 Okay, so this is the continuation of the previous lecture and we were just discussing Hurwitz's theorem which we need. So let me again just recall what we are trying to show is the following if you have a sequence of analytic functions on a domain in the complex plane and suppose you assume that the sequence converges normally to a limit function and you allow this exception that the limit function can take values the value infinity then what can happen is either the limit function is completely an analytic function or it is completely the constant function infinity namely the function that takes every point infinity, okay. So in other words and of course here when I say I am including functions that take the value infinity it means that I am not just taking complex valued functions I am taking functions values in the extended complex plane which is the complex plane along with the point at infinity, okay denoted by the symbol infinity and of course the convergence whenever the point at infinity is concerned the convergence should be taken with respect to the spherical metric, okay which is a spherical metric on the Riemann sphere transported via the stereographic projection to the extended plane, okay and you must also remember that the convergence in the spherical metric is the same as the convergence in the Euclidean metric on the complex plane so far as domains of the complex plane are concerned, okay, alright. So in the proof of that result we needed Hurwitz's theorem you see I wanted to understand the beauty of that result is the following see when you do a first course in complex analysis and you are not worried about the value infinity, okay you are only worried about complex functions then you know that if you take a uniform limit of analytic functions you will get an analytic function and more generally you need not take uniform limit but you can take a locally uniform limit which is for example ensued by taking a normal limit a normal limit is a locally uniform limit, okay. So even if you take a locally uniform limit or a normal limit of analytic functions you will get an analytic function this is what you study in a first course in complex analysis the proof involves just Cauchy's theorem and Moreira's theorem, okay but now what we have to do is you see we have we see our aim is to prove the great Picard theorem the little Picard theorem, so Picard theorems and the problem is that to prove that you will have to worry about infinity as an isolated singularity, okay and the problem is that you will have to worry about meromorphic functions and families of meromorphic functions you should do topology on the space of meromorphic functions and the fact is that if you allow the value infinity a meromorphic function becomes a continuous map into the extended plane. So if you take an analytic function and suppose it has a pole at a point, okay then as you approach the pole the modelers of the function goes to infinity, so the function goes to infinity, so the function of course becomes discontinuous in the usual sense but then if you allow the function to take the value infinity you think of infinity as actually a value and where it is the extra point you have added to get the one point compactification of the complex plane it is that it is a point at infinity in the extended complex plane mind you the extended complex plane is a nice topological space it is a compact matrix it is a compact matrix space it is a complete matrix space, okay it is because it is just equivalent topologically isomorphic to the Riemann sphere which has all these properties, okay and now the point and mind you the point infinity can be seen on the Riemann sphere as the north pole, okay so it is a you can think of it as a valid point, okay and now if you take a meromorphic function and think of it as taking value is not just in complex plane but also allow it to take the value infinity then the meromorphic function becomes a continuous function because what you will do is at a pole you will define its value to be infinity and this definition is continuous, okay it is continuous for the topology on the extended complex plane or for example if you want to use the topology and that is of course the topology induced by the spherical metric on the extended plane, okay so now you see by allowing infinity to be a value, okay and taking continuous functions with which can take the value infinity you are also allowing meromorphic functions. Now you see what you have done is you have jumped from holomorphic functions on a domain to all the way to meromorphic functions on the domain which means you are allowing even for functions with poles, okay and then you go one more step further and say that you also allow the constant function infinity may be the function that assigns every variable to the value infinity, okay it is a constant function infinity and mind you constant functions are continuous in any sense of the term, okay so now since you all this has happened now if you again take a normal limit of analytic functions a normal limit of holomorphic functions then you know of course if everything is happening in complex in the complex plane you are only worried at complex values and if this is really a usual convergence, okay then you will get the limit function to be analytic there is nothing more but the point is now you are allowing the extended complex plane you are allowing the value infinity then why could it not happen that a sequence of analytic functions converges in the limit to a meromorphic function so suddenly a pole some poles can pop up in the limit that can happen right you would not expect it to happen by continuity, okay but we have already seen that you can get a function which is identically infinity for example