 Okay, so let me have your attention you see we are now more or less you know comfortable with thinking of the point at infinity of functions behaving at infinity, analyticity at infinity, singularities at infinity okay. So you know we wanted to do that because we needed to look at functions on the extended plane okay and this view point is very very important because it is not normally when you do complex analysis say for example in a first course you are worried about only functions on the plane okay on the complex plane you are not worried about the point at infinity okay and but if you include the point at infinity you get more information that is the point. So I explained this last time but let me again repeat it, so I am saying for example look at let us take an entire function okay let us take an entire function take a non-constant entire function okay. Then if you look at the little Picard theorem what it will tell you is that the image of the whole plane is going to be either the whole plane or the whole plane minus one point which is one value which it may not take and that is the that is the best possible I mean it cannot miss two values what it means is that if an entire function misses more than one value it has to be a constant okay. So you see and of course you have but then you know if I ask you now take this entire function and take the exterior of a circle of sufficiently large radius okay and take its image what will the image look like okay then you know you do not have an answer you do not have an answer. So whereas for example if you knew that the that if you can think of infinity as and you know a singular point and in fact if infinity is an essential singular point okay then you can apply the great Picard theorem which will tell you that the image of every deleted neighbourhood of infinity will be either the full plane or the plane minus a point okay. So you can use that which is a more powerful theorem to say that if you have an entire function and suppose it is not algebraic namely that infinity it is a transcendental entire function that is infinity is you know essential singularity then if you take if you take the exterior of a circle of sufficiently large radius for that matter if you take exterior of any circle okay because exterior of any circle is going to be a deleted neighbourhood of the point at infinity in the extended plane and if you apply Picard the big Picard theorem you get the information that the image of the exterior of any circle is always is always going to be the whole plane or the whole plane minus a point okay and it will also give you the additional information that all the all the values it takes it takes on infinitely many times okay. So you see the you see the advantage of thinking of the point at infinity okay you get more information so so along these lines you know that you know for example you can characterize entire functions as transcendental or algebraic based on whether infinity is an essential singularity or not if infinity is not an essential singularity then it is a pole or a removable singularity if infinity is a removable singularity then it will by Lewis theorem the function has to be constant if infinity is a pole then the function has to be a polynomial and otherwise it is a transcendental entire function infinity is an essential singularity okay. Now what we need to do is see we are the whole aim of this all this is we are trying to prove the big Picard theorem and as a corollary deduce the little Picard theorem okay and you know theory required for that is to do analysis on compact spaces of meromorphic functions okay. So you have to study the topology and you have to understand do some analysis using compact families of meromorphic functions so we need to worry about meromorphic functions so let me so let me start by telling you what a meromorphic function is I have told this before but let us go into that a little bit more detail because that is something that gives you a link to algebra okay so the reason why lot of complex geometry is connected to algebra is because of the fact that all the meromorphic functions they form a field naturally and that field is an extension field of the complex numbers because the complex numbers always can be thought of as constant functions every complex number corresponds to the constant function given by that number so you know and constant functions are of course meromorphic okay they are analytic so you see that the field of meromorphic functions as a field extension of the complex field is the properties of this field extension in algebra for example in Galois theory and things like that that determine a lot about the geometry of the domain on which you are studying the meromorphic functions okay and this is extensively used for example in classifying Riemann surfaces and more generally if you want to classify manifolds and so on complex manifolds you can use this theory okay it gives you an interface between complex analysis and topology and algebraic geometry complex geometry okay and algebra in that sense okay so what is a meromorphic function the definition is pretty simple it is a function which has which is analytic on a domain okay but the only singularities it has are poles okay that is the definition of a meromorphic function. So let me write that down so the field of meromorphic functions so here is the definition let D in the extended complex plane be a domain okay so I am looking at a domain in the extended complex plane by mind you it means that it is an open connected set and of course we always assume that it is non-empty okay a meromorphic function on D is either an analytic function analytic function on D or a function that is analytic on D except for poles okay so meromorphic function by definition is a function that might that could be defined on D or D minus points where those points will be of course isolated they will be of course singularities okay and the point is that they have to be isolated in fact they have to be poles okay this is the definition okay so we say that we often use the abbreviation I mean we use this short phrase analytic except for poles that is the that is what meromorphic