 So you see there are few things which I want you to notice at this point the first thing is that you know we have we have the extended complex plane which consists of the complex plane plus the point at infinity okay and the point at infinity how it is thought of concretely as a point on a space is by identifying it with the north pole on the Riemann sphere via the stereographic projection which gives you a homeomorphism between the Riemann sphere okay which is the unit sphere in 3 space centered at the origin okay and the complex plane along with the point at infinity that is the external complex plane. Therefore see therefore you are able to think of you are able to think of the point at infinity as the as a north pole under this correspondence which is given by the stereographic projection. Now so you know if you just imagine only the complex plane the point at infinity seems to be an invisible point okay it seems to be far away and out of you know out of sight if you want it is something that you cannot visualize okay but then the way you should really think about it is you must remember that the point at infinity is an interior point okay of the extended complex plane mind you of course for that matter any point in a topological space an interior point because that point is always contained in the whole topological space which by definition is an open set a point in a topological space is called an interior point of a set if it contains there is an open set which contains that point in which is contained in the given set okay. So in that sense of course every point in any topological space is certainly an interior point of the topological space but what I want you to understand is that in general an interior point you should be able to you should be able to talk about a deleted neighborhood of an interior point and the fact is that if you take the extended complex plane and you take the point at infinity if you throw away infinity what you get is a complex plane and the complex plane itself is a deleted neighborhood of infinity that is what I want you to understand. So when you are looking at the complex plane you are looking at the deleted at a deleted neighborhood of infinity that is what you must understand okay and of course if you are looking at as we define just now if you look at the exterior of a circle then you are actually getting a smaller neighborhood of infinity and the neighborhood becomes smaller and smaller as the circle becomes larger and larger okay that is how it goes and this one so you must get used to thinking of infinity as an interior point with a deleted neighborhood of infinity being thought of as the exterior of a circle in the complex plane okay and all this is not just imagination it is concretely correct okay because this is what you really see when you translate it in terms of the stereographic projection and look at what is happening on the Riemann sphere the point at infinity appears as on the Riemann sphere as a as a honest point it is a north pole okay and a neighborhood of infinity is going to be an exterior of a circle on the complex plane and that appears honestly as a open neighborhood of the north pole on the Riemann sphere and for that matter if you take the whole complex plane under the stereographic projection it will go to the whole Riemann sphere minus the north pole. So saying that the whole complex plane is a deleted neighborhood of the point at infinity amounts to same in terms of the stereographic projection that the whole sphere minus the north pole is a deleted neighborhood of the north pole which is correct okay. So what I am trying to tell you is that the stereographic projection allows you to really think concretely okay so and this is very very important because you see if I want to think of infinity as a point okay then where is that point how do I see it so the idea is that you see it as a as a north pole on the stereographic projection okay and then there is one more thing normally when you are studying properties of a function when you do analysis normally what you do is that you study the properties of a function at an interior point and the reason is because you want to basically what you do in analysis is that you study limits and to study limits you will have to take limits in all possible directions so if you want to study the behavior of a function at a point then you should be able to study the limit of the function as you approach that point in all possible directions and namely you have to look at function values close to that point in all possible directions around that point so basically you want the function to be defined in a deleted neighborhood of that point okay otherwise you cannot do this okay and that is the reason why you need an interior point if you want to really do analysis okay and of course taking a limit starts with continuity and then even derivative is a limit and so on and so forth okay now you see what about the point at infinity you can ask is it also true for the point at infinity that you are able to approach it from all directions and the answer is yes you see you take if you look at the complex plane the point at infinity is not visible okay but you must always think in terms of the Riemann sphere and the north pole as representing the point at infinity under the stereographic projection so you know how do you approach how is it correct to say that you know you are able to approach infinity from all directions so you can think of approaching infinity by going along a curve on the complex plane which going to an unbounded part of the curve okay and for example you can take straight lines passing through the origin and then as you move away from the origin on the line you can think