 So you see what we did last time was to try to define when the when an analytic function is having infinity as a point of analyticity okay and so as I told you if you recall one of the standard ways of defining analyticity at a point is to say that the function is differentiable at that point and also in a neighbourhood at every point in a neighbourhood of the given point okay. So this is how you define analyticity at a point in the complex plane okay but since we are worried about analyticity at infinity okay what we do is that we tend to look at the point at infinity as a singularity, a singular point of an analytic function which is defined in a deleted neighbourhood of infinity which means that the function is analytic for mod z greater than r for r sufficiently large outside a circle of sufficiently large radius the function you are given a function which is analytic and then infinity the point at infinity becomes an isolated singular point and then you want to say the function is analytic at infinity so it would not help to say that the function is differentiable at infinity because if you write the differential limit it does not make sense at infinity. So what is the way out? The way out is to actually get the draw inspiration from Riemann's removable singularity theorem. Riemann's removable singularity theorem tells you that if you are looking at an isolated singularity of an analytic function at a point in the complex plane then saying that the function is analytic at that point namely that the which is essentially saying that the function can be extended to an analytic function at that point including that point okay which is one of the definitions of what a removable singularity is okay is equivalent to requiring that the function has a limit at that point it is which is equivalent to the continuity of the function at that point in fact it is also equivalent to the function being bounded in a neighbourhood of that point bounded in modulus of course. So this is so the so the model of the story is that you can use these conditions to define the function to be analytic at infinity okay what you can say is that though it will not it does not make sense it will not work to say that the function is differentiable at infinity you can always say that the function has a removable singularity at infinity in the sense that the function either has a limited infinity so the limit as z tends to infinity f z exists okay that is one condition the other condition is the function is bounded in bounded at infinity that means there is a deleted neighbourhood of infinity where the function the modulus of the function can be made less than a positive constant okay and these conditions are all one and the same okay and why these conditions are one and the same is because of this other important philosophy that the studying the function at infinity studying f of w at infinity is the same as studying f of 1 by w at 0 okay. So and I told you that the rational or the justification for that is that z going to w equal to 1 by z is actually a homium of some of the Riemann extended complex plane onto the extended complex plane which interchange is 0 and infinity okay and further if you throw out the point 0 and infinity then you get the punctured complex plane the complex plane punctured at the origin this map is it going to 1 over z which is equal to w is going to be an analytic holomorphic isomorphism of the punctured complex plane onto itself okay and under such an analytic isomorphism the nature of the singularity at 0 and the nature of the singularity at infinity which is the image of 0 they should correspond okay this is the philosophy that we use now what I want to say is that I want to go ahead with go ahead with this and so now that we have defined a function being analytic at infinity okay and the weakest definition for a function being analytic at infinity is that it is bounded at infinity namely there is a point I mean there is a small neighbourhood of infinity okay which should be thought of as all z such that mod z greater than r for r sufficiently large then mod fz should be made you should be able to make mod fz less than m okay for some positive constant m okay so that is exactly what you want to do so what I want to do next is I want to worry about how I want to worry about what it means to say that a function has a removable singularity at infinity okay so we want to analyse this right so let us take the let us take the so we will look at the case of an entire function alright but even before that I wanted to say that you know this one more aspect that we need to actually look into okay and this aspect is about this aspect is about the Laurent series okay so you see you know the in some sense how do you study the isolated singularity of a function at a point in the complex plane at a point z not in the complex plane one of the ways of studying this is by looking at the Laurent expansion of the function centred at z not that means you expand the function in positive and negative powers of z minus z not that is the Laurent expansion you get the the coefficients are the Laurent coefficients and the Laurent theorem says that there is such a Laurent expansion okay which is in general valid in an annulus alright and then you know that the nature of the Laurent expansion to be more specific the nature of the principal part or the singular part of the Laurent expansion will tell you what kind of singularity z not is so you know that if the Laurent expansion has only negative powers I mean it has only finitely many negative powers of z minus z not you know it is a pole if it has no that is if the principal part has only finitely many terms then it is a pole okay if the principal part does does not exist namely if the principal part is 0 then it is a removable singularity okay and if the principal part has infinitely many negative powers of z minus z not then it is an essential singularity okay this is this is the trying to classify singularities based on the Laurent expansion okay which is something that you know now the question is what is the analog this of this when z not is a point at infinity