 Yeah, so let us continue with our discussion which is about trying to see what kind of results you can expect for a function which is analytic at infinity. So I told you last class I mean in the last lecture that you know if a function is analytic at infinity then you cannot expect Cauchy's theorem to work okay and the easiest demonstration of that is taking the function 1 by z okay which is analytic at infinity but if you integrate it over a simple closed curve in a neighbourhood of infinity you are going to get there is a possibility that you do not get 0 okay. So Cauchy's theorem will fail could fail but what works is and before that we saw that you know if you have a function is analytic at a point in the plane then you know that all its derivatives are also the all the derivatives exist derivatives of all orders exist and they are also analytic at that point that anyways true for a function which is analytic at infinity if a function is analytic at infinity then all its derivatives are also analytic at infinity and we in fact those derivatives will add the derivatives at infinity will be 0 and the crucial point to note that is that you do not define actually the derivative at infinity it does not make sense but you deduce that the derivatives are all 0 at infinity okay. And then the last thing that I said was about Morera's theorem okay and that Morera's theorem actually works okay even for a function which is analytic in a domain in the external complex plane okay so let me explain that so suppose f of z is analytic in a domain in infinity which is external complex plane okay and integral over gamma f of z dz is 0 for every simple closed curve in that domain so then I should not say analytic which is it should be it should be the conclusion suppose f of z is continuous so I should so let me remind you that the whole point about Morera's theorem is that you are trying to get a converse of Cauchy's theorem okay and the Cauchy's theorem is that the integral over a closed curve is 0 okay integral over a simple closed curve is 0 and you want to say that if and that is true for an analytic function for a function is analytic on and inside a simple closed curve then the integral over that curve is 0 but you want a converse that is you want to say that if a function if the integral over every curve is 0 every simple closed curve is 0 then the function is analytic then you will have to add this condition of continuity because if you do not add the condition of continuity then you can get into trouble okay. So that is also the same statement in the case of a domain including the point at infinity suppose f is continuous in a domain including the point at infinity and you know the integral over any simple closed curve is 0 then f is analytic in that domain that is the statement but the point is I want to stress that when you say integral over a closed curve and you are saying for a curve in the domain which contains a point at infinity you must understand that this is not a curve that will pass through infinity okay. So when I say when you say a curve in a domain okay it could in principle pass through any point of the domain but when I say a curve in a domain in the external complex plane the domain in the external complex plane could contain the point at infinity and in the case when it contains the point at infinity when I say a curve in the domain it necessarily means that I am not thinking of a curve passing through the point at infinity it does not make sense okay. So whenever we talk about simple closed curve whether it is in the complex plane or in the external complex plane a simple closed curve is always a simple closed curve in the complex plane okay even if you are looking at a domain in the external complex plane which includes a point at infinity a simple closed curve always means just a simple closed curve in the usual complex plane the point at infinity is not involved okay. So here is a conclusion the conclusion is that if f of z is continuous in a domain in the external complex plane and integral over every simple closed curve in that domain of f is 0 then f is analytic in that domain. So this is Moray-Ram's theorem and the fact is that and the proof is pretty easy as they was saying by words last time the proof is the proof uses the usual Moray-Ram's theorem and then and it uses the definition of being analytic at infinity okay. So let me write this f is analytic on the domain so you know so let me call this domain as let me call this domain as D so f is analytic on D minus the point at infinity okay throw away the point at infinity f is analytic on that by the usual Moray-Ram's theorem of course by usual Moray-Ram's theorem I mean the version of Moray-Ram's theorem that you study on the complex plane okay it is analytic because it is continuous if you even if you throw if it is if a function is continuous on a domain then it is also continuous on every subset of that domain. So if a function is continuous on the domain D then if you remove something from D it continues to be continuous okay the restriction of a continuous function is always a continuous function. So the function continues to be analytic in the domain with the point at infinity thrown out okay and that is a domain in the usual complex plane and integral over every close curve is 0 that is one of the conditions of Moray-Ram's theorem. So usual Moray-Ram's theorem by that I mean the version of Moray-Ram's theorem