I told you take the domain which is the exterior of the unit circle and you take the functions z, z square, z cube etc. So the nth function is z power n now that sequence of functions it is a sequence of analytic functions mind you in fact entire functions that sequence converges if you look at it in the usual sense it will not converge because you take any value z with mod z greater than 1 z power n will diverge because mod z power n will go to infinity because mod z is greater than 1 but if you now include infinity as a value and think of the extended complex plane three months we are in disguise okay then this z power n will tend to infinity it tends to a point a value in your set and the sequence of functions z power n that converges to the constant function infinity at every point outside the unit circle and lo behold this convergence is even normal this convergence is even uniform on compact subsets with respect to the spherical metric that is the amazing thing so you once you include the value infinity even you get normal convergence okay but the point is that it is converging to the constant function infinity now what is the guarantee that something instead of converging at all points to infinity why cannot it just converge at some points to infinity why cannot it converge on to infinity only at say an isolated set of points that means you are going to get a meromorphic function okay or why cannot it converge to infinity on some subset which is for example not even isolated why cannot such strange things happen okay so that is the theorem we are trying to prove what the theorem we are trying to prove is that either what happens is normal what you now expect normally that the limit function is actually a nice complex valued analytic function or the other extreme happens namely the limit function is always infinity it is a constant function infinity you do not get something in between you do not get the meromorphic functions with poles they will not come in between okay though you so you do not get that that is the so it does not go you know wrong in that sense and that is the theorem we are trying to prove okay see we have to be worried about all these things because we are allowing the value infinity okay once you allow the value infinity anything can happen and then you will have to be very careful and you have to prove things carefully see these are all basically ingredients that you really need to understand very well if you want to understand the Picard theorems the proof of the Picard theorem that is the reason why I am doing this pretty slowly okay so now in order to prove that so how will we prove that what we will do is we will take a sequence of functions on some domain in the complex plane and assume that the sequence converges to a function uniformly on compact sets okay that is normally and of course this convergence will be you are allowing the function you are allowing the limit function to take the value infinity so the limit will be in functions with values not in the complex plane but functions with values in the extended complex plane plus of course the convergence is going to be normal alright and what you want to show is that suppose the limit function is not the function which is identically infinity you have to show that the limit function is actually analytic okay so it is the limit function is either analytic which means it is a harness complex valued analytic function or it is infinity that is all there are only these two cases there is nothing in between okay that is what we are trying to prove so what we will do is essentially for that you know for proving that we need two facts one fact is the fact is the important property that the spherical metric is invariant with respect to inversion so the map z going to 1 over z that defines an automorphism a self isomorphism of the extended complex plane okay it is a homeomorphism in fact it is isomorphism in the topological sense and that when you translate it to the Riemann sphere it simply becomes rotation by 180 degrees around the x axis that is what I told you last time and of course distances on a sphere are not going to change if I rotate this sphere okay that is obvious so moral of the story is that this tells you that the spherical metric is invariant under the inversion okay that is one fact that we need the other fact that we need in the proof is Hurwitz's theorem okay which I think some of you must have seen in a first course in complex analysis probably some of you have not seen but anyway I will tell you what it is see roughly this is what I was trying to tell you at the end of the last lecture and I am now continuing roughly the philosophy of Hurwitz's theorem is the following. Suppose you have a uniform limit of analytic functions and suppose that the limit function is also a honest analytic function okay then if you take a 0 of the limit function mind you a 0 of the limit function has to be isolated because the limit function is analytic and you know zeros of an analytic function are isolated and you know this is equivalent to the identity theorem in if you have seen it in a first course in complex analysis. So the limit you take a 0 of the limit function then Hurwitz's theorem says that you see since you have sequence of functions converging to a limit function then the 0 of the limit function also comes as a cluster point or as a limit point of zeros of functions which converge to that limit function. So that means that there is a so you know so you know geometrically what it says is suppose you have a 0 of the limit function at a point z0 then there is a small neighborhood around z0 where that will be the only 0 this is because it is zeros of an analytic function are isolated and the limit function is