not a not a when z not is a point in the complex plane but when z not is a point at infinity so how do you get a handle on this how do you get a handle on this so you see so let me write that down so my point is what is this Laurent series Laurent series at infinity which is rather funny thing again you know the you must remember that the way you deal with infinity is you have to be very clever there are certain things you can do with infinity there are certain things you cannot do with infinity so for example when you want to define a function to be analytic at infinity you do not take the root of trying to define it differentiable at infinity because derivative at infinity is not defined it is not easy you cannot define it so easily right of course you can go to another level of abstraction called what is called of what is called a Riemann surface okay and you can make the you can make the complex plane along with a point at infinity namely the external complex plane into Riemann surface and once you have a Riemann surface and you have a holomorphic function then you can talk about derivative at any point by using local coordinates okay but then we do not want to get into that amount of generality you can look up that point of reasoning if you look at my video course on Riemann surfaces which is available on the web okay but then let us not go to that level of abstraction okay at this moment at this level of our exposition we do not try to define derivative at infinity but trying to say it is analytic infinity we get away by using the drawing inspiration from Riemann's removable singularity theorem okay by simply for example just requiring the function is continuous at infinity okay so in the same way if you look at the Laurent series at infinity how do you define this so you know that see the idea of so you have to think like this the idea of what is a Laurent series see the whole point about Laurent series Laurent series is a generalization of the Taylor series okay and what is the Taylor series a Taylor series is trying to express an analytic function in terms of simple analytic functions functions is a simplest possible analytic functions so you know if when I say Taylor series of an analytic function at a point Z not in the complex plane I am simply expanding the function in powers of Z minus Z not and I know Z the powers of Z minus Z not they are this they are simple functions they are simple polynomials okay and I am trying to expand I am trying to write the function the given analytic function as a limit of polynomials after all a power series for that matter any functional series by definition when it converges is just the limit of partial sums okay and if you take a Taylor series or a power series then the partial sums are all polynomials okay and it is the limit of these polynomials that gives you the given function okay which is the limit of the series. So the purpose of a Taylor series is to expand a function as a series in terms of simple analytic functions that is the idea okay and this is this is at a point where the function is analytic okay but suppose a point your in question is a point not a point of analyticity suppose it is a point where the function has an isolated singularity then what comes in is the Laurent expansion. The Laurent expansion gives says that well you will all you can still get a expansion of the function in the form of a series but then now you will have to allow also negative powers okay now so in general we think of philosophically we think of the Laurent series as a as a generalization of Taylor series and the guiding philosophy is that it number one is that it allows you to expand the function in terms of simple functions that is that is point number one. Point number two is that the Laurent series can be broken up in two pieces okay there is one part of the Laurent series which consists of positive and zero powers of z minus z0 where z0 is a point where you are looking at the centre of the series that is called the analytic part of the Laurent series okay and then there is also the part of the series that involves the negative powers of z minus z0 which you call as the singular part or the principle part of the Laurent series. So you see the general idea of the Laurent series that it breaks the function into two functions it breaks the functions into two pieces as a it expresses the function as a sum of two pieces one piece is the analytic part of the function at that point in the neighbourhood of the point the other piece is the principle part of the function at that point which is not analytic at that point okay now using these two guiding philosophies you can also define what a Laurent series at infinity means okay and well the point is that the point is as follows so you suppose f of z is analytic at z equal to infinity or let me not even start with analytic at infinity let me just say let me say it is analytic in a neighbourhood of infinity suppose f of z is analytic in a neighbourhood of infinity say in and you know for obvious reasons let me do the following thing let me not use z let me use w okay say in mod w greater than r okay. So I am using w as a variable because I will always you know when I want to study w at infinity I will rather study you know there is one of the tactics that we have been using is that you study w equal to 1 by z at 0 so that is why I want to reserve w for 1 by z okay so suppose f is analytic in a neighbourhood of infinity say in mod w greater than r so of course by this I do not mean that the function is analytic at infinity mind you okay so you know you have to be a little careful when I say a function is analytic in a neighbourhood of a point in the complex plane it is understood that the function is also analytic at that point okay but then when I am saying f of w is analytic in a neighbourhood of infinity I am not necessarily meaning that it is also analytic at the point at infinity the point at infinity could is a singularity okay it is an isolated singularity I do not know whether it is analytic or not okay. So let me state that we do not know we do not know if f is analytic at infinity okay now