means okay and now we need to make a few observations the first thing is that you see the number of poles okay the set of poles of the function it may be of course empty in which case a function is actually analytic or holomorphic but if it is not analytic then it has poles at least one pole and the fact is that set of poles in D will be a countable set so the first thing is that you cannot have an uncountable set of poles so a meromorphic function will have only countably many poles and what is the reason for that the reason for that is because of second countability okay see the complex plane is just R2 and you know Euclidean spaces Rn they are all second countable because they have a countable basis so you can take all the open balls centred at points with rational coordinates and with radii given by rational numbers okay and use the fact that the rational numbers form a countable set okay and then you get this collection of open balls centred at rational points with rational coordinates and having rational radii as a countable collection of open sets and they will form a basis for the topology okay in the sense that any open set is a union of such sets okay and the intersection of any two sets like that of this type again is a union of sets of that form okay so it is a basis for the topology so the topology has a countable basis this is second countability and now in particular for example the plane R2 or the complex plane as topologically is second countable so you know what you can do is suppose you have so and of course any subspace of a second countable space is also second countable because you all you have to do is that you have to take the countable base countable base for the bigger space and intersect it with the subspace to get a countable base for the subspace okay so if you take the domain you take any domain in the complex plane or extended complex plane it will be second countable okay and by the way if you are looking at the extended complex plane of course it is homeomorphic to the you know the Riemann sphere and the Riemann sphere is a subset of R3 real 3 dimensional space and that and since real 3 dimensional space is second countable the Riemann sphere being a subset is also second countable okay so the extended plane is also second countable okay after all you are just adding one more point at infinity and if you want you can take all the discs centre at infinity but of course you should not say finite radius okay unless you are looking at the images in the Riemann sphere with the point at infinity corresponding to the north pole okay but in any case if you are taking a domain in the extended plane it is going to be second countable and now if you take a function which has poles in that domain then the set of poles the poles by definition are isolated poles are by definition isolated singularities so you have an isolated subset of a second countable set of a second countable space an isolated subset will always be countable okay the reason is because since the points of the subset are isolated you can cover each of those points by a member of the countable basis okay and you can choose different members from the countable basis because the points are separated from each other because they are isolated and in this way you get a mapping an injective mapping from the set of isolated points in this case these are the poles to the countable base okay and the moment you get an injective map what it tells you is that whenever you get a map from a set to an injective map from a set to another set which is which is countable then your original set is also countable because a subset of a countable set is countable okay therefore you see the first observation is that a meromorphic function will have only countable in many poles and then if you are looking at a meromorphic function on the whole Riemann sphere okay which means you are taking the domain in the extended complex plane often I keep saying Riemann sphere because I think of the extended complex plane as Riemann sphere they are one and the same so if your domain is the whole extended complex plane and you are looking at a meromorphic function on the whole extended complex plane you get more what you get is not only is a set of poles countable it is actually finite because an isolated set of points in a compact set second countable set is going to be only finite okay because you know one property of compactness is that if you had an infinite subset it should have an accumulation point okay therefore the moral of the story is that if you are looking at a meromorphic function on the whole extended plane then it should have only finitely many poles okay so let me write that down so here are remarks so remarks number 1 the set of poles in D is countable by second countability okay so I am just saying that I am just using the fact that the poles are of course isolated and I am using second countability okay and the second remark is that if D is the whole extended complex plane then the set of poles is finite and that is because of compactness of the extended plane okay and then there is yet another fact if you see the set of poles they form of course they are set of isolated points because poles are by definition isolated but on the whole some of them might converge see they mean isolated points but they may get closer and closer and closer okay so like the sequence of real numbers 1 by n okay the sequence of real numbers 1 by n consist of you know it is a set of isolated points but it converges it converges to 0 okay so you could have a set of poles okay you could have a convergent sequence of poles and then the fact is that this convergent sequence of poles has to converge only on the boundary it cannot converge in the interior. The reason is that if it converged in the interior you are going to get a point which has to be in the domain of f so it has to be either singularity or it has to be a point of non-singularity but then you know it cannot be a point of non-singularity it cannot be a point of analyticity because it is approached very as close as you want by poles and if a point is a non-singular point there should be a disc around that point where there are