremum norm okay so there is a supremum norm which induces metric and its convergence with respect to that metric okay but of course when we are considering the external complex plane we are using the spherical metric that is one thing that you must always remember okay the external complex plane is a compact metric space and the metric you are using on that is a spherical metric okay which is actually a spherical metric on the Rayman sphere transported to the external complex plane okay by the identification of the Rayman sphere with the external complex plane using the stereographic projection okay. Now the point is that this is the this is what you will get if you are looking at you know continuous functions but if you are looking at analytic functions the rule is that you cannot expect a uniform convergence on unbounded sets you can expect uniform convergence only on compact subsets and this is called normal convergence okay so when you are in the context of analytic functions or in the context of meromorphic functions okay then you have to worry only about uniform convergence on compact subsets and that is called normal convergence okay and you have to do everything normally okay and the point is therefore in the context of analytic functions or meromorphic functions the correct notion of compactness is not sequential compactness but it is sequential compactness restricted to compact subsets and that is called normal sequential compactness and Montel's terminology is that you do not say normally sequentially compact you simply say normal okay so this is one you know this is something about terminology that you must know. Then the other thing is of course about the subtlety in talking about normal sequential compactness the subtlety is you see when you say normally sequentially compact it means that it is sequentially compact when restricted to compact subsets and what does sequentially compact I mean if you blindly read for any property P normal P means the property P is supposed to hold when restricted to compact subsets. So if you go by that normally sequentially compact will mean that you know it will mean that it is sequentially compact when restricted to compact subsets. So what does that mean it means that you give me a sequence okay then if you restricted to compact subsets I will get a convergence of sequence okay but if I restrict to different compacts if I start with a given sequence okay of functions and if I restrict to different compact subsets I may get different convergence of sequences okay but the fact is there is it is more than that you can get uniformly a subsequence which will work on which will converge on every compact subset and this is actually you know another diagonalization argument that we used if you check. So actually you know this normally sequentially compact is as strong as sequentially compact with respect to normal convergence okay. Sequentially when you say sequentially compact with respect to normal convergence what you mean is that given a sequence I can find a subsequence which is whose convergence is normal normally convergent subsequence which means that same subsequence will converge on when you restrict to any compact subset but when you say normally sequentially compact you might interpret it as you know give me a sequence for every compact subset I will get a convergent subsequence but it looks as if you change the compact subset the convergent subsequence could change but the truth is that there is not much difference because you can use always a diagonalization argument okay and you can use a sequence of compact subsets that fill out an increasing sequence of compact subsets that fill out your domain okay that is the argument that we used. So there is really no confusion in say normally sequentially compact and sequentially compact with respect to normal convergence there is really no difference okay that is because of this diagonalization argument that you can apply on a sequence of compact increasing sequence of compact subsets that can cover your domain okay. So that is one subtlety then the then here comes the other technical issue the technical issue is that you know again because you are dealing with the point at infinity we have defined normal normality in two pieces we have defined normality outside infinity that is a first requirement and the second requirement is normality at infinity okay. Now think of it for a moment what does it mean it means suppose I start with sequence in my family the normality outside infinity will give me a subsequence which will converge on compact subsets outside infinity okay and what will happen is separately for the same sequence of functions I will get another subsequence which will converge normally at infinity okay. Now I seem to be getting two different subsequences okay and I do not seem to be getting a single subsequence which will converge both outside normally outside infinity and also at infinity I do not seem to be getting that and the fact is that you can do this okay it is not much of a discrepancy because you see suppose you have a family script F which is normal in this sense okay what you do is start with the sequence in the family okay first go to a neighbourhood of infinity okay go to this go to the second go to the second condition namely you go to a neighbourhood of infinity which is thought of as a neighbourhood of 0 with the variable inverted there you first pick a subsequence which is you know which converges normally at infinity okay then what you do this subsequence is also anyway a subsequence of the original family which is normal outside infinity okay so take the same subsequence and now apply it to outs the domain outside infinity and you get a further subsequence which will converge normally outside infinity so that is this new subsequence that you picked up that will be one which will converge both outside infinity and at infinity okay so even though you are doing it piecewise everything is everything works up fine okay so you do get a global given give me a give me a sequence you do get a global subsequence okay which will converge normally both outside infinity and at infinity okay there is no confusion alright only thing is you have to do this twice alright so let me write that down that is there is no ambiguity in the conclusion of normality for f for if f1 f2 and so on is a sequence in f which is assumed normal on D then we first pick a subsequence fn1 fn2 and so on that converges normally in a neighborhood of infinity which means that fn1 of 1 by w fn2 of 1 by w and so on converges normally in a neighborhood of w equal to 0 this is what it means okay and then we pick a further subsequence of this subsequence so fn i1 fn i2 and so on which also converges normally on D minus infinity then this