 Alright so let us continue with whatever we were doing. See our aim as I told you is to prove the Picard theorems for which we need to look at families of meromorphic functions. So we need to do topology on a space of meromorphic functions okay and so I told you in the previous lectures that you know if you want to topology on a space of functions and you go and try to draw inspiration from usual topology, then for example if you look at real valued or complex valued bounded functions, continuous functions on a topological space then the topology is done by looking at uniform convergence okay. So we define convergence in the space by uniform convergence but now if you try to take inspiration from this and come to complex analysis and then you are working of course with holomorphic functions that are that is analytic functions and then otherwise we want to be a little bit more general and we want to work with meromorphic functions and meromorphic functions you know are analytic or holomorphic functions except for an isolated set of points where they have only pole singularities okay. So you want to work with such functions but the point is that even if you take analytic functions okay and you look at a sequence of analytic functions converging okay normally what happens is that you do not get uniform convergence okay but you get only normal convergence. So when you come from topology to complex analysis you must remember that you should not work with uniform convergence but you should work with uniform convergence restricted to compact sets and that is called normal convergence okay that is the first point. Then the second point is this pathology that you know you can have a domain in which you can have a decent sequence of analytic functions which goes to infinity everywhere okay. Now so for example I was discussing about this domain which is the exterior of the unit disc and I said you take the you know the sequence of functions given by z power n. So the first function is z the second function is z squared and so on and then this sequence you know it converges point wise to the function which is infinity okay. So and the point is that this convergence normally if you are only thinking of a first course in complex analysis we will say this sequence diverges okay because if you take any complex number z with modulus greater than 1 and you look at the sequence z power n that is not going to go to a limit it is going to go to infinity okay you will simply say the sequence diverges but the fact is that we want to think of this as a convergent sequence and there are 2 reasons for this the first thing is that it converges to the constant function which is infinity at every point outside the unit disc and we have to allow the value infinity because we want to think of meromorphic functions because see if you are thinking of meromorphic functions then you can define the value at a pole to be infinity because that is the limit that you get as you approach the pole okay. So we have to allow the value infinity and once you allow the value infinity then you must also allow the constant function which is infinity and if you go by that reasoning then this sequence z power n actually tends to the constant function infinity and that is the reason that we say that z power n tends to the constant function infinity it tends to the constant function infinity outside the unit disc okay and in the exterior of the unit disc alright and of course unit circle is not included we are looking at the exterior of the unit circle and then the point is that if you want to talk about this convergence in terms of a metric okay then you will have to worry about giving a metric on the extended complex plane and I told you that that metric that you can use is the spherical metric there is a so called spherical metric and what is the spherical metric you take two points in the extended complex plane take the images in on the Riemann sphere under the stereographic projection and then use the spherical metric on the Riemann sphere the spherical metric on the Riemann sphere is just the it is a length of the minor arc of the big circle passing through those two points on the Riemann sphere okay. So in this way you get a spherical metric on the extended complex plane and then with respect to this spherical metric I can actually define the distance between any point any two points in the extended plane so for example I can have one point which is a point in the complex plane I have I can have the other point to be a point at infinity and I can still have this notion of distance okay and it is with respect to this metric the spherical metric that the convergence is actually normal okay so here this is the point that we have to understand if you take the sequence of functions z power n in the domain mod z greater than 1 then this sequence converges normally to the constant function infinity okay and this is the view point that is very important okay and what I am trying to tell you today is that you know this is the only pathology that will that that can occur if you take a domain in the complex plane and suppose you have sequence of analytic functions which converge normally to a function on the domain okay and assume that you allow this function the limit function to take the value infinity suppose you assume that you