no singularities but since this limit is being approached by poles this cannot be a non-singular point it cannot be a holomorphic point or analytic point it has to be therefore necessarily a singular point but then you assume that the singular points are all going to be poles and here you have gotten a non-isolated singular point that is a contradiction therefore if the set of poles of the meromorphic function has a convergent subset then it will converge only on the boundary okay and of course this cannot happen if in the case when the domain is the whole extended plane because when the domain is the whole extended plane okay the boundary is empty alright and there are only finitely many points okay. So well so let me write this down if the set of poles has convergent sequence of distinct points then it has a limiting point in the boundary of T okay that point cannot it cannot converge inside T okay so you know the so the moral of the story is following the moral of the story is when you are looking at meromorphic functions and you are of course if you are looking at meromorphic functions on the whole extended plane then there is not anything we are going to see that these are exactly the rational functions okay we are going to prove that rational functions which are given by you know quotients of polynomials they are exactly the same as the meromorphic functions on the extended plane there is nothing more okay but if your domain is not the extended plane okay then you then the meromorphic functions if you look at for example if you are looking at meromorphic functions on the unit disc okay then it is possible that you may have a sequence of poles that is converging and it will converge to a point in the boundary of the unit disc okay and behaviour at that point of this of this meromorphic function or this family of meromorphic functions is very very important topologically okay so this is very very important that you need to study the boundary points also okay especially when you have sequence of poles tending to the boundary okay and of course what the remark says is that the if there is a sequence of poles which is converging it has to go only to the boundary it cannot come cannot converge to your interior point okay well now so let me go ahead with this fact that I told you that you know rational functions of meromorphic and the converse holds a rational function is given by a quotient of polynomials f of z is equal to P of z by Q of z okay. So you know now I want to tell you something I am now all this time you know I was using the variable w I was writing f of w g you know the reason why I was using the variable w is because I wanted to study the point w equal to infinity and the way I would do that is by studying g of z equal to f of 1 by z which is f of w by making the transformation w equal to 1 by z okay but now we have now by now we understand how to deal with the point at infinity so I am switching back to the variable z okay so z will be our complex variable from now on so rational function is just a quotient of polynomials so P Q polynomials in z with coefficients in complex numbers of course you know quotient of polynomials of course includes even constants okay you can take Q to be 1 okay constants are also treated as polynomials of degree 0 okay in fact in particular 0 is also considered as a polynomial constant polynomial and therefore you must understand that constants are also meromorphic functions okay and of course you know of course a function rational function like this is certainly a meromorphic function on the extended plane that is very clear because if you take a function of this form if you take a function which is quotient of polynomials okay it is going to where is it going to where is it going to have problems with analyticity it is going to have problems with analyticity where the denominators vanishes so you take the those point and where the denominator vanishes is just the set of zeros of a polynomial that is only finitely many points okay and maybe some of these maybe the numerator polynomial also has zeros at some of these points so some of these zeros may cancel out okay but then the fact is that this function cannot have poles at points more than the set of zeros of the denominator polynomial okay and at all other points it is analytic so it is an analytic function defined on the whole you know extended plane with having only you know finitely many points which are going to be poles so it satisfies the definition of a meromorphic function so you see therefore a rational function is certainly a meromorphic function okay and the fact the theorem is that the converse is also true okay and so that is what I am going to talk about rational function is analytic on C union infinity except possibly at the zeros of the denominator polynomial hence it is meromorphic on the extended plane so the theorem is that the converse holds the converse holds so here is essentially here is a theorem if f of z is meromorphic on the extended plane then f of z is a rational function okay so there is no difference between meromorphic functions and quotients of polynomials okay there is absolutely no difference and you know now this should you should now think of it like this you if you look at the polynomials in one variable in say in the variable z with complex coefficients that gives you the polynomial ring in one variable over the complex numbers C square bracket z okay and you know that is an integral domain if you have studied that in algebra and then this integral domain has what is known as a field of fractions a quotient field this is just like you get the rational numbers as a field of fractions when you look at the integral domain consisting of the integers okay so if you look at the field of fractions of the polynomial ring in one variable over complex numbers you are going to get just quotients of polynomials okay and these are precisely the meromorphic functions on the extended plane so the moral of the story is that if you look at the extended plane okay the set of meromorphic functions automatically is a field it is none other than the field of fractions of the polynomial ring in one variable over C