subsequence fn i1 fn i2 so let me use is a subsequence that converges normally on D okay so there is no problem alright you are able to get one subsequence that will work both outside infinity and at infinity okay so this is a little fact that you need to know so you know we see this is just part of a usual philosophy in mathematics so there are two things I want to say usually what happens is good properties of functions hold on open sets the other thing is to verify that a good property is true you verify it only locally that means you can verify it at each point in a neighborhood of each point or on an open cover okay for example this is the case with continuity okay or analyticity and so on okay good properties if you want to check a function is analytic you check at each point or in a neighborhood of each point if you want to check a function is continuous it is enough you check at each point okay so in the same way you see this is normal the idea of a normal family is also local okay if you say that a family is you know normal in pieces okay that is it is normal on an open cover alright then it continues to be normal so for example what we are saying is that if a family is normal outside infinity and if a family is normal in neighborhood of infinity okay this outside infinity and neighborhood of infinity together constitute a cover okay and for the whole domain and when you say the function is normal outside infinity and at infinity okay then you are getting it is normal on the whole domain okay so you see normality is a local property that is what is happening alright and it is a good property and all good properties are usually local properties you can verify them locally and they are valid on open sets okay so this is something that you should remember as a philosophy now so this should have been normally alright okay so now what I am going to do is now let us go on with now you know with this background you know it is very easy to write out the analog of Montel's theorem and Marti's theorem for domains in the external complex plane okay so here is Montel's theorem so here is Montel's theorem so let me put it as for domains in the external complex plane what is the theorem? The theorem is let script f be a family of holomorphic or analytic functions on a domain D in the external complex plane then script f is normal if and only if script f is normally uniformly bounded okay so this is the this is Montel's theorem okay where you say that so where you say that normality of the family is the same as non-normal uniform boundedness of the functions in the family alright and mind you normality in the family means that it is normally sequentially compact that is it sequentially compact with respect to normal convergence alright every sequence admits a subsequence which converges normally that is a subsequence which converges uniformly and compact subsets alright and normally uniformly bounded is uniformly bounded on compact subsets alright and this normally uniformly bounded on a domain in the extended plane again how do you define that you define it piecewise so what you say is that if you want to say a family of functions is normally uniformly bounded on a domain in the extended plane what you do is that you first say that it is normally uniformly bounded when you throw out infinity so it should be normally uniformly bounded on D minus infinity and then you say that the corresponding family with the variables inverted okay is normally uniformly bounded in a neighbourhood of 0 for the inverted variable which corresponds to the neighbourhood of infinity of for the original variable okay so this normally uniformly bounded also needs to be defined when you consider the point domain which contains the point at infinity but again you define it in two pieces you define you make one definition outside in infinity and then at infinity the definition you make is by inverting the variable Z and so that you change Z equal to infinity a neighbourhood of Z equal to infinity is a neighbourhood of infinity you change that to a neighbourhood of W equal to 0 where W equal to 1 by Z or Z equal to 1 by W okay. So this is Montel's theorem and you can see that you know the proof of this theorem just comes from the usual Montel theorem because finally what you have done is you have dealt by to deal with the point at infinity you are actually going back to 0 by inverting the variable alright so the moral of the story is that by this device you are able to deal with the point at infinity so the proof of this Montel theorem will follow from the usual Montel theorem okay and of course you have to worry about one thing namely that when you come from normal uniform boundedness to normality okay what you will get is that you will get separately normality outside infinity and you will get separately normality at infinity but I just now we just now discussed that normality is a local property so you will get normality on the whole domain okay so everything works out fine alright. The only thing that I want you to remember is that the at the back of all this how does this differ from the usual Arzela-Ascoli theorem the usual Arzela-Ascoli theorem says that you know if you want compactness then it is the same as sequential compactness and that is equivalent to you know uniform boundedness and equicontinuity okay so you need equicontinuity alright and of course usual Arzela-Ascoli theorem is for functions defined on compact metric space but then here you are working with functions on a domain certainly not a closed set okay it is an open connected set and therefore what you will have to do is that you will have to go from usual convergence you have to go to normal convergence so you have to you must instead of expecting uniform convergence on the whole domain you should expect only uniform convergence on compact subsets of the domain and then the big deal is as usual you know the you do not have to worry about equicontinuity because equicontinuity is automatic in the case of analytic functions you know the if the functions are original functions are bounded then the derivatives also become bounded and that is because of the Cauchy integral formula and the Cauchy estimates for the first derivative which can be expressed as an integral of the original function okay therefore if you have uniform normal uniform boundedness of the functions then you have normal uniform boundedness of the derivatives and normal uniform boundedness of the derivatives always implies equicontinuity okay so you get equitinity for free okay and therefore the only important condition is the uniform boundedness normal uniform boundedness of the family okay and that is the whole point about Montel's theorem and then the extension of this theorem to Merom-Offic functions is of course