what do you do with this of course you know the let us go by the philosophy that you to study f of w at infinity you study f of 1 by z at 0 okay the behaviour of f of z f of w at w equal to infinity is the same as the behaviour of f of 1 by z which is g of z mind you f of 1 by z is the same as w f of w where w is equal to 1 by z at at z equal to 0 okay note that g g of z is defined in defined and analytic defined and analytic or holomorphic in a deleted neighbourhood of the origin which is just given by 0 less than mod z less than 1 by r which is actually writing this mod w greater than r in terms of z okay putting w equal to 1 by z okay. So and 0 less than 0 strictly less than mod z strictly less than 1 by r is a deleted neighbourhood of the origin which is circle I mean this is the interior of a circle with the origin to remove radius 1 by r okay. Now but then you know now you are looking at the point 0 in the complex plane and we have Laurent's theorem since now 0 is an isolated singularity of g of z okay and the idea is that the nature of the singularity of g at 0 should be the same as the nature of the singularity of f at infinity okay that is the idea okay and of course you know why that is correct because z going to w which is z going to 1 by z is an isomorphic okay of deleted neighbourhoods alright. So now so g has a Laurent expansion okay so g has a Laurent expansion has a Laurent expansion so what is the Laurent expansion it is g of z equal to sigma n equal to minus infinity to infinity a n a z power n this is the Laurent expansion of g okay and mind you I am the centre of the expansion is the origin normally if the centre is the point z not then you have to use powers of z minus z not but here z not is 0 so use powers of z and the point that it is Laurent expansion is that there are negative powers of z included as well that is why this summation is running from minus infinity to plus infinity and well so this is the this is g as it is now you know what you must understand is that you know if you look at if you look at this what does it mean for f so what this will tell you is that see this is valid for mod z less than 1 by r z not equal to 0 okay and if I plug in see of course instead of z I can put 1 by w okay instead of z I can put 1 by w and g of 1 by w is just f of w okay so this is the same as writing this is equivalent writing f of w is equal to sigma you know n equal to minus infinity to infinity a n w power minus n okay so I can simply replace z by 1 by w so z power n becomes w power minus n and summation will run from again minus infinity plus infinity the only thing is that because I have changed my variable the powers the power of the nth power of the variable has a negative subscript okay right so this is well this is the Laurent expansion but now so you know the point is that the Laurent expansion so this gives you a clue as to what you should call the Laurent expansion of f at infinity you can very well call this expression that you have written for f as a Laurent expansion at infinity okay because it is in line with the philosophy that it is expressing f as a series okay in terms of simple functions the functions are just powers of z okay so you can very well call this thing on the right this expression for f w as a Laurent expansion of f at infinity that is fair enough but then there is a little bit more to be seen you see if you write if you look at g of z the Laurent expansion splits into 2 pieces as I told you the Laurent expansion splits into a principal part which is a singular part plus an analytic part okay and the principal part of the singular part consists of negative powers of z so you know let me write this like this so this is sigma n equal to minus infinity to minus 1 a n z power n plus sigma n equal to 0 to infinity a n z power n okay there are 2 pieces and this fellow here is the so what I have written here is the principal part is the principal or the singular part at the at the origin okay and this part is the analytic part at the origin okay so you know if you want let me give symbols to these things so this is so if you want this is gs of z gs of z is the singular part okay and this is ga of z ga of z is the analytic part alright now let us make this change of variable which is z going to w which is equal to 1 by z okay and you know now watch carefully what is our philosophy our philosophy is that if you change variable from z to 1 over z okay then the behaviour at 0 at z equal to 0 should correspond to the behaviour of w equal to 1 by z at infinity therefore if you go by this if I transform gs of z to w which is 1 by z what I should get should be the singular part at infinity okay because gs of z is a singular part at z equal to 0 if I transform gs of z by putting z equal to 1 by w what I should get is a singular part at infinity okay and similarly if I transform ga of z by putting z equal to 1 by w I should get the analytic part at infinity so what do I get see basically what I get is I get f of w you see is so I will get sigma n equal to minus infinity to minus 1 An so I will get w to the minus n plus sigma here I will get n equal to 0 to infinity An w to the minus n okay and if you watch carefully so let me use a different colour at this point if you watch carefully now you see this guy here this corresponds to what is this this is just gs of 1 by this is gs of 1 by w okay because I put z equal to 1 by w and but gs was a singular part and therefore gs of 1 by w should also be a singular part so this should be in principle this must be equal to the singular part of f at f of w and what are you and watch carefully the singular part of f at w has what powers of w it has positive powers of w okay it has positive powers of w because n is negative so w to the n minus n is positive therefore the moral of the story is that if you write a if you write a Taylor series if you write a positive power series in a variable that at infinity corresponds to a singular part okay and that is that is very believable because as you go to infinity okay the partial sums which are polynomials are going to go to infinity so infinity is a pole actually the for