okay so you can see that so well so let us try to prove that theorem let me use another colour so prove so you see so I am given a function f z the function is supposed to be considered on the extended plane okay which means you have also considered the point at infinity okay and then you the only information that is given to you is that the function is meromorphic which means you know that it has only poles and since the set in consideration is the extended plane is compact you know I have already told you they are going to be only finitely many poles okay and it is possible that infinity may be a pole or not okay so first let us first deal with infinity first okay and so you look at the laura expansion of the function at infinity alright and you know that the laura expansion of the function at infinity will consist of both positive and negative powers of z okay and of course the constant term and you know the singular part at infinity is the part that consist of positive powers of z okay and you know since so now you see infinity there are 3 choices for infinity see infinity can be either removable singularity okay or it could be a pole okay and of course it cannot be an essential singularity because we have assumed the function is meromorphic so it cannot have any singularities other than poles okay now if infinity is a removable singularity okay it means that the if you take the laura expansion at infinity okay the principal part which consist of positive powers of z or there is a singular part has no terms okay and if you assume so it consist of only the constant part and the negative powers of z okay that is if you assume infinity is a removable singularity if you assume infinity is a pole which is only other possibility then you know that the principal part of the singular part consisting of positive powers of z has to be finite so it has to be a polynomial of positive degree okay without a constant term and of course the degree will be the order of the order of the pole at infinity okay so in any case you are going to get the singular part as either 0 or a polynomial that is what I am trying to say at infinity and if you take the function and remove that singular part whatever is left is going to be analytic at infinity that is what you must understand so you see this is something that we use repeatedly what is the point about the laura expansion at a point of a function the laura expansion consist of a singular part or principal part and an analytic part and if you take the function and subtract the singular part what you will get is only a Taylor series which will be actually the Taylor series of the analytic function which is given by the difference of the function and its singular part the moment you take away the singular part the principal part the function that the function becomes analytic okay so this is a trick that you always use if you want to extract the analytic part what do you do you take the function and subtract the singular part or the principal part okay so let me write that down so let us first deal with the point at infinity the point z equal to infinity if the laura expansion of f of z at infinity is f of z is equal to sigma n equal to minus infinity to infinity An z power n that is sigma so let me write it like this sigma n equal to 0 to minus infinity An z power n plus sigma n equal to 1 to infinity An z power n z power n of z is this and now let me call this fellow as P infinity of z okay this P infinity of z is what this is the singular or principal part at infinity okay so let me so this is the this is the singular or principal part at z equal to infinity okay and of course whatever is left out here this is the this is the analytic part okay so you see of course you should take this laura expansion to be valid for mod z greater than r for r sufficiently large so it is so valid for mod z greater than r are sufficiently large so this is very important okay and so if you take so the point is that so if you take f of z minus P infinity of z okay this is going to be this is analytic at infinity okay this is going to be analytic at infinity okay because f of z minus P infinity of z will only consist of the constant term A not and negative powers of terms involving negative powers of z and negative powers of z behave well at infinity okay and of course I should tell you something if you look at the point at infinity since we have assumed that f is meromorphic this P infinity of z is not actually a power series it is only a polynomial okay P infinity of z will be 0 if infinity is a removable singularity that is a point of analyticity for f and it will be a polynomial of positive degree equal to the order of the pole of f at infinity if infinity is a pole okay so let me write that down note that P infinity of z is 0 if infinity is a removable singularity of f and is a polynomial of positive degree equal to the order of f the order of the pole of f at infinity there is no other possibility you do not have the situation when infinity is an essential singular point okay because we have assumed f is meromorphic the only singularities that are allowed are poles okay fine so now what you do is you look at now let us look at the let us look at the other singular points see I have assumed f is meromorphic so it has only finitely many poles because it is meromorphic on the on the external plane which is compact that is very very important okay there are only finitely many poles in the usual complex plane now I forget the point at infinity because I have already dealt with it I now want to keep track of the points on the plane where f has poles there are only finitely many let me call those points z1 through let us say zn okay so let us write that down let z1, z2 and so on zn be the poles of f of z in the complex plane okay and again let me stress this is very very important that you are getting finitely only finitely many poles because you are you have assumed that f is meromorphic on the extended plane the compactness of the extended plane is doing a big job here otherwise you need not get finitely many poles okay. So so you take these poles now you know you whatever you did at infinity you do at each of those