 Okay, so what we are looking at is you know we are trying to understand the behaviour at infinity okay and somehow the idea is that if you want to study f of z at infinity then it is same as studying f of 1 by z at 0 okay. So the question was why is this why is this justified and well I was trying to explain that in the towards the end of the last lecture so I will take off from there. So let me let me go back to what I was saying so you see so the idea is the idea is the following the idea is the following the idea is that you know whenever 2 objects are isomorphic okay then their property should correspond alright and it is not only there for example if 2 topological spaces are isomorphic okay then you expect both of them to have the same if to have the same topological properties. So if 1 has a certain topological property you should expect the same thing to happen of the other okay so for example if 1 if 2 topological spaces are isomorphic 1 is connected then the other should also be connected. So a disconnected topological space cannot be isomorphic to a connected topological space because continuous image of a connected set is connected okay and so on and so forth. So this is a very general philosophy in mathematics for example if 2 vector spaces are isomorphic you then they will have the same dimension okay so their properties like dimension properties like in topology properties like connectedness compactness these are all intrinsic properties okay and they are not supposed to change under isomorphisms. Now therefore so that is one level of thinking the other level of thinking is that especially when you are looking at an isomorphism of spaces okay it is not only that the properties of the spaces geometric properties of the spaces should correspond okay namely if 1 of the spaces has a geometric property then the other should also have and by geometry here of course I mean you know geometry is something that includes both I mean it includes topology it includes algebra it includes analysis it includes an interplay of all these properties okay. So we expect if 2 objects are isomorphic we expect them to behave in the same way whether you no matter whether you look at them topologically or algebraically or analytically okay and of course you should also assume that the isomorphism is also going to be compatible with whatever structure you are looking at if you are looking at topological properties the topological structure then you expect the isomorphism to be a topological isomorphism which is a homeomorphism if you are looking at algebraic properties you should expect the you should think of an isomorphism which also you know behaves well with the algebraic structure and if you are looking at analytic properties for example if you are looking at manifold theoretic properties then you should expect the isomorphism to be an isomorphism that respects that structure. So now you see the big deal is that whenever 2 spaces are isomorphic then the functions on those spaces are also isomorphic okay so this is a very very important idea so and functions are isomorphic in the sense that you know properties of functions carry over so that is what I was trying to explain last time so look at this thing so x phi from x to y is a homeomorphism of topological spaces and you are given this f which is a function on the topological space y with values in another topological space z okay and I am just saying it is a function I am not saying that to begin with I do not know whether it is continuous or not okay but the point is that the moment you give it is very natural in set theory that whenever you give me a function on the target set you can compose it with the given map from the source set to the target set to get a function on the source set so that is called the pullback of the function. So you know f is a function on y you can compose it with phi to get this function g which is now a function on the source which is x okay and then the question is that what is the connection between f and g in fact f and g should correspond to one another in this isomorphism of spaces which induces an isomorphism of functions okay so in fact you know what is happening is that there is a map from functions from y to z to functions from x to z that is a pullback map and what is happening is f is going to g okay and this is an isomorphism because because phi can because phi has an inverse okay this is an isomorphism and the fact is that under this isomorphism properties functions with properties particular properties coincide so far they correspond so for example if f is continuous then g is continuous and conversely if g is continuous f is continuous because you can get f from g and g from f because phi is invertible okay now this is at the topological level okay then I am saying that let us look at it at the level of complex analysis for example so suppose D1 and D2 are domains in the complex plane and phi from D1 to D2 is an analytic isomorphism namely it is a holomorphic map it is an analytic map which is injected okay and you know an injective holomorphic map is an isomorphism and then our philosophy should tell us that the functions on D2 correspond they are in 1 to 1 correspondence with functions on D1 okay in particular if you are looking at complex valued functions on D2 