allow that then it is very beautiful that either the limit function is completely analytic or it is completely infinity you do not get anything in between. So you know you do not get this this kind of horrible pathology that you have sequence of analytic functions and they tend to a for example a meromorphic function okay that will not happen you cannot have a sequence of analytic functions on a domain they converge normally okay that is they converge uniformly on compact subset of the domain and in the limit you get low behold you suddenly get a meromorphic function that will not happen. So this is a very very important thing it tells you that things are going on well as you know as per usual intuition something does not you must always see whenever you work with some extraordinary cases you should always be careful. So for example when I am allowing a function to take the value infinity the function can very well be a meromorphic function because meromorphic function takes the value infinity at the poles. So it can happen that I have a sequence of analytic functions on a domain it is converging normally to a limit function now if I allow the limit function to take the value infinity okay then the limit function could very well be even meromorphic okay. But the fact is that does not happen what happens is either the limit function is analytic that is the good thing and the and the and in the bad case the worst thing happens the limit function is always infinity okay and the reason for this is 2 important facts one fact is the theorem of Hurwitz for analytic functions the other thing is the so called symmetry of the spherical metric with respect to inversion okay. So these are the 2 facts and this is what I want to concentrate today upon so I will start with the following thing let maps D, C union infinity so look at this set let this be the set of maps or just functions from D which is it is a domain D is a domain in the complex plane to the external complex plane. So this is the important thing I am allowing the value infinity alright and so what I am going to do is there are many subsets of this which I am interested in so let me write them down among this is inside this is this is the I will put this C of D C union infinity and what is this C of C this is the set of continuous maps this is the set of continuous maps okay and you know when I say this is set of continuous maps I am looking at all those maps which are continuous from this domain D to C union infinity and mind you C union infinity has a topology. So C union infinity has one point compactification topology and under this with this topology it is homeomorphic to the Riemann sphere under the stereographic projection and in fact it is in fact even a metric space okay you can take either the spherical metric or the chordal metric on the Riemann sphere and transport it to C union infinity via the stereographic projection and so C union infinity becomes even a metric space it becomes a complete metric space it is compact it is very nice okay so it does make sense to look at continuous maps into a topological space or metric space so it is continuous in that sense okay. So the point I want you to understand is that now if you look at this set of continuous maps into C union infinity then the constant map which is equal to infinity okay that is the map that takes every point to infinity that is also continuous because mind you always constant maps are always continuous okay so the function which is infinity uniformly that is a continuous function mind you. So this is the point that you have to carefully notice this is what we introduce and in particular what happens is that this contains the this set M of D the set of meromorphic functions on D the set you take any meromorphic function on D what is a meromorphic function a meromorphic function is a holomorphic function on D minus an isolated set of points for which the given function has poles at which the given function has poles so you take a meromorphic function on D okay it is a honest function outside and it is a honest holomorphic function outside a you know isolated set of points but at each of those isolated set of points in D the function has a pole but I can define the value of the function to be infinity at a pole because that is the limit I get as I approach a pole okay that is the definition of pole okay and therefore this a meromorphic function becomes a continuous function into C even in infinity that is the point that is another critical point you have to understand okay by including the value infinity you are making the meromorphic function continuous on the whole domain you are making it continuous even at the poles okay and do not confuse that continuity with the usual continuity because the usual continuity is with respect to complex numbers the target is complex numbers and you do not allow the value infinity okay but this continuity is different this continuity is with respect to the extended complex plane okay this is something you have to clearly distinguish okay because you know if a meromorphic function in the usual sense cannot be continuous at a pole because at the pole it becomes infinity okay but and that is because you do not allow it to take the value infinity you do not think of infinity as a value if you are only thinking of complex values