analytic okay and then you can choose in this neighborhood sufficiently small so that in that neighborhood all the members of your original sequence which converge to this limit function beyond a certain stage that means for all sufficiently large indices subscripts okay the functions in your sequence also have zeros in that disk okay and the number of zeros is also equal to the order of the 0 of the limit function at that point and these zeros you know these zeros as you make the disk smaller and smaller and smaller these zeros actually converge they converge to the 0 of the limit function. So in other words what it says is that you know if the limit function has a 0 at a point then all your functions in your sequence beyond a certain stage should also have zeros in a neighborhood of that 0 of the limit function okay so a limit function cannot get a 0 just like that okay it cannot happen that you know all your limit functions never have any zeros then suddenly you know in the limit suddenly the limit function a 0 pops up out of out of the blue out of nowhere that does not happen okay. So you see intuitively this is very nice to believe okay but the problem mathematics complex analysis or mathematics is that you have all these intuitive things you believe that such things should not happen by continuity you believe that such nice things should always happen but then to prove them is the 0 that is where the meat lies you have to really sit down and work it out and that is why all this analysis is being done okay. So the key to who which is theorem is the so called argument principle okay which I will try to recall so what is this argument principle so the argument principle is as follows what is this argument principle so the situation is like this you are in some domain so there is a domain here and well and inside the domain there is some there is a simple closed contour okay and of course the interior of that contour also lies in the domain okay so both gamma and the interior of gamma okay they lie inside the domain and you have a function f which is meromorphic on the domain it is meromorphic on the domain means that it is analytic with the exception of an isolated subset where it has poles that is what it means okay you have a meromorphic function and assume that so you know the function f has only singularities it has a poles okay and of course it will have zeros also but you know anyway zeros of an analytic function are isolated you know that so there is some subset of the domain which is disjoint from the subset of poles which where the function has zeros okay now what you do is you make sure that this contour gamma does not pass through a zero or a pole so you assume that f is not equal to zero on gamma f has no pole on gamma okay now what is the argument principle if you recall it so the argument principle is if you integrate over gamma the logarithmic integral of f okay which is by definition integral over gamma f dash of z by f of z dz okay because you know d log is a suggestive notation the derivative of log derivative of log f is 1 by f times the derivative of f so it is f dash by f so integrating d log means you are integrating f dash by f alright so you calculate this integral of f dash over f you calculate this integral over gamma okay mind you this integral is very well defined because you see notice that the integrand is f dash by f the only problem is for the integrand is where f has zeros because when f has zeros then f is in the denominator so f dash by f will have poles okay but I have not allowed any zeros of f to lie on gamma mind you when you integrate something the variable of integration is varying only on the region of integration here the region of integration is gamma which is the contour and on the contour f is not going to vanish so that function I have written down f dash by f there is going to be no problem with the denominator and the numerator there is going to be no problem because you see if a function has a pole at a point then its derivative will have a pole of order 1 more at the point okay because if a function has a pole at a point z0 it locally looks like some g of z by z minus z0 power n where n is the order of the pole at z0 okay and if you take this g of z by z minus z0 power n and differentiate once you will get a z minus z0 power n plus 1 in the denominator so that means you know if it has a pole at a point then you its derivative will have a pole of higher order one higher order at that point so the only way this integrand can get into trouble because of the numerator is because of the poles of f but then I have also told you that none of the poles of f should lie on the on gamma so the integrand has got nothing has no problems on gamma so this integral is well defined okay and argument principle is that this integral is going to give you 2 pi i times the number of zeros minus the number of poles inside gamma okay that is the argument principle so this is equal to 2 pi i times number of zeros minus number of poles and here of course n zeros is number of zeros of f inside gamma and n poles is number of poles of f inside gamma okay. So what I am saying here is that when you say number of zeros or number of poles I want you to realize that you have to count them with multiplicity that is very very important so for example if you have pole you may have only 3 poles inside gamma but each pole may have different orders okay suppose you had 3 double poles inside gamma then the number of poles will become 6 because you have to count a double pole twice even though its one in the same point okay similarly zeros should be counted with multiplicities okay for example if you take z power n at 0 the multiplicity is n as a 0 if you take 1 by z power n at the origin