for the usual complex plane that works okay and f becomes analytic and because f becomes analytic on D minus infinity what it tells you is that infinity is a singular point okay it tells you that infinity is a singular point because D minus infinity is a neighbourhood of infinity okay and of course you know in all this I am assuming that infinity is in the domain if infinity is not in the domain there is nothing to prove okay because it is usual Moray-Ram's theorem which we assume okay so I am assuming that infinity is in the domain and if infinity is in the domain then infinity is an interior point and if I say that outside infinity the function is analytic that means the analytic function is analytic in a neighbourhood of infinity the domain itself any domain which contains infinity is a neighbourhood of infinity and f is analytic in that neighbourhood of infinity in the deleted neighbourhood of infinity and but there is also this extra assumption that f is continuous at infinity because f is continuous on the domain it is continuous at every point in the domain so infinity is also there in the domain so f is continuous at infinity but then you know be cheated by saying that continuity at infinity is same as analyticity at infinity and this is kind of inspiration we drew from the removable singularity theorem in the usual complex plane. So what you get is that first you get that if you throw the point at infinity out the function is analytic okay and that will tell you that the point at infinity is a singular point is an isolated singular point and then continuity at infinity will tell you that it is also analytic at infinity and therefore it is analytic at all points in the domain and your new version of Maria's theorem works okay. So let me write this down by usual Maria's theorem I mean that is for domains in the complex plane because D minus infinity is a domain in the complex plane thus infinity is an isolated singular point of f but f is continuous at infinity so it is analytic at infinity so that is the proof and you see the what you see that is that you know it is just being one is just being very clever when I say infinity is an isolated singular point and then when I say f is continuous at infinity you know that means I am actually saying that f is infinity is a removable singularity and infinity being a removable singularity was a clever way in which we defined f to be analytic at infinity okay and the inspiration was taken from the removable singularity theorem because you know the straight forward way of trying to say that the function is analytic at a point if it is differentiable at that point and also in a neighbourhood of the point will not work for the point at infinity because you cannot define the derivative at infinity at the point for the point at infinity that is the problem okay fine. So Maria's theorem works now what I want to tell you about is about a problem that I want you to try so here is a problem for you to try let D be a domain in C or C union infinity let f f of z be analytic in D let f of z be continuous in D suppose there exists an n greater than 1 such that of course when I say n greater than 1 it is an integer n greater than 1 n is not a real number okay greater than 1 suppose there is an integer n greater than 1 such that f of z to the power of n is analytic in D show that f itself is analytic in D okay so this is a problem which I want you to try okay so the idea is very simple you have a domain of course non-empty connected open set the domain you can take the domain either in the complex plane or you can include the point at infinity if you want first maybe you first try it for domain in the complex plane and then you will see that the same arguments will work for the point at infinity suppose f is continuous in D and there exists an integer power of f greater than 1 which is analytic then f itself has to be analytic so we can try this now what I am going to do is well we have now more or less seen something about the infinity being a point of analyticity or is the same as saying infinity is a removable singularity the next kind of singularity is that of a pole okay so let us go to the study of infinity as a pole okay so let me go to the next page infinity as a pole so here is a situation you have a function f which has infinity as an isolated singular point okay that means it is defined in a deleted neighbourhood of infinity which you should think of as a the exterior of a sufficiently large circle on the complex plane okay and the question is that when is infinity a pole okay so how do you define this so the of course there are the way we do it is that we call the independent variable as W we write the function as f of W and then we say the behaviour of f of W at W equal to infinity is the same as the behaviour of f of 1 by z at z equal to 0 because you are plugging in W equal to 1 by z okay so this is a philosophy which we have justified earlier because W equal to 1 by z is a is a homeomorphism of the extended complex plane onto the extended complex plane which is an analytic isomorphism of the punctured plane onto itself okay so under an analytic isomorphism the type of singularity must coincide okay so let me so let me do that so how do you define f f has f of W has a pole at W equal to infinity this is the definition this is I will put d f definition if and only if g of z which is f of 1 by z has a pole at z equal to 0 so this is the way we do