the parties for the partial sums okay it is not bounded at infinity so it is correct okay so the whole point is that when you look at the so the Taylor series at infinity should be thought of I mean the when you look at the Laurent series at infinity the principle part should it will look like a Taylor series at the origin it is because it is exactly that by the transformation is it going to 1 over z okay so this is this is a singular part of f and then and then this guy here this is f a of w which is g a of 1 by w and this is the analytic part at infinity okay so you see f of w is also split into a singular part f s w plus f a of w which is an analytic part at infinity okay and mind you the analytic part at infinity contains all the negative powers including the constant because I have put n equal to 0 is also here so a not is here in the analytic part at infinity and so you see when you look at the variable at 0 okay then the analytic part consists of the non-negative terms and the singular part consists of the negative terms but when you look at the variable at infinity the analytic part consists of the negative terms and the including the constant and the principle part consists of the positive terms this is the this is exactly what happens and it is correct okay now why do you think that this analytic why do you think that the negative powers are the portion of the expansion which involves the negative powers is analytic at infinity that is correct because you see as a variable approaches infinity the negative powers approach 0 so you see that the it is bounded essentially okay so it is analytic and because our definition of analytic at infinity is either that it should be bounded or it should tend to a limit okay and no positive power of the of a variable will ever tend to will ever be bounded or will ever tend to 0 if you let the variable go to infinity okay so everything is fine so you know so here is the so here is our definition our definition is you take a function analytic you take a function is analytic in a deleted neighbourhood of infinity okay write out its laura expansion okay basically the laura expansion is the laura expansion of by gotten by changing the variable to its reciprocal okay and then what you do is you take the positive part of the laura expansion okay in the in the in the in the original variable and call that as a singular part okay and the negative part including the including the constant term is what is called the analytic part okay so now we have this clear definition of what a laura series at infinity should mean okay fine now once you have made this definition of laura series at infinity what you need to know is that whether this fits in well with our with with the theory that you do on the finite complex plane so for example you can ask this question suppose of you know for a point z0 in the finite complex plane a function is analytic at that point if you assume to begin with that point is an isolated singularity the function is analytic at that point if and only if you write the laura expansion about that point it has no singular part its principal part is 0 okay so if you go by that philosophy for a function f which is defined in a neighbourhood of infinity I mean for which infinity is an isolated singular point the function will be analytic at infinity if the laura expansion of at infinity has no singular part that is what should happen now does that happen it does because you see here is my function fw defined in neighbourhood of infinity its singular part is this is this part which consists of positive powers of w okay if that singular part is not there okay that means that I can express f only in terms of negative powers of say w then that is of course analytic at infinity because all these go to 0 as w goes to infinity okay so the moral of the story is that the our definition is correct so you know the point is that sometimes you may have to make definitions based on certain philosophy and then you have to check whether it matches with what what happens what you expect to happen as morally correct okay so if from this it is very clear that a function is analytic at infinity if and only if its singular part at infinity vanishes okay then if you take the laura expansion at infinity okay then its singular part vanishes so let me write a few things so this is the this is called the singular part of fw at infinity and this guy here is called the analytic part of fw at infinity and now you know if you look at it in a very simple way you know when you are looking at infinity the good functions are negative powers of w because they go to 0 and the bad functions are positive powers of w because they go to infinity so if you expect a function to be good at infinity it should be expressible only in terms of negative powers of w and that is why the negative powers of w along with the constant that part is the analytic part at infinity okay so the so the definition is very clear and with this definition you see that Rayman's removable singularity theorem is also valid in its various forms at the point at infinity a function namely a function which is which has infinity as an isolated singularity has that singularity as removable singularity if and only if it is bounded in a neighbourhood of infinity if and only if it tends to a limit at infinity and that is also equivalent to saying that the Lorentz series at infinity has no singular part they are all equivalents okay so you get the same version of the theorem as you would get in the case of a finite point a point in the usual complex plane okay so everything fits well the only thing that does not work is trying to define a derivative at infinity that does not work okay fine so now what I am going to do is I am going to ask this question as to what it means to having a removable singularity at infinity for example for a good function for example for a function like an entire function okay so what does it mean and you will see that that will throw up connections with Lieuwell's theorem and so on so see so let us analyze this suppose that f of w has w equal to infinity as a removable singularity okay so