the multiplicity of the pole is n you should think of it as n poles the point is geometrically it looks like only one point but actually algebraically there are n of them because there are n factors alright so when you write number of zeros minus number of poles you have to count multiplicities it is not just the number of points which are zeros minus the number of points which are poles that is not correct okay now well this is called the argument principle and you know the point is that if you are if your functions are if your function is actually analytic they are going to be no poles okay and what you are going to just get is 2 pi i times number of zeros okay and why is this called the argument principle the reason why it is called the argument principle is that if you take this number on the right side and divide it by i you will get 2 pi times an integer okay and this 2 pi times an integer is you know it is actually an angle 2 pi n is an angle so 2 pi itself is one full rotation okay so actually this 2 pi times is integer is actually the it is an angle and what is that angle that is the that is the change in the argument of that is the change in the argument of f okay as you go around alright that is the change in the argument that you get as you go around and in fact it is a change in the argument of f as you go around and the way to see the way to see it is like this you know if you if you if you naively write log f as a ln mod f plus i times argument of f suppose you write it like this okay then you know you can see that if I if I if I put a d to this then I will get d log f is d ln f d ln mod f plus i d r f this is what I will get okay and now if I integrate this over gamma okay if I integrate this over gamma what you will get is so I will get integral over gamma d log f is equal to integral over gamma d ln f d ln mod f plus i times integral over gamma d r f now you see the what is this this this will be 0 see d ln mod f is it will be an exact differential it will be 0 and basically the you see it is a it is a derivative of ln of mod f and ln of mod f is a nice real valued function okay so it is it is the exact it is the exact so the first integral is exact and for you know for an exact integral by fundamental theorem of calculus the integral will be final value minus the initial value of the antiderivative the antiderivative for the first integral is ln mod f so it is ln mod f final value minus ln mod f initial value but this is a closed loop so final value will be the same as initial value okay therefore it will be 0 first integral is just 0 and what is the second integral the second integral d r d r f is a change in the argument of f it is a change in the argument of the function f of z as z goes once around gamma okay and that is what this 2 pi i times number of zos minus number of poles okay and now you see that you know there is this i here there is this i here alright if you get rid of this i you now see the reason why it is called the argument principle it is a what it actually says is that integral over gamma d r f is actually 2 pi times number of zos minus number of poles so it what it actually says is that if you integrate the logarithmic derivative what you are going to get is just the change in the argument of f and what is that change in the argument of f it is 2 pi times an integer and what is that it is a change in the argument of f is mind you it is multiples of 2 pi it is a multiple of 2 pi and what is that multiple that multiple is actually number of zos minus number of poles that is what the argument principle says okay and I one needs this for Hurwitz's theorem and how does one get the proof of Hurwitz's theorem from this it is very simple you just use uniform continuity. Let me go to proof of Hurwitz's theorem I am just giving a short sketch details can be filled in later so here is the proof of Hurwitz's theorem so what is the situation in Hurwitz's theorem you are given that f n converges to f normally f in in in a domain d this is in a domain d in the complex plane and f n and f r analytic okay and f has a 0 is it not in d of multiplicity multiplicity otherwise order n okay this is given and what is Hurwitz's theorem Hurwitz's theorem is that all the f n also will have n zeros okay it they will have n zeros counted with multiplicities in a small neighbourhood of z0 and these as n tends to infinity these zeros will come closer and closer and closer and in the limit they will all coalesce to this 0 z0 that is Hurwitz's theorem so Hurwitz's theorem says that the 0 of a limit just does not pop up like that it pops up as a limit of zeros of the original functions which gave that limit okay that is what Hurwitz's theorem say. So how do what do we do see we we prove this just by using the argument principle it is pretty easy so see you have z0 here alright and then what you do is you choose a sufficiently small you know you choose a sufficiently small disc centred at z0 radius delta okay so this is say radius delta mod z minus z0 less than delta is contained in d you take a sufficiently small disc and in fact you also make sure that I will also put mod z minus z0 less than or equal to delta is in d which means that I am also including the boundary the reason is I want compactness by including the boundary I am making it a closed disc is compact because it is closed and bounded and why do I need that because I can now use normal convergence because normal convergence means that whenever you have a compact subset it is uniform convergence so inside this subset inside this closed disc and in particular on the boundary of the closed disc which is a nice circle centred at z0 of radius delta mind you that is a compact set that is also closed and bounded even on the circle I have