that you are approaching infinity and you can think that you know as you change the lines you are approaching infinity in different directions okay so the truth is that all this will agree if you translate it in terms of the stereographic projection okay so if you look at it in terms of stereographic projection you will see that you are actually approaching the north pole you know from various directions okay so if you want just to help you think about it so here so let me draw this again so you know basically you have the complex plane here these are the x and y axis and this is the complex plane thought of as a x y plane in R3 okay and of course as we have done before we are not going to call the third axis as z we are calling it as u because z is supposed to stand for x plus i y okay and the stereographic projection is basically you draw the circle I mean you draw the sphere centered at the origin radius 1 unit and look at the surface of the sphere and so basically I have here this is my circle on the complex plane this is the unit circle on the complex plane centered at 0 radius 1 unit and then I have this sphere which is a Riemann sphere well in fact this is supposed to meet on the x axis so let me see where I can change that fine so here is how it is and now you see yeah so on the complex plane if I want to approach the point at infinity what I do is that I can go along the ray okay and move away that moves away from the origin okay and as you move further and further along the ray I am supposed to be going to infinity okay and well and the direction of the ray is supposed to give one of the direction in which I can approach infinity okay so and why is this correct why is this correct this is because you see if you take the image of this ray under the stereographic projection okay what you actually get is that you will basically get you will basically get this under the see mind you so let me remind you see you have the north pole here okay the stereographic projection is done in the following way you give me give me any point or give me any point on the sphere okay then that point is mapped by the stereographic projection to its image phi of p and that is done by simply join the north pole to that point and looking at the point where this line hits the plane okay so here is so phi is the stereographic projection okay and now you can see that under the stereographic projection the origin goes the south pole of the sphere goes to the origin okay so the origin corresponds to the south pole the south pole on the sphere corresponds to what point on the plane it corresponds to the origin because it is supposed to correspond to the point on the plane which is gotten by joining the south pole to the north pole by a line and looking at where that line hits the complex plane and so the south pole corresponds to the origin okay and now you can see that the image of this ray is going to be this circular arc like this this is what I am going to get okay on the you know that is what I am going to get on the rayman sphere okay and mind you if I move this phi of p further and further towards infinity the point p moves further and further along that great circle to the north pole okay so you see basically what is happening is that you are going like this so this is how it is going okay and the fact is that if you had taken another direction okay then you will see that you are that does correspond to going approaching the north pole on the sphere in another direction okay so for example so let me draw that you can easily imagine it but it is good to draw a few diagrams now and then so you know if I for example where to take a line like this suppose I were where and mind you this green this green line in this this violet line that I have drawn they are in fact supposed to lie on the x y plane mind you they are lines on the plane they are rays on the plane and I am drawing them to indicate that a point is moving towards infinity okay so if you look at this this violet line that I have drawn okay that is going to the left of the diagram then what is its image under the stereographic position that corresponds to again you know a circle like this a great circle like this which is going like this okay and that is again another so you know if I take a point Q here it will under the stereographic projection it will go to phi of Q so it will go to phi of Q and you know as phi of Q moves to away and away from the origin which is supposed to be thought of as going to infinity then Q on the Riemann sphere is going to move closer and closer to the north pole okay so you see that if you look at points on rays starting from the origin and moving towards a point at infinity namely moving away from the origin they do correspond to different directions of approaching the north pole on the Riemann sphere okay under the stereographic projection therefore what I am trying to say is that I am trying to say that this the point at infinity can be approached in all directions okay this is the kind of justification you can give thanks to the stereographic projection okay. So if you let the variable z to go to infinity along an arc or a path or a ray or even a curve which is unbounded and going to infinity then you are going to infinity okay in a certain direction and you can do this in many ways and that amounts to really approaching infinity from all directions and the way you see it is that you see it on the Riemann sphere okay using the stereographic projection. So you must understand concretely that you know thinking of infinity as a concrete point, thinking of a neighbourhood of infinity, thinking of small neighbourhoods of infinity okay and then thinking of being able to approach infinity from different