they should be in bijective correspondence with complex valued functions on D1 okay and that is again by the pullback so if you give me a function f on D2 with complex values then you know I get the pullback function g okay and f is analytic if and only if g is analytic and the reason for that is again the fact that phi is invertible phi is analytic and you use the fact that the composition of analytic functions is analytic okay so now what I want to do is that you can go one step further and you can discuss singularities so you know so let me write this down so let us assume that you again have this you have a you have a map from D1 to D2 phi is a map from D1 to D2 phi is a homeomorphism okay and I am assuming that D1 and D2 are well they are domains in the complex plane okay and of course mind you I am being a little careful I am saying that phi is a homeomorphism I am not saying it is an isomorphism it is a holomorphic isomorphism analytic isomorphism but I say the following thing suppose phi takes a point Z0 in D1 to a point W0 in D2 okay so I am writing the function phi as W is equal to phi of Z okay Z is the independent variable it is supposed to vary in D1 okay and W is phi of Z it is the dependent variable it is supposed to take values in D2 okay and I am assuming that phi takes Z0 to W0 okay and suppose suppose that you know phi from well suppose phi from D1 to D2 is actually an analytic isomorphism. So now I am you know I am assuming something more I am assuming that phi is not just a homeomorphism I am assuming it is an analytic isomorphism which means in particular it is a homeomorphism okay mind you an analytic map is continuous okay and therefore analytic isomorphism is stronger than being a homeomorphism okay well now you see now you can do the following thing see suppose you already know that if you have a function on D2 I can pull it back to give you a function on D1 but suppose I had a function on D2 with a singular point isolated singular point a singularity at W0 okay then by composing with phi I will get a function on D1 which will have an isolated singular point at Z0 okay. So you see suppose so the situation is like this you have D2 minus the point W0 and that is being carried over by phi to D1 minus Z0 because you know phi is after all bijective it will it will take the complement of Z0 to the complement of W0 and Z0 will go to W0. So basically what it is doing is that it is taking the punctured domain punctured at Z0 and mapping it holomorphically isomorphically analytically isomorphically onto the punctured domain at D2 punctured at W0 okay and suppose you have a function suppose you have a function f here complex valued function okay which is analytic okay then you know if I compose this compose it with phi I will get this function g which is first apply phi then apply f okay and of course g will also be analytic okay and mind you this inside this is an open subset this inside this is an open subset the complement of a point is always open and this diagram also commutes okay and well the fact is that what this tells you is that f has an isolated the point W0 I do not know whether at the point W0 f is analytic or not okay but if there is a deleted neighborhood of W0 for example the domain D2 minus W0 it is a deleted neighborhood of W0 and there I know f is analytic so W0 is an isolated singularity of f okay and what this diagram tells you is that the function g that you got by pulling back f via phi is also having an isolated singularity at Z0 okay and now you can you can believe that if you believe in this philosophy that under a pullback of functions by an isomorphism okay properties of function should coincide you can believe that the nature of the singularity of f at W0 should correspond should be exactly the same as the nature of the singularity of g at Z0 okay so it is natural to expect that if f has say a removable singularity at W0 then g should have a removable singularity at Z0 if f has a pole at W0 of a certain order then g will also have a pole at Z0 of that order and if f has an essential singularity at W0 then g will have an essential singularity at Z0 and the converse will also code okay so properties of f the nature of the singularity of f at W0 should correspond to the nature should be exactly the same as the nature of the singularity of g at Z0 okay so this is something that is very easy to understand and why is this why is that why is that true that is just true because D1 to D2 is an analytic isomorphism okay it is because it is an analytic isomorphism but this is happening see so let me write this in a let me use a different color okay so well so you see so let me write this here nature of the singularity I am abbreviating it as sin g of f at W0 is equal to the nature of the singularity of g at W0 okay and see this is this happens basically because you can you can see this in a moment you see so why is this true so you know let us look at 3 cases suppose f is suppose f has a removable singularity at W0 okay then what it means is that you know by Riemann's removable singularity theorem you know that the limit of f as W tends to W0 exist is also the same as saying that the f extends to an analytic function at W0 okay and is equal to saying that f is continuous at W0 okay and if f is continuous at W0 by composition