but now since I have added the extra point at infinity the meromorphic function becomes also continuous at infinity that is the point you must notice and then of course there is of course the the further subset of H of D this is our this is a subset of all holomorphic functions okay or analytic functions and of course you know holomorphic or analytic functions are also by default they are also included as meromorphic functions so the meromorphic function is a function which can either be analytic or if it has singularities the singularities must only be isolated and they must be only poles that is the definition okay. So the definition of meromorphic includes the definition of holomorphic okay so let me write that these oops so this is so this is M of D is meromorphic function and H of D is holomorphic functions and so you know basically what we want to do is we want to do topology on this set the set of meromorphic functions alright on the domain D and I told you this topology has to be done with respect to I mean we do we say that we do topology by trying to study convergence okay so the convergence that we should think of is normal convergence so the topology corresponds to working with normal convergence okay and what we saw last time is that if you take the domain to be the exterior of the unit circle okay if you take the domain to be exterior of the unit circle and you take the sequence Z n, Z power n okay namely Z, Z square, Z cube then that is a sequence of that is a sequence in H of D that is a sequence of holomorphic functions and it converges normally to what it converges normally to the to the constant function infinity which is which is in this set above this script C D C union infinity it is here okay and so I want to say the following thing you must have studied this in a first course in topology but the idea is essentially the same namely that whenever you have a uniform limit okay the limit function is also continuous okay so that is the first thing that I want to that is the first thing I want to prove or recall at least and mind you here we are not working with uniform convergence on the whole domain whatever convergence we are working with is only normal convergence that is the that is this but that is good enough because you know normal convergence will also give locally uniform convergence so locally you can still do the same thing that you would do if you had uniform convergence everywhere okay and that is good enough for local properties like continuity, analyticity, etc that is good enough okay so here is a here is a lemma here is a lemma if f n is a sequence of continuous maps from the domain D into C union infinity and f n tends to f which is thought of as a map from D to C union infinity so that means when I say f n tends to f point wise so this is point wise on D okay that means f n of z tends to the value f of z as n tends to infinity for each z in D okay and this convergence is normal on D then f is also a continuous map from D to C union infinity so I am I am just saying that a normal limit of continuous maps is continuous and why should this be true because you know uniform limit of continuous maps is continuous and if you have a normal limit it is actually a uniform limit locally okay and therefore the limit function is continuous locally but continuity is a local property therefore if something is continuous locally then it is continuous if you have a global map which is locally continuous on every on sufficiently small open sets okay which cover the domain then it is continuous okay so this is a very this the proof of this lemma is just trivial I mean if you assume the fact that the uniform limit of continuous function is continuous okay so this is something that you can easily deduce what you so let me let me pinpoint the important fact here if you take a point in D okay then you can choose a sufficiently small disc surrounding that point to lie in D that is because D is an D is a domain mind you D is an open connected set so D is an open set so if you give me a point of D there is a whole disc surrounding that point which is lying inside D and if I make the radius of the disc small enough I can ensure that even the boundary of that disc lies inside D okay now if I include the boundary to the disc then it becomes a compact set because it is a closed and bounded set and on a compact set I know that this convergence is not is actually uniform because I have been given normal convergence normal convergence means that whenever you look at the convergence on a compact subset it is uniform okay therefore if you give me any point I can find a sufficiently small disc on whose closure the convergence is uniform. So in particular it is also uniform on sufficiently small discs okay that means that the limit function on sufficiently small discs is continuous but then I can cover the whole space by such small discs so the limit function becomes continuous everywhere okay that is the proof alright. So I want to write down the proof I have told you in words now comes a very important theorem so here is the theorem the theorem is that so let me tell you in this lemma we have been only worrying about continuous maps alright now you can ask this question because in the you know in the previous slide I have if you look at the previous slide I have given you