directions all this is concretely possible and it is because of the stereographic projection that is the important thing okay fine. So now with all this let me again continue with this string of definitions which I wanted to give. So one definition must tell you that I have already done I have defined what limit as z tends to infinity of f of z means okay. So I have defined what it means to take a limit at infinity okay which is what I did last time so if you want if you look at it so here is a definition we define limit z tends to infinity for z in the external complex plane as going to z as allowing the modulus of z to become arbitrarily large okay which is written by this condition mod z greater than R and that is essentially you know it is a exterior of a circle and that you know as R becomes larger and larger corresponds to a smaller and smaller neighbourhood of infinity if you look at it via the stereographic projection and therefore you know we define limit z tends to infinity f of z equal to lambda where lambda is a complex number to mean the obvious thing that as z approaches infinity f of z should approach lambda so which means that the distance between f of z and lambda namely this quantity can be made as small as you want provided you make z close to infinity sufficiently close to infinity and that is reflected by choosing this delta okay and the hypothesis is that there is a delta given an epsilon okay so now this is how you define limit z tends to infinity so this corresponds to getting a finite limit at infinity okay you are getting a limit at infinity but the limit is finite okay now I am going to tell you the other thing the other two possibilities I am going to look at this situation when you can get an infinite limit at a finite point of the complex plane that is one more case and then the third case is when you get an infinite limit at the point of infinity okay so there are three definitions and all of them are important so let me make that definition we define limit z tends to z not f of z is equal to infinity okay if given epsilon greater than 0 there exists a delta greater than 0 there exists a delta positive such that the distance between f and infinity can be made lesser than epsilon provided I make the distance between z and z not lesser than delta so let us translate that so let me write it in quotes distance between f and infinity f of z and infinity can be made smaller than epsilon whenever mod z minus z not is less than delta okay and now you know it is a little bit I am just trying to say that f the values of f namely f z they get closer to infinity and I want to say that they are closer to infinity by epsilon but then you know if you want to really talk about distance you must have some notion of distance between the point at infinity and a point on the plane okay only then you can talk about distance between infinity and a finite point okay. Now it happens that it can be done okay what you can do is you can define a distance between the point at infinity and the point in the complex plane in the following very clever way and of course the key is the stereographic projection gives you everything so what you do is you take the point at infinity and you take any other point take their images on the stereographic projection you will get the north pole and some other point on the sphere and then you take the distance and again for the distance you can you have two choices one is either you can take the distance in R3 of those two points on the surface of the sphere or you can take the you can take the geodesic which is supposed to be the it is going to be the circular arc of shortest length that connects those two on the sphere okay. So these respectively will give you the R3 the usual metric in R3 and the other one is called the usual metric in R3 is called the chordal metric because you are taking two points on the sphere and then you are joining them by a straight line segment that is going to be a chord and you are just measuring the length of the chord and the other thing is called the spherical metric the spherical metric is you are moving on the surface of the sphere and you are taking the shortest distance and that shortest distance is going to be the length of the circular arc along the circle the circle the great circle that passes through these two points okay. So and then what you can do is you can take that as your definition of distance of two points in the external complex plane and the beautiful thing that happens is that with this the external complex plane in fact even becomes a metric space okay and whether you put the chordal metric or whether you put the spherical metric you get isomorphic metric spaces okay and in fact the topology induced by these metrics is the topology on the external complex plane given by the one point compactification which we see which we saw last time. So these are all facts that you can really write down and verify okay because you are in three dimensions you can even write down the stereographic projection in terms of x, y and u coordinates okay you can write down the stereographic projection you can verify all these statements okay but you know here is where we will not be so concerned about distance okay we will interpret this fact that the distance between f and infinity can be made sufficiently small by simply saying that f is in a sufficiently small neighborhood of infinity okay. So you know that this is where the this is where you try to escape from having to do the more complicated job of trying to give a distance between a point between a point infinity and a point in the complex plane. What you do is that you say that the you say that f of z values of f are closer to infinity are as close as