with phi it is clear that g is also continuous at Z0 okay so so it is very clear that if f has a removable singularity at W0 then g has a removable singularity at W0 and the converse is also true if g has a removable singularity at Z0 then f will have a removable singularity at W0 because you can if you compose g with phi inverse you get f okay so this tells you that f has a removable singularity at W0 if and only if g has a removable singularity at Z0 okay now what about the case of W0 being of pole if f has a pole at W0 then the limit as W tends to W0 of f is infinity okay and you know you see by continuity it should happen that g will also have a pole at Z0 is continuous at Z0 okay and g being continuous at Z0 is the same as saying that g has a removable singularity at Z0 okay you have this now what is the situation when f has a pole at W0 f has a pole at pole at W0 if and only if you know say of order n greater than or equal to 1 what is the condition for a pole it is well limit one condition is of course limit Z tends to I mean limit W tends to W0 of f of W should be infinity this is this is one of the conditions and that is also the same as saying that limit W tends to W0 of W minus W0 to the power of n times f of W is non-zero okay this is exactly the condition that f has a pole of order n okay and you know if you translate this limit and realizing that as W tends to W0 if and only if Z tends to Z0 because f is a homeomorphism okay and you know under continuous map the image of a convergent sequence is again a convergent sequence okay the image of a limit is again a limit alright so you know so if you translate that you will get you will actually if you change variables this will tell you that Z minus Z0 to the power of n G of Z is non-zero if you let Z tends to Z0 okay. So the easiest thing to do is the following thing what you do is you take this this is the easiest probably the easiest thing to do what you do is you put W is equal to phi Z okay if you put W equal to phi Z you will get limit phi Z tends to phi Z0 f of phi of Z but f of phi of Z is GZ okay and limit phi of Z tends to phi of Z0 is the same as limit Z tends to Z0 so you get limit so this is the same as saying that limit Z tends to Z0 G of Z is infinity okay see this is plainly equivalent okay so this is what I want so this is what we want so this is plainly equivalent to limit EZ tends to Z0 G of Z is equal to infinity it is just you just make a change of variable from W to Z so you will have to so you put W equal to phi Z okay then you will get f of phi Z f of phi Z is just GZ by definition okay and W tends to W0 will read phi Z tends to phi Z0 but phi Z tends to phi Z0 is equivalent to Z tending to Z0 because phi is a homeomorphism so this is the same as this and this condition limit Z tends to Z0 GZ is infinity is will tell you that it is a pole Z0 is a pole of G okay and you will have to do a little bit more work to for example compare Laurent series to say that the pole is exactly the same order on both sides okay so you see therefore essentially I am just using the fact that phi is a homeomorphism I am not using anything more than that alright so well so let me write this here G has a pole Z0 and a little bit more work will tell you that the pole will also have order n okay fine so so you know you can if you try to if you try to if you try to prove this thing below okay that may not be so easy to do at the face of it okay fine so anyway what this tells you is that f if f has a pole at W0 then G has a pole at Z0 and conversely okay and of course you know you can get the converse because instead of phi you can use phi inverse which is also a homeomorphism okay and well and then what is the last case is the left out case f will have an essential singularity at W0 if and only if G has an essential singularity at Z0 this is just by tautology it is just by logic because essential singularities are singularities which are neither removable nor poles okay and you already shown that removable singularities correspond to you have shown that poles correspond therefore essential singularities which is the complement of these two should also correspond okay so let me write that down it follows that f has an essential singularity at Z0 at W0 if and only if G has an essential singularity at Z0 okay this is very clear and in fact you know if you want you can also say it in another way what is the condition that a function has an essential singularity at that point one condition for example is that the limit as you approach that point does not exist okay if limit W tends to W0 f W does not exist then f must have an essential singularity at W0 and again you know if you use the substitution W equal to phi Z and remember that phi is a homeomorphism it is very clear that the limit as W tends to W0 f W will not exist if and only if the limit as Z tends to Z0 G of Z does not exist and this will tell you that W0 being an essential singularity for f is the same as Z0 being an essential singularity for G okay so essential singularities correspond but of course here I am using the fact that the limit does not exist and where did that come from that basically came from an application of actually if you go back it is an application of Reimann's removable singularity theorem which says that if the limit