these two subsets there is one subset which is M of D which is the meromorphic functions and then there is this other subset which is H of D which is the holomorphic functions and then you can ask that ask what will happen to the limit function if the given sequence of functions is in M of D or H of D. So you can ask so what we have just now proved in this lemma is that a normal limit of continuous functions is continuous alright that is what the lemmas is now you can ask is a normal limit of meromorphic functions meromorphic that is one question you can ask then the other question you can ask is the normal limit of holomorphic that is analytic functions analytic you can ask that but you know you already have a we have an we have seen an exception we have seen an example of a normal limit of holomorphic analytic functions which is going to the constant function infinity okay. Now this is the only exception that is the beautiful thing so in the case of analytic functions and that is the theorem okay so if you are looking at sequence of holomorphic or analytic functions suppose it converges normally to a limit function then the limit function is either completely holomorphic that is completely analytic or it is completely infinity it is a constant function infinity and there is nothing in between okay. So this is very good behaviour so what it means is that you know if you are the exceptional whenever you are working with respect to the spherical mention whenever you are that is you are working with respect to the extended complex plane namely you are allowing the value infinity okay then you must allow the exceptional case that is sequence of holomorphic function tends to the constant function infinity uniformly on compact sets okay that is the only exception that is what the theorem says. So for example you do not have this further pathologies like you know you have a sequence of analytic functions and in the limit you get a meromorphic function suddenly a pole pops up in the limit you know such kind of horrible behaviour does not happen okay now that is very very important that tells you that you know it is behaviour is very good right. So otherwise you would have been worried if you can find a sequence of analytic functions okay which is converging normally to a limit function and the limit function suddenly starts having poles you know you of course intuitively you do not expect that to happen but how do you prove that such a thing does not happen and that is what the theorem says okay. So let me write that down theorem if f n is a sequence of holomorphic functions on D H of D and f n converges to f normally on D then either f is also holomorphic or f is identically infinity that is all you do not get the intermediate case of f being meromorphic okay. So this is a theorem and this is I mean intuitively this looks fine but you the big deal with serious mathematics is to guess some statements which are intuitive and then the bigger deal is to really prove them giving proofs is very important part of mathematics. So let us go to this well so now so the proof requires a couple of things the first of all the proof requires the so-called invariance of the spherical metric with respect to inversion so let me explain that. So that is one ingredient in the proof the other ingredient in the proof is so-called Hurwitz theorem which probably you have seen it in a first course in complex analysis but I do not expect many people to have seen it in a first course in complex analysis so I will tell you what that theorem is it is pretty simple theorem it is got to do with uniform convergence and it is got to do with the argument principle which you might have seen okay. So let me start with this start with the invariance of the spherical metric with respect to inversion so let us again recall so let me put this here maybe I will use a different colour so well invariance of the spherical metric with respect to inversion this is a very important fact so here is my so here is what I am going to say so you see you have this you have this so let me again draw this stereographic projection so here is the this is this is X Y plane this is a complex plane which is identified with X Y plane and then you have this you have the well you have the Riemann sphere again okay and of course there is this third axis which so here is the third one this is U because of course you know I have been I am I am reserving I am reserving Z to be X plus I Y so I am calling the third axis as U and now you see you take 2 points Z 1 and say Z 2 okay on the complex plane and in fact you know they then you get their stereographic projections on the Riemann sphere so you get these points so you get this point P 1 and you get this point P 2 okay and of course you know you get P 1 by so this is a north pole and you join north pole to P 1 then it should go and hit Z 1 you join north pole to P 2 and it has to go and hit Z 2 okay so this is the this is the definition of the stereographic projection and what is the spherical distance between Z 1 and Z 2 this is actually the spherical distance on well let me call this as oops let me call this as S 2 this is the real 2 sphere this is topological standard topological notation so I will let me put this put S 2 here between P 1 and P 2 and what is the what is the thing I have written