uniform convergence because it is a compact subset of t okay that is the reason why I am including the circle and now you see you can of course choose this disc in such a way that z0 is the only 0 of f that is because you know f is an analytic function zeros of an analytic function are isolated which means that if you give me 0 of an analytic function I can find a sufficiently small disc surrounding that 0 where there is no other 0 okay so you make that also so let me write that z0 is the only 0 of f in the above disc and of course I am not assuming that there is any other 0 even on the boundary of that disc okay mind you alright then what is your now what is your now let us write let us write out the argument principle integral over mod z-z0 is equal to delta d log f is going to be 2 pi times n this is what I will get I think I will also get n I will also get an i okay 2 pi i n this is what I will get okay if I write the change in argument then I have to remove the i if I am just writing the logarithmic integral then I will get 2 pi i okay actually it is 2 pi i times number of zeros minus number of poles mind you the limit function f is analytic so it has no poles okay so it is only zeros and the only 0 it has is at the centre which is z0 that is because I made sure that there are no other zeros nearby the zeros of an analytic function are isolated and what is the order of that 0 at z0 it is n okay and that is what I have used here okay that is one part of the story now look at the look at the following thing fn if on the other hand on this side suppose I write integral mod z-z0 less than delta is equal to delta is equal to delta and if I write d log fn suppose I write this what will I get I will get number of zeros minus number of poles of fn times 2 pi i where I count zeros and poles of fn inside the inside that disk inside that open disk with multiplicity okay that is what I am going to get and you know essentially what will happen is you see so this is going to be some you know let me use a better notation okay so let me put this is capital N sub n small n okay times 2 pi i and again you know I will not have any poles because you know the fn's are all anyway analytic functions there are no poles normally I should write number of zeros minus number of poles times 2 pi i but there are no poles here okay so I get 2 pi i times nn and now notice this is the big deal if I take the limit as n tends to infinity of this integral okay now you see this is the point fn converges to f normally on d and the set on which I am doing that I am worried about is the set where I am doing this integral it is this circle because you see I am worried about this integrand the integrand the variable of integration varies over the region of integration the region of integration is the circle so I am worried about this circle but this circle is compact it is a compact subset of d and on this circle therefore there is uniform convergence now you know because of uniform convergence I can interchange limit and integral okay now this is one of the important properties of uniform convergence uniform convergence allows you to do things like changing limit and substitution which is continuity and then limit and differentiation which is differentiability term by term and it also allows you to change limit and integration which allows which is the same as saying that you can integrate term by term okay so because of uniform convergence I can push the limit inside okay and when I push the limit inside what will I get I will get the thing on the right I will get simply integral mod z minus z0 is equal to delta limit n tends to infinity d log fn and that is equal to actually this limit n tends to infinity d log fn is just d log f okay so what I get is look at the moral of the story the moral of the story is that limit as n as small n tends to infinity limit as small n tends to infinity capital n sub small n is capital n this is what I get finally this is just by applying argument principle but you see what does it say capital n sub small n is a sequence of integers okay and a sequence of integers is tending to a constant means that the sequence becomes constant beyond a certain stage okay see what does this mean this means that for n sufficiently large n n is the same as capital n sub small n is the same as capital n okay see sequence means it should come closer and closer but when these are integers the closer means either closer means that they are the same literally okay if you say one integer is within an epsilon of another integer they have to be the same if epsilon is less than 1 alright so n n is equal to n for n sufficiently large what does that mean it means this fn they have what is this n capital n sub small n it is the number of zeros of fn inside that disk and what is what have you got you have gotten that beyond a certain stage all the fn have exactly n zeros and these n zeros and that n is the same as the number of zeros of is the same as the multiplicity of the zero of the limit function f at z0 that is what it says and now mind you whatever I have done here will work if I make delta smaller after all if I make delta smaller the right side is not going to change because I am always going to get only the order of the zero of f at z0 so therefore this whole argument will work if I make delta smaller and smaller and smaller and smaller so that means what all these zeros of fn beyond a certain stage you see they are all you know they are going to go and cluster they are going to go closer and closer and closer they are going to cluster and they are going to finally coalesce into the zero at z0 and that is what it says it says that the zero of a limit normal limit of analytic functions that zero does not just pop up just like