it okay but what are these conditions so you know so what is a condition that a function g has a pole at a finite point of the complex plane there are 3 equivalent conditions you know one is based on the limit of the function as we approach that point the limit should tend to infinity that is one way of looking certifying that something is a pole. The other way is to look at the Laurent series at that point and then see that there are only finitely many terms in the singular part and the third one is to say that you know think of your pole of certain a pole has to have a certain order okay a pole has to have a certain positive integer order and that positive integer is actually the power of the variable minus the centre which you have to multiply with the function to neutralize the pole because it is like if a function has a pole of order n at z not okay then multiplying the function by z minus z not power n should neutralize the pole okay so you neutralize the pole by multiplying by the by killing the 0 of the denominator okay so I will write down all those 3 conditions so the first thing is there exists an n greater than 0 such that you know z power n g of z limit z tends to 0 is equal to is non-zero okay is non-zero and of course in fact it is a non-zero complex number it is not when I say it is non-zero it cannot it cannot be I do not I do not want this to be infinity so this should belong to the punctured complex plane okay so this is one definition of pole of order n g this is one definition of g having a pole of order n at z equal to 0 okay had it been a pole of order n at z equal to z not you know you will have to modify it to limit z tends to z not z minus z not to the power of n times g of z is a non-zero complex number okay so this is one definition okay then the other definition is of course it is it is a definition that has got nothing to do with the order of the pole it is the fact that the function approaches infinity as you approach the pole which means that the function in modulus approaches infinity as you approach the pole so that second condition is limit z tends to 0 g of z is infinity okay and mind you you should think about this in terms of what in terms of a limit being infinite that we have defined earlier okay so the way this should be made sense of is that as z comes closer and closer to 0 that is if z is restricted to a small deleted neighbourhood of 0 then the values of g a z will lie in a small deleted neighbourhood of the point at infinity which is supposed to be the exterior of sufficiently large circle in the complex plane okay so that is what that is how you should think of this and well the third condition is that the principle part of the singular part of the Laurent series of g at the origin has only finitely many terms okay and the highest negative power is of course going to be you know the subscript is going to be this n which is the order of the pole okay or rather minus of that subscript okay so or rather let me say this the you take minus of the highest negative power of z that should be n okay. So the Laurent series of g of z at 0 has only a finite singular or principle part and what you must remember is that this is the same as saying that it has only finitely many terms involving negative powers of z okay now the fact is that each of these definitions have their analogues for f at infinity okay having a pole at infinity and well you know I just want to point out few subtleties so you know for example the one in the middle goes through very easily limit z tends to 0 g of z is infinity is the same as saying limit w tends to infinity f of w equal to infinity this is direct because basically z is 1 by w and z tends to 0 is equivalent to w going to infinity and the limit going to and so I just make a change of variable from z to w by putting w equal to 1 by z or z equal to 1 by w okay. So these two are one and the same is obvious okay and especially you know again you are using the fact that w is equal to 1 by z which is z going to 1 by z is a homeomorphism okay that is there that is implicitly used here okay essentially because basically under continuous map the image of a convergent sequence is a convergent sequence okay so this is very very clear so if you want to test of whether a function is having a pole at infinity you just have to see whether it goes to infinity as you approach the point at infinity okay so and what does it mean in terms of the topology of the point at infinity it means that if mod z is sufficiently large then mod I mean if mod w is sufficiently large then mod f w is also sufficiently large you can make mod f w as large as you want provided you choose mod w as large as you want that is just a verbal restate statement of saying that f tends to infinity as w tends to infinity okay. So the one this and the center is pretty easy the one in the top needs a little bit of thought you see if you want to neutralize this pole of order n of g at 0 mind you a pole of order n should be thought of as a 0 of order n of the denominator or of the reciprocal of the function so if you want to neutralize a pole you should multiply a power by a power of the variable okay or the power of variable minus the center at the point which is a pole okay but to do it at infinity you can imagine you have to divide okay if you want to utilize a pole at infinity okay you will have to divide so that you can see by simply substituting z equal to 1 by w in this