on the right this ds superscript S 2 is actually the you know the geodesic distance it is the distance along the great circle passing through P 1 and P 2 you know if you have a sphere and you take 2 points on the sphere if they are 2 distinct points there is only 1 big circle circle of largest radius that passes through those 2 points and lies on the sphere that is called the great circle passing through those 2 points okay and then what you do is that you have these 2 points on the circle and they divide the circle into a smaller portion a larger portion in general and you take the length of the smaller arc okay so that is what this distance here is and that is defined to be equal to the distance of the distance spherical distance between the point Z 1 and Z 2 okay and well this is this is spherical distance now the point is that you know I need also to write something here so let me do the following thing let me take another color so you have also so if I write it out in a well so let me write it somewhere here yeah so you have S 2 and you have this stereographic projection with C union infinity so C union infinity is extended complex plane there is a point at infinity added to that the set of complex numbers and this infinity goes to the north port mind you under stereographic projection so what I have written here is a stereographic projection and see the point so this is so here P 1 goes to Z 1 and P 2 goes to Z 2 okay and this is this is the situation that we have but on C union infinity there is an automorphism okay there is a self isomorphism there is an isomorphism at least as a set and what is that isomorphism that is inversion okay so on C union infinity you have the map Z going to 1 by Z Z going to 1 by Z makes sense on C union infinity where you declare 0 to go to infinity and you declare infinity to go to 0 so it interchanges 0 and infinity but for points which are different from 0 and infinity you know for a complex number which is different from 0 and infinity of course when you say complex number infinity is not allowed so if it is a non-zero complex number then it is reciprocal is also a non-zero complex number so the point is that on this right side here which is C union infinity there is this map Z going to 1 by Z and this map Z going to 1 by Z is in fact a homeomorphism because you know it is continuous and it remains continuous even if you give C union infinity the of course you know Z going to 1 by Z is continuous in the punctured complex plane that is if you take the complex plane and remove the origin Z going to 1 by Z is actually you know holomorphic isomorphism because it is an injective holomorphic map and you do not have to worry go too deep because it is inverse is itself for Z going to 1 by Z the inverse map is itself okay you apply the map twice you get identity okay so it is a holomorphic isomorphism it is an analytic isomorphism from C minus 0 to C minus 0 but the point is if you include 0 then you have to also include infinity and you have to send 0 to infinity and you have to send infinity to 0 and in therefore you get a bijective map of the extended complex plane and that map is actually a homeomorphism okay with respect to the 1 point compactification topology on C union infinity you can check that so but you see this so Z going to 1 by Z which is inversion is a homeomorphism on the extended plane now if you if you if you transport that via this stereographic projection you will get an you will get a homeomorphism of the real sphere S2 which is the Riemann sphere okay because you know if if 2 spaces are isomorphic then if 2 if 2 topological spaces are isomorphic that is homeomorphic if on 1 space you have a automorphism homeomorphism self-homomorphism then this isomorphism will transport it and give rise to a homeomorphism self-homomorphism of the other space okay so this inversion which is a homeomorphism of the self-homomorphism of the extended plane will give you a self-homomorphism Riemann sphere okay and guess what is what it is guess what it is you know what it is it is you can check it it is actually nothing but rotation of the Riemann sphere okay it is rotation of the Riemann sphere above the x axis okay and you can see that that is because you see look you take any you take any point on the on the x axis okay you take any point on the x axis say for example take the point 1 take the point 1 okay on the x axis that is the that corresponds to complex number 1 okay where does it go under inversion it goes back to 1 alright and the point minus 1 goes back to minus 1 and you know the stereographic projection is such that for every point on the unit circle the point on the Riemann sphere is the same as the point on the unit circle for the stereographic projection the stereographic projection induces a bijection on the unit circle because the unit circle is also on the complex plane it lies also on the Riemann sphere on S2 and this stereographic projection fixes point wise it fixes the unit circle. So this inversion it is going to induce some you know homeomorphism of the Riemann sphere that is going to fix plus 1 and minus 1 okay and look at what happens to look at what happens to the point at infinity under inversion infinity goes to 0 and 0 goes to infinity okay now what is what does this translate to the Riemann sphere you see on the