you want and now that means that you know you modify this statement to say that the f of z lies in a very small neighborhood of the point at infinity and what is a very small neighborhood of the point at infinity it is given by of course you know the exterior of a circle of sufficiently large radius and how large well you can make it to be inversely proportional to epsilon okay because epsilon is supposed to be sufficiently small if you want you can make 1 by epsilon to be the large radius. So you know so we will replace this you know to mean the following thing so we will make it mean that mod f z can be made greater than 1 by epsilon okay so you know this is the replacement I am trying to this is the replacement for the definition of trying to actually have a distance between a point at the value f z and the point at infinity and having that made less than epsilon okay so this is where you have to pay attention as to how the definitions and this is intuitively correct because you see if epsilon is sufficiently small okay then you know 1 by epsilon is pretty large okay and then mod f z greater than 1 by epsilon means that you are going to a sufficiently small neighborhood of the point at infinity this is what you have to understand okay so it is so here the fact is that you know epsilon is actually not the distance okay you should not think of it you know verbatim like that you must think of it as going to going outside a circle of very large radius and you if you want you can make that radius to be inversely proportional to epsilon okay so see so this is what it means for us okay. So with this definition you know what it means to say that so here of course I should say that z not is a complex number it is a finite complex number so this is how you define a function taking the value infinity at a finite point z not okay and this is something that you have already seen for example whenever a function has an isolated singularity which is a pole then the limit of the function as you approach the pole is infinity I mean this is what we always write and we interpreted to say to saying that the modulus of the function approaches infinity and what is another way of saying that the modulus of the function approaches infinity it is just saying this namely that the modulus of the function becomes arbitrarily large okay and that is what mod f z greater than 1 by epsilon says okay. So here is an example if you know f of z has a pole at z equal to z not in the complex plane then limit as z tends to z not of f of z is actually infinity okay and now you see you can really think of infinity as a point and you are saying that the function takes the limit the limiting value is that point okay. So this is how you get an infinite limit at a finite point okay then there is a third case which is when you get an infinite limit at the point at infinity okay that is the third case so and why is that important that is important because I want to think of infinity as a pole for a function okay you know at if you want to think of a point as a pole then you know the function should tend to infinity as you approach that point so if I want to think of infinity as a pole then the function should tend to infinity as I approach infinity so I want an infinite limit at infinity okay so that is the motivation for the and the necessity for the next definition so let me use a different colour okay. So just a moment yeah we define limit z tends to infinity f of z is equal to infinity now how do you define this so again the I mean well you know if you really try to use epsilon definition you will say that I want the values of f of z to come to within an epsilon distance of infinity given any epsilon however small for z sufficiently close to infinity okay so here again you know you have to so the moment you say distance from infinity mind you the way out is that there is no distance from infinity as it is either you will have to do the hard work of defining distance from infinity to a point by going to the stereographic position taking either the chordal metric or the spherical metric and instead of doing all that we can be very topological and we can say that whenever you want to say you are going close to infinity you are going to a neighbourhood of infinity and that is very easy to define it is the exterior of a circle of sufficiently large radius. So you know in that sense what does this mean this means that you know given you can get the values of f of z can get as close to infinity as you want provided z gets as close to infinity a sufficiently close to infinity that is what it means okay so you know so how do you write it down by now you should have got it so let me write this down provided given epsilon greater than 0 that exist delta greater than 0 such that mod fz is greater than 1 by epsilon okay this is supposed to mean that f is going close to infinity because mind you you must think of epsilon becoming smaller so 1 by epsilon is becoming larger and mod fz greater than 1 by epsilon is saying that you are in a smaller neighbourhood of the point infinity okay whenever mod z is greater than 1 by delta so this is again when you say mod z greater than 1 by delta mind you that you are trying to find a sufficiently small delta if you take find a sufficiently small delta 1 by delta become sufficiently large if 1 by delta become sufficiently large mod z greater than 1 by delta is a sufficiently small neighbourhood of the point infinity this is the point you have to be careful about okay so this is how you define an infinite limit for a function at infinity okay and for example this will help you to define when infinity is a pole for a function okay for example if you take any polynomial right so now we are through