so this is equal to limit w tends to infinity f of w by w to the power of n is a non-zero complex now okay. So this is the point that you have to notice to neutralize a pole at a finite point in the complex plane a finite point z not in the complex plane you have to multiply by a sufficiently high power of z minus z not okay and in fact it is a particular power of z minus z not okay and that is the order of the pole okay at z not but if you want to neutralize the pole at infinity you have to divide by the variable by the correct power of the variable okay so for example the simplest case is take the identity function f of w equal to w okay that has a pole at infinity because f of w equal to w if I take limit w tends to infinity I am going to get infinity limit w tends to infinity f of w is just infinity okay so infinity is a pole and it is a pole of order 1 because I can utilize it by dividing by w you take the function f of w equal to w divide by w I will get 1 now that is constant function which even at infinity remains 1 okay so you can see that in this way you can see that if you take a polynomial of degree n if you take a polynomial of degree n in w polynomial of positive degree in w that has a pole at infinity and the order of the pole will be simply the order of the degree of the polynomial okay so if you have polynomial of degree n in w if you divide it by w power n and then take limit as w tends to infinity you will see that you will end up with you will end up with a nonzero complex number okay so polynomial of degree n has a pole of order n at infinity okay that is something that you can see and coming to this last condition on the left which is that the Laurent series of g at 0 has only a finite singular or principal part what will that translate to it will translate to the fact that the singular part of the Laurent series of f at infinity okay will have only finitely many terms okay but mind you the singular part of a function at infinity is supposed to be the part that consists of positive powers of the variable okay see you must remember that the singular part and the analytic part they will interchange if you move from 0 to infinity okay so at 0 it is the positive powers of the variable that behave well and the negative powers of the variable do not behave well at infinity it is the other way around at infinity it is the negative powers of the variable that behave well and they form the analytic part along with the constant term okay and the positive powers of the variable misbehave and they form the singular part so saying that the right way to say the function has a pole at infinity is to say that if you take the Laurent expansion at infinity then there are only finitely many positive powers of the variable okay and that is equivalent to saying that the singular part at infinity or the principal part of it at infinity is a polynomial okay so let me write that down the singular part or principal part consisting of positive powers of W of the Laurent series at infinity is a polynomial of course you know the constant term is not included the constant term is taken to be part of the analytic it is part of the analytic part so the constant term which is the 0th power of the variable okay the question of the 0th power of the variable and then and the other question and the other term which involve the negative powers of the variable okay powers of 1 by W these form the analytic part at infinity okay and the positive powers of the variable okay they form the singular part at infinity and that has to have only finitely many terms which means that it is it has to be a polynomial okay a polynomial constant term 0 okay and of course the degree of that polynomial will be the degree of the pole okay so now and because of all this what you can say is that therefore you know these three are you know these three are equivalent so let me write let me put that here this is equivalent to this and this is equivalent to this and I will put therefore the point is that somehow you may have to go to the behaviour at 0 and mind you even at that these three conditions for a pole in the finite complex plane that they are equivalent if you have tried to write down a proof of the equivalence you will see that somewhere you might use a removable singularity theorem okay so that is something that you should try to do okay if you have not done it fine so so so the so the model of the story is you think of a function having a pole at infinity essentially to be polynomial plus something which is analytic at infinity okay so the model for the function having a pole at infinity is a polynomial so polynomials are functions which have a pole at infinity okay and the degree and the order of the pole at infinity is just the degree of the polynomial okay and therefore you must understand that you know and mind you these are entire functions polynomials are entire functions so the model of the story is that if you take an entire function okay and if it has a pole at infinity you should expect it to be only a polynomial okay and more generally this is not exactly correct you what you could get is that you could get quotient of polynomials also with the numerator polynomial bigger than the denominator polynomial okay that is also possible because that is also something that will as an improper rational fraction will work out to something that is a proper fraction plus a polynomial and the proper part will be analytic at