Marty's theorem so that also works okay but the point is that you know when you go to Marty's theorem you will have to worry about not using the usual derivatives because at what you are working with Merom-Offic functions and at a pole they are not differentiable okay and then you know the trick is to not use the usual derivatives but use the spherical derivatives okay and then you get the analog of Montel's theorem for Merom-Offic functions okay and that is Marty's theorem so let me write this down so here let me just mention that the proof follows from the usual Montel's theorem for domains in the complex plane and our remarks above okay and then now let me go on to Marty's theorem in Montel's theorem you know you can allow the functions to take only complex values because the analytic functions okay there is no question about taking the value infinity alright so whereas if you go to Marty's theorem you will have you are considering Merom-Offic functions and to make such a function continuous at a pole you define its value at the pole to be infinity so you will have to necessarily look at functions with you know values in the external complex plane okay so here is a statement similar to the statement of the Montel's theorem so in Montel's theorem we said f is a family of holomorphic functions on a domain D in the external complex plane now you will say f is a family of Merom-Offic functions on a domain D in the external complex plane okay but taking values in the external complex plane you will have to say that okay and then you also say that then of course the statement is that the family is normal if and only if the family of spherical derivatives is normally uniformly bounded that is the theorem okay so let me write that out. Let script f be a family of Merom-Offic functions a domain D in the external complex plane taking values in the external complex plane then this family is normal if and only if the corresponding family of spherical derivatives namely you take the spherical derivatives of the functions the original family is normally uniformly bounded okay so this is Marty's theorem. Again let me pinpoint a couple of things the first thing is that the reason why I am repeating this several times is I want you to note that there is a I want you to note these important differences see when you are looking at Meromorphic functions necessarily you will have to use values in the extended plane for two reasons the first reason is that you want the Meromorphic function to be continuous at a pole so you define the value at the pole to be infinity okay that is one thing. The second thing is that in the target space you want to have a metric okay so you know I want to a function if you are looking at a Meromorphic function at a pole it can take the value infinity okay so I will have to compare infinity the value infinity with finite values with finite complex values and the only way I can do that is by using a spherical metric which is available on the extended plane so it is very important that you have to take values in the extended plane and you have to use a spherical metric okay this has to be done if you are working with Meromorphic functions okay and then the other important thing is that you need to worry about spherical not the usual derivatives but the spherical derivatives usual derivatives of course will not work because usual derivatives will not they will not even be defined at a pole okay so you will have to work with the spherical derivative which is defined even at a pole and I told you there is no problem with the spherical derivative it is always continuous and even at a pole it is defined by continuity the pole is a simple pole then the spherical derivative is 2 divided by the modulus of the residue at that simple pole for that function and if the pole is a pole of higher order than the spherical derivative is 0 at that pole so and this is done in a very continuous fashion alright so the point is a spherical derivative is a continuous function continuous non-negative real valued function and that is and these are the functions that we are we have to ensure the uniform boundedness of this of the spherical derivatives okay when restricted to compact subsets okay now that is normal uniform boundedness of spherical derivatives is what we want and that is equal in condition to the original family being normal namely that the original family you know is normally sequentially compact okay so this is this is marty's theorem right. Now so you know this brings us to a very important point namely at this point you are able to get all the theorems that you want okay by you know including functions which can take the value infinity okay namely meromorphic functions and you can also have functions defined at infinity okay so what you must understand is that all these lectures all the all the tools that we built where to tackle two things first of all you wanted to tackle the function in a neighbourhood of infinity okay that means you wanted to study function in the neighbourhood of the point which is infinity for example you that is why we initially were worried about trying to define when infinity is an isolated singularity and if it is an isolated singularity is it what kind of singularity it is okay and the trick in all these cases was to replace z equal to neighbourhood of z equal to infinity by the neighbourhood of w equal to 0 where w equal to 1 by z okay and you know you must understand that this z going to w that is kind of it is a homeomorphism okay of the Riemann sphere onto itself if you want it is a home it is a it is a self-homomorphism of the extended plane onto itself which you can think of it also as a isomorphism of self-homomorphism Riemann sphere onto itself and what happens is that the infinity is you know is identified into the north pole origin 0 is identified with the south pole okay and it is just this the inversion is just switching the north and south poles of the Riemann sphere okay and this inversion is a very nice thing it does not change the spherical distance because essentially inversion corresponds to a rotation of the Riemann sphere by 180 degrees with respect to the real axis the x axis okay so the point is that you can deal with the point at infinity okay that is one aspect then the other aspect is you can deal with these functions which have poles namely meromorphic functions by allowing the value infinity okay so on the whole we have built up all these tools to deal with meromorphic functions even so you can look at a family of meromorphic functions even defined in a neighbourhood of infinity and work with it okay that is the generality to which we have defined things and now we will use all this then in the next few lectures to prove the Picard theorems okay so I will stop here