Riemann sphere infinity corresponds to the north pole 0 corresponds to the south pole okay the point 0 corresponds to the south pole because you see the 0 is here and you know if you what is the stereographic projection of 0 it is you have to take the line joining the north pole to 0 and then look at the unique point on the Riemann sphere where this line hits and that will be the south pole okay so 0 on the complex plane corresponds to south pole on the Riemann sphere so you know this inversion which send 0 to infinity and infinity to 0 on the extended plane when you translate it to the Riemann sphere it will send north pole to the south pole and south pole to the north pole now you can imagine this map what it is doing it is switching the north and south poles and it is fixing plus or minus 1 so it is a rotation about the x axis that is what is happening okay so this you can use your you know analytic geometry and you can actually write out equations and check that this is actually a rotation by 180 degrees of the of the sphere with respect to the x axis okay and now what does this tell you this tells you that if I take the point z1 and z2 okay and I take the spherical distance between them and if I take the inverse points 1 by z1 and 1 by z2 and take the spherical distance between the inverse points that will be the same because the spherical distances are being measured by looking at the corresponding points on the sphere and the inversion corresponds to rotation of the sphere but any if you take any two points on the sphere the spherical distance between those two points that is not going to change if I rotate the sphere that is invariant under rotation of the sphere. So the moral of the story is that the spherical distance between z1 and z2 is the same as spherical distance between 1 by z1 and z2 in other words the spherical metric is invariant for the inversion that is a very important fact which we are going to use in the proof okay. So let me write this down so let me say let me write somewhere here so diagrammatically what we are going to have is that so this is z1 and let us say that this is z2 sorry this is so if this is z1 then this is so 1 by z1 is going to lie somewhere here and if this is z2 then 1 by z2 is going to lie somewhere here and this 1 by z1 and 1 by z2 are going to correspond to points P1 prime and P2 prime on the Riemann sphere and the fact is that the spherical distance between the spherical distance on s2 between P1 prime and P2 prime is the same as the spherical distance on s2 between P1 and P2 because P1 prime P1 prime and P2 prime are just gotten from P1 P2 by rotation by 180 degrees okay and so in other words I am saying that you know see this if I draw this arc here and if I draw this arc here they are of the same length mind you this is a perspective drawing so they do not really look to be of the same length when I draw it on this but they are of the same length essentially and well so but you see this guy here on top is the spherical distance between 1 by z1 and 1 by z2 and this guy here in the bottom is the spherical distance between z1 and z2 okay and this the spherical distance is invariant and reinversion and the way I have drawn it I have taken z1 z2 in the complex plane but you can make one of them or even both of them infinity it still works it still works okay. So the moral story is therefore so let me also mention this P1 goes to z1 P2 goes to z2 and of course so let me write it here P1 prime P1 prime goes to z1 prime P2 prime corresponds sorry so z1 prime is actually 1 by z1 and P2 prime is goes to z2 prime which is 1 by z2 okay. So let me write the statement here the holomorphic that is analytic isomorphism z going to 1 by z from c minus 0 to c minus 0, c minus 0 extends to a homeomorphism c union infinity to c union infinity and this is inversion where you know you send infinity goes to 0 goes to infinity and the homeomorphism the homeomorphism that it induces via the stereographic projection on s2 s2 to s2 that it induces via the stereographic projection is just rotation of s2 about the x axis axis by 180 degrees okay. So this is the fact that you need to check you can write it out it is a very simple you know exercise in analytic geometry if you want okay but the point is that distances on the sphere will not change if you rotate the sphere so the upshot is thus for all points z1, z2 in c union infinity the spherical distance between z1 and z2 is the same as a spherical distance between 1 by z1 and 1 by z2 so this is the fact that we want this so it uses the fact that if you distances on the sphere will not change if you rotate the sphere and the rotation of the sphere by 180 degrees about the x axis corresponds to actually it translates to inversion for the extended plane okay. So the spherical metric is inversion invariant so let me write that in words i.e the so ds okay see the spherical metric ds is inversion invariant okay so this is one fact that we need to use okay I need to use this fact this fact will be used in trying to prove our theorem that you know if you have a sequence of analytic functions on a domain either it converges to an analytic function or it converges to the constant function infinity provided you assume the convergence is normal uniform and compact subsets okay so this is one fact the other fact is the