exists then it is removable okay and if the limit exists and it is infinite then it is a pole okay and if the limit does not exist then it is an essential singularity and all these ifs are actually if and only ifs okay fine now having said all this now how do we deal I want to get back to try to you know tell you the story that saying that f of Z studying f of Z at infinity at Z equal to infinity is same as studying f of 1 by Z at 0 okay and what is the why is that justified in light of in the light of this argument so it is justified in the following way so let me write that down well you know justify behaviour of f of Z let me use f of W because for so that I am consistent with my notation behaviour of f of W at W equal to infinity is the same as that of f of 1 by W at W equal to 0 okay so this is so if you look at if you have gone through a first course in complex analysis and behaviour at infinity was covered then you would see people saying that f has a has a the nature of the singularity of f at infinity is same as the nature of singularity of f of 1 by Z at 0 okay and why is that true it is true in the in the light of the following argument which is based on what we have been saying you take this map from C star let me note your C star take this map from C union infinity which is extended complex plane to C union infinity this is extended complex plane to the extended complex plane okay so now we are making use of the point at infinity and we are also making use of the topology of the extended complex plane so and you know what is a map the map is just Z going to 1 by Z you take this map so this is my map phi so here my phi of Z is 1 by Z so W is phi of Z which is 1 by Z so W is 1 by Z this is my map and this is a well defined map you see the point is that you have to be you have to you have to just send infinity to 0 and 0 to infinity this is the obvious thing that you will do and so you know so let me write that down you send 0 to the point at infinity and you send infinity to the point at 0 okay and mind you when you send when you make these definitions it phi continues to be a homeomorphism okay see limit Z tends to infinity phi of Z is what limit Z tends to infinity of phi of Z is just limit Z tends to infinity of 1 by Z which is 0 okay and what this will tell you so limit Z tends to infinity phi of Z is 0 and that is phi of that is exactly phi of infinity as per our definition because we have sent infinity to 0 and what does this tell you this tells you phi is continuous at 0 at infinity okay and the same kind of argument will tell you that phi is also continuous at 0 so if you take limit Z tends to 0 phi of Z is limit Z tends to 0 of 1 by Z and this is infinity which is phi of 0 okay and mind you this is we have defined what limit Z tends to infinity means we have defined what when a limit is infinite we are using all we are using all those definitions we are using the fact that we are thinking of infinity actually as a point okay and we are using and we are also thinking of infinity as a value okay do not a finite value. So the fact is that if you look at this map now from extended complex plane to the extended complex plane Z going to 1 over Z this is actually a homeomorphism this is the important point it is a homeomorphism and if you throw away both the origin and the point at infinity okay you get C star which is punctured complex plane C minus 0 okay and C minus 0 goes to C minus 0 and if you take if you restrict this map to C minus 0 it is an analytic it is a holomorphic isomorphism that is the point this map is a holomorphic isomorphism which extends to infinity okay so in a continuous way okay. So let me write that down so let me draw the diagram again so I have the C union infinity here and I have this homeomorphism to C union infinity and this is a map phi which is sending Z to 1 over Z okay which is W right and what is sitting inside this is C star which is C minus 0 with the punctured plane and on this side also I have C star that is the image of C star because if Z is a non-zero complex number then 1 by Z is also a non-zero complex number and this correspondence is and Z going to 1 by Z is analytic if Z is not 0 because it has the derivative minus 1 over Z squared you know that pretty well Z equal to 0 is a pole of it is a simple pole of order 1 for 1 by Z okay so the point is that if you restrict phi to C star what you get here is not just a homeomorphism it is a holomorphic analytic isomorphism so this is so let me write that here it is a holomorphic or analytic isomorphism this is what you get and now watch suppose I have a function which is defined in a neighbourhood of infinity okay suppose f is a function is defined in a neighbourhood of infinity what is the neighbourhood of infinity a neighbourhood of infinity is the exterior of a circle of sufficiently large radius so you know if I have a function defined on mod Z say greater than r sufficiently large this is what a neighbourhood of infinity is this is a neighbourhood so let me write that neighbourhood of infinity in where this is the neighbourhood of infinity in the external complex plane mind you that is how the topology on the external complex plane has been given okay and in fact it is if you look at it in the external complex plane then you are including the point