that it comes from zeros of the original sequence of functions beyond a certain stage okay that is the that is what it is a zero okay so now we will have to use this to give the proof of the fact that if a sequence of analytic functions does not converge to an analytic function then it has to completely converge to infinity the constant function infinity. So now let us go ahead with let us go ahead with trying to prove whatever we are trying to prove so so proof of so we go back to the old problem that we were worried about the statement that we first gave so let fn be a sequence of analytic functions on D which is in the complex plane it is a domain so fn is in our notation it is in h of D, h of D stands for the analytic functions or holomorphic functions on D and mind you this h of D is contained in M of D, M of D is a set of meromorphic functions on D and that is further contained in the set of all continuous functions from D to C union infinity okay and mind you when I and mind you here when I am writing it like this I am treating the meromorphic functions also as functions which can take the value infinity and the meromorphic function the value at a pole is defined to be infinity mind you okay and the sequence fn is in here and you assume that fn converges to an f point wise okay so fn converges to f normally with respect to the spherical metric okay mind you I have to allow the spherical metric because I have the point at infinity also as a value alright and mind you this is sitting inside the set of all maps not necessarily continuous from D to C union infinity okay and this script C is continuous maps this is this is continuous maps alright and the mind you let me again let me again insist this is convergence with respect to the spherical metric okay point wise convergence with respect to the spherical metric why I need the spherical metric is because the limit function at one at some point can be infinity then I will have to measure distance with respect to infinity and I can do that only with this spherical metric okay now I already proved a lemma last time it is something that you already know that whenever you have a normal limit of continuous functions it is continuous so that is quite clear because a normal limit is a locally uniform limit and therefore and you know a uniform limit of continuous functions is continuous and the function which is locally continuous is also continuous because continuity is a local property so f this f this limit function f is certainly continuous okay f is continuous so f is actually here f is a continuous map from D to C union infinity and what is it and what is the claim this is a big theorem that we are trying to prove the claim is if f is not analytic then f is identically infinity that is our theorem that is what we are trying to prove okay so we have to show that if f is not analytic or let me write it the more logical way if f is not identically infinity then it is analytic this is what we have to show okay so this so let me point out you do not have a situation where so the limit function f either lies it is either the function function here which is the constant function infinity or it is here itself it is in H of D itself it cannot go into this it cannot become maromorphic that is it cannot become honestly maromorphic okay so you know it is like saying that what is the meaning of saying it cannot be honestly maromorphic it means that suddenly a pole cannot pop up okay in the limit you have sequence of analytic functions it is converging to a limit function a pole simply cannot pop up out of the blue of course the original sequence does not have poles because they are analytic functions okay by continuity you should expect this also to happen but again you know this is the point this is this has to be proved and you can see it is like it is there is already the flavor of Hurwitz's theorem he says that you know when you have a limit of analytic functions 0 of the limit just does not come just in power does not just pop out of the blue it comes from a limit of 0 so the original functions so that is what we are trying to prove so assume that f is not identically infinity then show that f is analytic okay and what does it mean it means that if f does not is not going to be the identically the function infinity it cannot assume infinity even at a single point mind you it has to assume only fine only complex values at every point that is the strong thing there if it assumes value infinity at a point and if that point is isolated point where it is assuming the value infinity it means it is a pole okay but that does not happen that is what it is alright that is what we have to prove so how does one see that well you see you let you take this subset D sub D infinity okay this is a set of all z in the domain where f of z is infinity okay and this is a proper subset of the of your domain it is a proper subset of the domain because you assume that f is not identically infinity if f is identically infinity then D infinity that is the same as saying D infinity is D okay but when you say f is not identically infinity it means that the limit function f has some point where it has some finite complex value a value which is different from infinity in the extended complex plane okay. Now mind you this is a closed set actually okay because it is the inverse image of infinity under f which is a continuous map I have already told you that f is continuous and infinity single point is always closed okay in the one point compactification so if you take the inverse image of infinity that will be D infinity so D infinity is just D infinity is just f inverse infinity and that is a that is closed inside it is a closed subset just by continuity of f alright so I will stop here.