infinity okay so so let me let me let me let me say that more in more detail so here is so here is a remark if f of w is an entire function with infinity as a pole then f is a polynomial in w and conversely okay so the converse is something that we have just seen if f is a polynomial then a polynomial is of course entire and a polynomial is does have a pole at infinity of course you know in all these things you have to assume that you are not working with a constant function because a constant function should always be excluded so because even constants are thought of as polynomials so let me just put non-constant entire function so let me write non-constant here okay let me write non-constant here mind you if you have an entire function which is analytic at infinity it has to be constant okay this is a non-constant entire function will have a problem at infinity it will have a singularity at infinity and that infinity that singularity cannot be removable okay that is just another avatar of Liouville's theorem okay so so how do you prove this proof is very very simple well let me say it in words f has a pole at infinity so if you write the Laurent expansion at infinity the singular part is a polynomial okay and then there will be an analytic part which consists of constant and it will consist of negative powers of w okay but then this is also supposed to be analytic at 0 so you cannot really have any negative powers of 0 because they will negative powers of the variable because they will simply the negative powers of the variable will simply misbehave at w equal to 0 so all the coefficients of the negative powers of variable have to vanish which means that f is essentially a polynomial so it is pretty easy to see that this is true okay so let me write that down for the Laurent expansion at infinity has only a polynomial as a singular part as the singular part and the analytic part at infinity cannot be non-constant since terms involving 1 by w will render f singular at 0 okay so that is the proof okay so let me again repeat it you write f of w you write it as positive negative powers of w including a constant then the positive powers of w that can be only finitely many because that is what will happen if f has a pole at infinity so the positive powers of w they will be polynomial okay and the constant and the negative powers of w that will be the analytic part at infinity okay but then this expansion should be also valid at 0 so at 0 negative powers of w cannot be allowed okay so there are no negative powers of z as a result you see that the analytic part of f is just a constant okay so the analytic part is a constant the singular part is a polynomial when you add it together you get a polynomial so f has to be a polynomial okay so the only functions entire functions non-constant entire functions which are analytic which have a pole at infinity are polynomials okay and of course if you relax the condition that f is entire but you add the condition that f is meromorphic okay which means that it has only isolated singularities which are poles in the plane then you can prove that f has to be rational function okay it has to be a quotient of polynomials with the degree of the numerator polynomial greater than the degree of the denominator okay so this is something that you can show but we will come to it because you know eventually we have to come to meromorphic functions that is what I want to do and you know just to connect things back in our discussion you know we are trying to prove the Big Picard theorem and the Big Picard theorem involves looking at functions which have essential singularities at infinity okay and or probably functions which have poles at infinity and basically you want to study meromorphic functions and families of meromorphic functions you want to do topology on and in fact study compact families of meromorphic functions compact spaces of meromorphic functions so but in all these things you want to be able to work with infinity very easily you want to work with the point at infinity in a very easy way and that is the reason why we are analyzing the point at infinity in so much detail okay fine so now that we have this so let me write that as well so here is another remark suppose f of w is a function which is meromorphic on the complex plane which means that it has only poles as isolated singularities in C and no non-isolated singularities okay. So this is what a meromorphic function is it has only the only singularities it has they are all isolated so it has no non-isolated singularity and every isolated singularity is a pole okay this is what a meromorphic function is and then you also assume that infinity is an isolated singular point and assume that infinity is a pole okay then the claim is that f is quotient of polynomials with the numerator polynomial having degree greater than the denominator polynomial okay so let me write this then f has pole at infinity if and only if f of w is equal to p of w by q of w where p, q are polynomials with degree greater than degree q okay and f has a removable singularity at infinity if and only if f of w is of the same form p of w by q of w but now the degree of p is less than or equal to degree of q with p, q polynomials degree of p is less than or equal to degree of q okay so basically this remark tells you the obvious thing namely you know if I if you take two polynomials okay and you take their ratio you take p of w and q of w