f w is f analytic so it has no singular part it has only the analytic part of its expansion and that is what this is writable as you know the analytic part of the expansion at infinity will involve negative powers of the variable and also the constant term so it will be n equal to 0 to infinity if you want I can call it as b n w power okay so this is analytic part at infinity now let us analyze what it means to say that the function is for example you know entire suppose so I am looking the following case suppose I have an entire function and suppose it is analytic at infinity what happens we will see that it will reduce to a constant okay and that is just another avatar of Lieuwell's theorem okay so how do you see that see suppose f has a removable singularity at w equal to infinity then f of w has this expansion which is analytic at infinity then you see g of z which is f of 1 by z where I put w equal to 1 by z what I will get is I will get sigma n equal to 0 to infinity I will get b n e z power n okay which is you can see that is clearly a Taylor series at the origin it is a positive it is a power series at the origin okay center at the origin so it has to represent an analytic function and that is correct because f is analytic at infinity f w is analytic at w equal to infinity if and only if g of z which is f of 1 by z that is analytic at z equal to 0 let us go back to Riemann's removable singularity theorem okay saying that f is analytic at infinity is same as saying that f is bounded at infinity okay so it means f is bounded in a deleted neighborhood of infinity alright so by if you want Riemann's removable singularity theorem f is bounded so I am using b dt as an abbreviation for bounded at w equal to infinity so f of w if you want. So there exists a positive constant m greater than 0 such that you know mod f of w is less than m for if mod w if mod w is greater than m okay so this is what bounded at infinity means in a neighborhood of infinity the function in modulus can be bounded by a positive constant okay and this is equivalent to infinity being a good point namely it is equivalent to infinity being a removable singularity okay now watch see for mod w less than or equal to R okay so mod w greater than R the modulus of the function is bounded by m okay and look at mod w less than or equal to R I am using the assumption that so I am putting this extra condition that f is entire and mind you I am saying f is entire as a function of w okay so I am trying to look at an entire function which is having a removable singularity at infinity so f of w itself is an entire function even for w finite okay so you see if f of w is entire okay then you know f is analytic at 0 so is analytic at 0 okay f is analytic at 0 because an entire function is supposed to be analytic at every point at every finite point so if I say f of w is entire as a function of w it should be analytic at for all values of w in the complex plane in particular it should be analytic at 0 and if it is an analytic at 0 then you know if you write out it should tend to a limit as w tends to 0 okay but then look at this expression look at this look at this expression this expression mind you normally this expression will be valid it is supposed to be a Laurent expansion at infinity so it should be valid only in a neighbourhood of infinity but since the function is entire it is valid everywhere it is valid on the whole complex plane so it is valid at 0 also okay and if it is valid at 0 you can see all the b n's for n non-0 they all should be 0 okay because the moment I get a negative power of w at w equal to 0 it is not going to give me a finite limit it is going to go to infinity because it is going to become like a pole okay so the moral of the story is that if you assume f is entire then f is analytic at 0 and this will imply that all the b n's are 0 for n not equal to 0 and this implies that f is a constant okay so that is very obvious so all I am trying to say is that if you have an entire function which has a removable singularity at infinity then it is a constant what is the counter positive of that the counter positive of that is suppose you have a non-constant entire function then infinity is certainly not a removable singularity for a non-constant entire function infinity cannot be a removable singularity because the only entire function entire functions which are analytic at infinity are the constants okay and you can the reason why I got into this stuff about modulus is because I wanted to say that this is actually another avatar of Lieuwell's theorem see because you see look at this stuff that I have written in between see f is analytic at infinity so outside the circle of sufficiently large radius mod w greater than r mod fw is bounded okay but if you look at the interior of that circle and the boundary of the circle I will get mod w less than or equal to r and you see mod w less than or equal to r is a compact set it is both closed and bounded and I have assumed f is entire so it is continuous you know a continuous function on a compact set is bounded because the image of a compact set under a continuous map is again compact and compact is will imply bounded so what this will tell you is that there is a bound for f even in mod w less than or equal to r that combined with the bound for mod w greater than r will tell you that f is an entire function which is bounded on the whole plane and then Lieuwell's theorem will tell you that f is a constant so that is the point I want to tell you I want to tell you that see this argument that an entire function which is analytic at infinity is constant is actually another avatar of Lieuwell's theorem okay it is actually another avatar of Lieuwell's theorem that is what you have to understand. So the moral of the story is that whenever you are looking at a non-constant entire function infinity is certainly a singularity it is a honus singularity it is not a removable singularity so it can either be a pole or it can be an essential singularity it cannot be a removable singularity okay and the only exemptions are constants which are very uninteresting okay so I will stop here.