with all these three definitions okay so what we have done is we are able to think of infinity as a concrete point you are able to have infinity in the domain of definition of the function namely you can allow you can write f of infinity okay you can write the value of f at infinity and if you want f to be continuous at infinity it had better be limit z tends to infinity f of z okay and you also allow infinity to be a value of the function okay so the function can take the value infinity okay and the way that is facilitated is because of these definitions okay now what we will see next is using these definitions how you can treat infinity as an isolated singularity and classify the kind of singularities namely look at what it means to say for a function to have an essential singularity or a pole or an removable singularity at infinity okay that is what we are going to do next okay so what we will do now is look at infinity as a singularity okay so let me write that down infinity as a singular point so you want to look at infinity as a singular point for a function okay and of course mind you we are only worried about isolated singularities okay so we are not going to the complicated case when infinity is a non-isolated singularity okay so if infinity is an isolated singularity for the function it means that whenever you say something is an isolated singularity of a function it means that there is a deleted neighbourhood about that point where the function is analytic so it means that your function should be analytic in a deleted neighbourhood of infinity and that means that your function should be analytic and you know by definition a deleted neighbourhood of infinity is just the something that should contain the exterior of a sufficiently large circle on the complex plane centred at the origin if you want, okay. And therefore your function should be first of all defined at infinity I mean it should be defined in the sense that it should be defined outside a circle of sufficiently large radius, okay. So that is a prerequisite, okay. Mind you because if you want to study function at a point that point even if it is not a good point for the function it may be a singularity for the function. The function should be defined in a deleted neighbourhood of that point I mean this applies even to very simple things like continuous functions. See if you want to talk about the continuity of a function at a point then you know you need to study the function close to that point and look at what happens to limit of the function as you approach that point. In a deleted neighbourhood of that point the function should be defined the same way, okay. The prerequisite for studying infinity as a singular point is that the function should be analytic in a deleted neighbourhood of infinity which means it should be defined on for mod z greater than r for r sufficiently large, okay. So let me write that down. We start with an analytic function, analytic or holomorphic, an analytic function f of z which is defined in a deleted neighbourhood so I am using n b d for neighbourhood of infinity that is defined and analytic for mod z greater than r for r sufficiently large, okay. Mind you this also includes the case of an entire function. All I am saying is that the function should be defined outside a circle or sufficiently large radius but I am not saying it need not be defined inside the circle, okay. So what I am interested is only the behaviour of the function outside at all points outside a circle of sufficiently large radius because that is that for me is what a neighbourhood of infinity is, okay. So by definition you know by the definition of singularity infinity becomes a singularity because what is a singularity? A singularity is a point for a singularity is defined only for an analytic function and our definition of singularity is that it is a point which can be approached by points where the function is analytic. So there must be and we are interested in isolated singularity so you see if you take the deleted neighbourhood of infinity mod z greater than r is a deleted neighbourhood of infinity if you take only the finite complex plane I mean only the complex numbers and of course if you take mod z greater than r in the extended complex plane you are also including the point at infinity, okay which by the way is also an open set with infinity as an interior point on the extended complex plane, okay. Now you see you would have done this so I am trying to bring your attention back to something that you should have seen in the first course in complex analysis normally the philosophy is that if you want to study f of z at infinity you will study f of 1 by z at 0 and for the obvious reason that as z tends to 0 1 by z goes to infinity which you can now make sense of because of our definitions, okay. Well you know the question is why is this why is this the right thing to do, okay so you should ask yourself why certain things are defined in a certain way or what the what is the philosophy behind these things so you can ask this question what is the justification for saying that studying f of z at z equal to infinity is the same as studying f of 1 by z at z equal to 0 you can ask this question. So this comes to this brings us to something interesting it has got to do with this idea that you know when you say 2 objects are isomorphic, okay then properties of the objects should also correspond, okay. So this is a very general philosophy in mathematics if you have 2 isomorphic objects, okay both of them should have the same type of property, okay because an isomorphism is supposed to preserve the property preserve all properties, okay. So for example if you have 2 groups and they are isomorphic you cannot expect