they are two polynomials and you take p of w by q of w you know that p by q will be an analytic function at all points where q is not 0 okay and the points where q is 0 they will be poles okay and so p by q will be a meromorphic function okay and this meromorphic function will behave well at infinity if the degree of p is less than or equal to degree of q okay and that is because if the degree of the numerator is less than or equal to degree of denominator as the variable goes to infinity you will get a limit okay whereas if the degree of p is greater than the degree of q this quotient will go to infinity as you go to infinity so infinity will become a pole okay and that is exactly what this remark says okay. So you can try this it is pretty easy to do but we will come back to it later now what I want to shift my attention is to do something connected with residues okay so you know one of the ways of trying to study a function at a singular point is by the so called residue or the function of that point why is the residue important the residue is important because the residue actually gives you the integral of the function along a simple closed curve that goes once around that point okay so the whole importance of the residue is because it will allow you to integrate the function around a singularity that is why residues are important and that is why you use residues to compute so many integrals in a first course in complex analysis you even compute many real integrals by converting them as real parts or directly into complex integrals and then applying the residue theorem or the Cauchy theorem okay so the moral of the story is that residues are important just because you can integrate a function around a singularity okay and you know so now if you think of the point at infinity as a singularity then it is natural that you will ask about the residue of the function at infinity okay so you can talk about the residue or the function at the point at infinity and what do you get so the answer to that is something very nice actually tells you in a way why Cauchy's theorem fails for the point at infinity the beautiful thing is that since Cauchy's theorem has failed for the point at infinity you think you have lost something but the point is you are getting it back in a different way there is a so called extended version of the residue theorem it is called the residue theorem for the extended complex plane which tells you that you should expect the Cauchy's theorem to fail at infinity okay so if I tell you in a nutshell if I want to tell you in a nutshell what the residue theorem for the extended complex plane is it is very very simple it says take a function on the extended complex plane which has only isolated singularities okay then mind you if it has only isolated singularities there are only finitely many okay because you are looking at an isolated subset of a compact set the extended complex plane is a compact set mind you the extended complex plane is a one point compactification of the complex plane it is a compact set home it is homeomorphic to the Riemann sphere okay which is compact okay and if you take a if you give if you take an isolated subset of a compact set you should get only finitely many points okay so the moral of the story is that if when I am looking at a function on the extended complex plane which has only isolated singularities there are only finitely many singularities okay and then one of the singularities could be the point at infinity so that function may have infinity as a singular point or it may not have infinity as a singular point but the beautiful thing is the residue theorem says now if you integrate over along any simple closed curve okay and assume that the simple closed curve goes around all the finite singularities okay all the singularities in the finite complex plane okay the residue theorem says the sum of all the residues okay inside the curve plus the residue at infinity will add up to 0 the total sum of residues is 0 that is the residue theorem for the which includes a point at infinity the residue at infinity has to neutralize the sum of the residues at the finite plane many points see that is the reason you see when I try to integrate 1 by W over a sufficiently large circle okay I get if I put positive orientation for this if I put negative orientation for the circle so that it the infinity lies in the interior of that circle okay which actually becomes exterior because of negative orientation then what happens is that I do not get 0 okay even though 1 by W is analytic at infinity I get minus 1 and what is that minus 1 that minus 1 is to it is to neutralize the plus 1 which is the residue at 0 of 1 by W and you see a sum of residues of 1 by W at 0 which is plus 1 and the residue at infinity which is minus 1 they add up to give 0 and this is exactly demonstration of the residue theorem for the external complex plane so the fact is that if you take a function which is analytic at infinity the integral at infinity does not vanish because it actually tries to you know it neutralize the sum of residues at finite at the finite points at the points of the complex plane which are singular points okay that is the reason why the integral at infinity is not giving you 0 that is the reason why Cauchy's theorem fails so Cauchy's theorem fails but it is not in vain you get the residue theorem for the external plane okay so I will elaborate on this next class okay.