at infinity and if you do not include the point of infinity it is a deleted neighbourhood okay so since I am looking at it in C star minus 0 it is a deleted neighbourhood of the point at infinity okay so this is a deleted neighbourhood of infinity infinity being deleted okay and you know under this map Z going to 1 over Z this should correspond to a deleted neighbourhood of the origin which is mod Z less than 1 by r you know r sufficiently large so 1 by r is sufficiently small mod Z is 1 is less than 1 by r is the so here I probably since my target variable is W I should not have used Z let me correct that this should have been W so I have reserved W for the target variable and W is so if you plug in there W equal to 1 over Z I will get 1 by mod Z is greater than r which is the same as saying mod Z is less than 1 by r that is so this corresponds to mod Z less than 1 by r which is a and you know of course this is a neighbourhood of the origin but if I but since I have not included the point at infinity on the right side the thing that I got to get on the left side is going to not include the origin because I am already you know see I am considering this as a subset here so infinity is not included and I am considering this as a subset here 0 is not included so this is a deleted neighbourhood of the origin so this is so let me write this deleted neighbourhood of this origin okay and mind you Z going to 1 over Z is still a holomorphic isomorphism of this small punctured disc centre at the origin radius 1 by r open disc with the exterior of the disc with radius r alright. So now suppose I have a function defined in a neighbourhood of infinity okay that means I have function f here I have my function f here okay and it is taking complex values alright then if I use so you know this diagram commutes basically this is just this diagram commutes means that I am just restricting the map phi that is all okay so this map from this punctured disc surrounding the origin to the exterior of the disc with radius r is just the holomorphic isomorphism Z going to 1 over Z which we have called as phi okay and you compose that with f you get as before you get g so g is just first apply phi then apply f as before but what is that so you know f is f of w okay and w and so you see that if I calculate g of Z g of z will then be f of phi of z okay but what is phi of z phi of z is 1 over z so g of z is nothing but f of 1 over z okay so what you see is that if you look at the map z going to 1 over z the pullback of f of w becomes f of 1 over z okay and now you go back to this philosophy that whenever you have an isomorphism of punctured domains and you have an analytic function on the target domain for and you pull it back to an analytic function on the source domain then the nature of the singularity of the function at the target on the target space is the same as the nature of the singularity in the source space. So you know if you apply that philosophy you can see that f is the nature of the singularity of f of w at w equal to infinity is must correspond must be the same as the nature of the singularity of g of z at z equal to 0 but what is g of z g of z is f of 1 by z okay so you know this justifies the statement that the nature of the singularity of f of w at w equal to infinity is the same as the nature of the singularity of f of 1 by z at z equal to z okay so you must so you should understand what is going on why this is a very natural thing to do okay fine. So the moral of the story is that we have justification as to why studying f at infinity f of w at infinity is same as studying f of 1 by z at z equal to 0 okay. Now let us go and try to look at what we are going to get okay and so we will you know we will get 3 cases as usual because we are you know trying to classify the singularity of f at infinity mind you in all these things to talk about to be able to talk about these kinds of things the function f should be defined in a neighbourhood of infinity which means the function should be defined the exterior of a circle okay for all values of the variable in the exterior of a circle of sufficiently large radius okay which is what a neighbourhood of infinity is. So what is the first case the first case is when will you say that f has infinity as a removable singularity okay or you know a removable singularity is the same as a point where the function is analytic okay that is exactly what the Riemann's removable singularity theorem says it says that if you take a function which has an isolated singularity at a point in the complex plane then it has a removable singularity at that point if and only if it can be extended to an analytic function at that point and of course the weakest condition is that it is even bounded in a neighbourhood of that point that is the strongest that is the strongest part of the Riemann's removable singularity theorem. So I would like to ask when will f be analytic at infinity okay now you know you must be careful when you think about the point at infinity because there are issues so for example you should not be tempted to say that f normally what is the definition of analysis at a point in the complex plane the definition of simplest definition of analyticity is that the function is differentiable at that point and in every neighbourhood in some