very important Hurwitz theorem okay which I will try to explain next. So the next thing that I need is Hurwitz's theorem that is another thing that I need Hurwitz's theorem okay so what is this Hurwitz's theorem so basically the theorem is very very simple what the theorem says is that it tells you something that is very very very believable what it says is that you know if you take a normal limit of analytic functions holomorphic functions then of course you know the limit function suppose the limit function is complex value not it does not take the value infinity so we are in the setup of an undergraduate course of complex analysis where all complex functions take only values in C okay and you are not allowing the value infinity okay so suppose the sequence of analytic functions converges normally to a function which takes values only in C then you know already that this limit function is already analytic that is something that you know I have already explained to you that a normal limit of analytic functions is analytic provided you are you make sure that the limit function takes values only in C okay this is again something that you have seen in the first course in complex analysis basically using Morera's theorem and Cauchy's theorem okay so the limit function is analytic now what Hurwitz's theorem says is that if you take a 0 of the limit function okay mind you the limit function is analytic and you know an analytic function has isolated zeros unless it is identically 0 and a non-constant analytic function if you take the zeros they are isolated that is a very important fact and it for example it is as powerful as the identity theorem which says that if two analytic functions are having the same values on a convergent set of points even at the limit including the limit which is also a point of analyticity then the functions have to be identically the same that is the identity theorem and that is equivalent to the this theorem that the zeros of an analytic function are isolated non-constant analytic function the zeros are isolated okay so you take a normal sequence of analytic functions and you take the limit function it is analytic and you take a 0 of the limit function then what Hurwitz's theorem says is that this 0 of the limit function comes from zeros of the functions of the original sequence. In fact if you take a 0 of the limit function it has to have a certain order you know zeros always have a certain order so you will have a 0 of order 1 or order 2 and so on so if the order of the 0 is capital N Hurwitz's theorem says that this because this limit function has come as a normal limit of a sequence what it will say is that beyond a certain stage all the each of the functions in your sequence will also have n zeros the same number of zeros with multiplicity as the 0 of the limit function that you are looking at and these zeros and this will happen in a neighbourhood of that 0 of the limit function and what will happen is slowly as n tends to infinity these zeros will cluster and cluster and cluster and come closer and closer and in the limit they will all coalesce to the 0 of the limit function. So basically what Hurwitz's theorem says is that the 0 of the limit function is actually coming as a limit of zeros of the original functions beyond a certain stage and the way this is done is that it is done very precisely in the sense that even multiplicities are taken care of that is essentially what Hurwitz's theorem is okay this is how you say it inverts but I mean but if you start of course when you technically write it down it looks pretty more difficult okay so now so let me write this let me write the statement of the theorem down suppose f n converges to f uniformly normally on D D domain and f n and f are all holomorphic functions on D and is it not belonging to D is 0 of f of order n greater than 0 then there exists a row such that f n of z has n zeros in mod z minus z not less than row for n sufficiently large so when I say n zeros counted with multiplicity in this disk for n sufficiently large and of course mod z minus z not less than row is in D so you have to choose row small enough that is such a row and further and these zeros and these zeros these 0 sets converge to z not as n tends to infinity so this is Hurwitz's theorem this is Hurwitz's theorem in notations okay so what it says is that the 0 z not of the limit function the limit analytic function f is the limit point of zeros of the functions in the sequence and the fact is that even the multiplicities are taken care of mind you multiplicities are very very important because you know multiplicities you have to count zeros with multiplicities you cannot just count zeros just as points it may be a 0 function may have 0 at a point but it may have a certain multiplicity that is the order of the 0 you have to count the order also okay so that is why multiplicities are important these n zeros they need not be n distinct points they could be lesser than n points with some points having zeros of higher order okay so multiplicity is very important so this is Hurwitz's theorem okay and the proof of this theorem essentially uses the argument principle okay it uses argument principle and it uses uniform convergence okay so I will try to give a proof of that in the I will give a sketch of that proof in the next talk.