poles okay so take any of those poles then in a in a deleted neighbourhood of those poles each of those poles f admits a Laurent expansion and you know if you take the Laurent expansion if you take the singular part it is going to have only finitely many terms okay and though and of course the highest negative power occurring will be equal to the order of the pole okay so so let the Laurent expansion of f of z around zk be f of z is equal to sigma does m, m equal to minus infinity and I will call the call I will call the question is a mk okay a mk and mind you it will be z minus zk to the power of m okay this will be the Laurent expansion the Laurent expansion centered at zk okay and valid in a deleted neighbourhood of zk and of course you know that will be actually a disc centered at zk and radius will be equal to the distance from zk to the nearest of the other zj's okay so valid valid in 0 less than mod z minus okay where of course rk I am not writing it rk is the distance of zk to the nearest of the other zj's j not equal to k okay and what is the principal part at zk it is going to have only finitely many terms okay and the highest negative power of z minus zk you are going to get that is going to be the order of the pole of f at zk so the principal part the principal part of f at zk is therefore pzk of f which is going to be let me let me write as pzk of z that is going to be sigma is going to be m equal to minus 1 to minus let me put tk amk into z minus zk to the power of m okay mind you where tk is equal to the order of the pole zk of f mind you these are all all the powers of z minus zk are in the denominator I am starting with m equal to minus 1 and going all the way up to minus tk and of course the a minus tk, k is not 0 a minus of tk, k is not 0 I mean this is the coefficient of the highest negative power of z minus zk right well so this is the principal part at zk and now what you do is that you do the same trick as before you use the fact that if you take the function and subtract away take away the principal part whatever you are going to be left with is going to be analytic because it is going to be a power series so it is going to represent an analytic function whose Taylor expansion at zk is exactly that power series which is the analytic part of the lower expansion see the law you must always remember this analytic part of the lower expansion is actually the Taylor series of what analytic function it is the it is the Taylor series of the analytic function which is given by the original function minus the principal part okay so let me write that down we need to use it f of z minus p zk of z is analytic at zk okay. So you see so now look at the look at the scenario the scenario is I have this meromorphic function f there are these poles finitely many poles z1 through zn okay and at each of these poles there is a principal part okay and if you take the principal part away what you get is something that is analytic at that point and of course there is also principal part at infinity okay. Now what I do is I take the function and remove all the principal parts okay you take the function, subtract the principal part at z1, subtract the principal part at z too and so on you subtract the principal part at infinity you subtract all the principal parts and what are you going to get? You are going to get an entire function on the whole Riemann sphere and what is it going to be? It is going to be a constant because of L'Huile's theorem. An entire function which is if you are going to get an analytic function on the extended plane which means you are getting an entire function which is analytic at infinity and entire function analytic at infinity by L'Huile's has to be a constant it is a bounded entire function analytic at infinity means bounded at infinity bounded at infinity means it is a bounded entire function and it is constant. So the moral of the story is you take the you take this Meromorphic function and subtract all the principle parts you are going to get a constant and now you push all the principle parts to the other side you will get that the Meromorphic function is a constant plus all these principle parts but each of these principle parts are rational functions and the constant is also a rational function so you have expressed the Meromorphic function as a sum of finitely many rational functions therefore it is rational and that proves the theorem okay so that is all. So let me write that down now consider g of z is equal to f of z minus sigma k equal to 1 to n p z k of z so these are the principle parts at those n poles and then also take away p infinity of z okay. Now what you get is g of z is analytic on the whole extended plane and by Liouville therefore it is a constant g is analytic on c union infinity hence a constant by Liouville so f of z is equal to constant plus sigma k equal to 1 to n p z k of z minus plus p infinity of z which is Meromorphic on the extended plane and that is the proof of that theorem that a function which is Meromorphic on the extended plane is just in fact what I want to say is not of course it is Meromorphic point is it is rational okay that is the point we wanted to show that a Meromorphic function on the we wanted to show that the Meromorphic function on the extended plane is a rational function okay. So what you have got is you have proved that a Meromorphic function f is constant plus these principle parts finitely many principle parts so this part p infinity of z is a polynomial okay it could be 0 and these are all involving negative powers of z minus z k's finitely many for each k okay and this is of course a rational function if you take LCM you will see that you will get a quotient of polynomials therefore it is a rational function and the beauty of this proof is that this proof also tells you that you get for every Meromorphic function you get a partial fractions decomposition the each p z k they are all the various terms of the partial fractions decomposition so this proof in one stroke tells you that a Meromorphic function has a partial fractions decomposition and is actually a rational function okay that is the advantage of this proof okay so I will stop with that.