neighbourhood of that point at every point in some neighbourhood of that point. Now you cannot have this definition at infinity you cannot say function f is analytic at infinity if it is differentiable at infinity and it is also differentiable in neighbourhood of infinity okay that the function is already differentiable in neighbourhood of infinity is given because it is already given to me because infinity is an isolated singularity. But trying to say that the function is differentiable at infinity will not make sense because derivative at infinity does not make sense. So what is derivative at infinity? If you really try to define f dash you know f dash of infinity if you want to define it like this and I will you will write limit z tends to infinity f of z minus f of infinity divided by z minus infinity which really does not make any sense. See f of infinity might make sense because if f for example f extends to something continuous at infinity f of infinity could be defined as limit z tends to infinity f z okay that is fine but this z minus infinity term is is absolutely absurd and trying to let z tend to infinity. So you know it is you know this is not the this is trying to make f differentiable at infinity is not going to help because there is no way to do it okay. So how will you do it? So trick is you do it in a in a rather you know indirect way you recall the Rayman's removable singularity theorem which says that if you look at a point in the finite complex plane then the function is analytic there at that point which is an isolated singularity if and only if it is if you want bounded at that point in a neighborhood of that point or if it has a limit at that point which means it extends to a continuous function at that point okay. So you do that so what you do is instead of trying to define the function to be analytic at infinity if it is differentiable at infinity which is wrong because you cannot define the derivative at infinity what you do is you say you define the function to be analytic at infinity if either it is bounded in a neighborhood of infinity or it is has a limit at infinity namely that it is continuous at infinity okay you make this definition and then you are in very good shape and it will also agree well with both these definitions will agree well with the earlier philosophy that the nature of the singularity at infinity of f of w at infinity is the same as the nature of singularity of f of 1 by z at 0. So you can see that so here is so define f is analytic at infinity okay so mind you I am so this definition of analytic is very very funny okay it is not the definition that the function is differentiable at that point in differentiable in a neighborhood at that point it is not that definition okay but it is a definition that point is a removable singularity okay that is an indirect way of defining it. So f is analytic at infinity so let me write f of w if well limit w tends to infinity f of w exists okay or limit or let me write this here f is bounded at infinity or f is continuous at infinity okay so and you know all these things are required these are 3 different ways of trying to define the function analytic at infinity the philosophy is that you are using you know you are using Rayman's removable singularity theorem and why are they equivalent they are equivalent because of the following thing because you see this is equivalent to saying that limit z tends to 0 f of 1 by z exists which is which we have called as gz exists because the map z going to 1 over z is a homeomorphism these 2 are equivalent okay and well f is bounded at infinity is the same as saying that f of 1 by z is bounded at 0 g of z is bound which is defined by f of 1 by z mind you g of z is by our original notation g of z is f of phi of z and phi of z is 1 by z which is w okay so g is bounded at 0 okay and the third thing is well g of z is equal to f of 1 by z is continuous at 0 and mind you all these 3 are in fact equivalent all these 3 conditions are equivalent for the function g of z why because now I am looking at a function g of z with 0 as an isolated singularity and all the 3 conditions are equivalent by Rayman's removable singularity theorem to saying that g is actually having a removable singularity at 0 okay so these 3 are these so all these 3 are equivalent and here this is Rayman's theorem I need to make more space to write it on so let me write it here so this is by Rayman's theorem on removable singularities and because these 3 are equivalent therefore these the conditions that I have written on the left side are equivalent that is the point I want you to notice see this is equivalent to this is equivalent to this this is therefore this therefore comes from the right side okay so so this will tell you that so you know now we have reconciled all the definitions f has a removable singularity at infinity if f of 1 by z has a removable singularity at 0 is the same as saying f of 1 by z is analytic at 0 f of 1 by z is continuous at 0 f of 1 by z is bounded in a neighbourhood of 0 and you see we have used 2 things we have used the fact that z going to 1 by z is a homeomorphism and we have also used the fact that we are using the Rayman's removable singularity theorem for a point singularity in the finite plane okay so we will continue in the next talk.