one of them to be abelian the other is the other non abelian, okay you cannot expect such things because an isomorphism carries an abelian group only to an abelian group, okay. So in the same way this also applies to spaces so if you have 2 spaces let us say topological spaces if 2 topological spaces are isomorphic which means that they are homeomorphic then all the topological properties of one space should agree with all the corresponding topological properties of the other space for example if 2 topological spaces are homeomorphic if one is connected then the other is also connected if one is not connected the other cannot be connected if one is compact the other is compact, okay and so on and so forth. So the fact is that there are properties which are supposed to be intrinsic properties these are called intrinsic properties for an object and they are called intrinsic because they will not change if you change the object up to isomorphism, okay. For example we say that the nature the abelian nature of a group is an intrinsic property because if you replace the group by an isomorphic group then it will happen that the replaced group will also have to be abelian because an isomorphism carries an abelian group to an abelian, okay. So we say the property of being abelian is an intrinsic property. So in the same way this also extends not only to properties of objects it also extends to properties of functions defined on objects this is the very important thing, okay. So I will give you I will give you an example. See so suppose phi from x to y is a homeomorphism which means that you know this is a so I am putting a tilde above the arrow to signify that this arrow is actually an isomorphism a topological isomorphism which is otherwise called a homeomorphism, okay. And of course you know as I just told you all properties of x should correspond to same properties intrinsic properties of y but what I want to say is that this also carries over to functions. So you know see suppose z is another topological space, okay and suppose f from y to z is a function, okay is a set theoretic map, okay. I can ask this question as to when f is continuous, okay. After all y and z are topological spaces and f is a set theoretic map and I can always ask when a set theoretic map between topological spaces is continuous, okay. Now the fact is that you know I can complete this diagram by into a triangle by drawing this arrow which is the composition of phi followed by f. So it is first apply phi then apply f and then I put a circular arrow like this to tell you that the circular arrow that I drawn inside the triangle is supposed to be supposed to be called as commutativity of the diagram, okay which is often used in algebra. It just tells you that you know if you go from x to z either via going first through phi and then through f or from x to z by the other map that I have written which is actually f circle phi they are one and the same, okay. So the advantage of this circular arrow is that sometimes I can write g here instead of writing f circle phi I can simply write g and then if I put the circular arrow it means that this g is supposed to mean f circle phi, okay that is what it means, okay. Now the question is that I am trying to state the obvious thing as you can expect you see that f is continuous if and only if g is continuous, okay the reason is because phi is a isomorphism it is a homeomorphism, okay see. So let me write that down clearly f is continuous if and only if g is and the reason is because you can use the fact that a composition of continuous function is continuous and you can use the fact that phi being a homeomorphism has an inverse its inverse is rather actually continuous, okay. So you can get from f to g you can get from g back to f, okay. So of course you know the way of going from f to g it is just g is just f circle phi and how do you go from g to f you apply first phi inverse and then you apply g you will get f, okay. So f is first applying phi inverse and then applying g, okay well if you write it down remembering that g is equal to f circle phi what you will get is that f it will just simplify to f circle phi circle phi inverse which is since composition of maps is associative I can change the brackets I will get f circle f circle phi circle phi inverse and that is f circle well phi followed by phi inverse is supposed to give me the identity on x. So this will be f circle identity on x and first if I apply identity on x oops no it is first applying phi inverse and then phi so it is not identity on x it should be identity on y and f circle identity y is just f, okay. So yeah so that is I am just trying to write out algebraically that f is just g circle phi inverse. So f is continuous if and only if g is that is obvious, okay. Now you know now this can you can take this over and do it to continuous functions you can do it to differentiable functions you can do it to analytic functions and so on and so forth. So for example you look at a different situation suppose phi is an analytic isomorphism, okay it is a holomorphic isomorphism of one domain in the complex plane to another domain in the complex plane, okay. So then a function on the target domain a complex valued function on the target domain is holomorphic or analytic if and only if the composition with phi is holomorphic on the or analytic on the source domain, okay. See if you look at this diagram what it says is that f is a function on the target which is y and f is continuous if and only if it is composition with the isomorphism phi is going to give you a continuous function on the source, okay. So we often in algebraic notation we always say that f is actually pulled back to g, okay we say f is the pullback of f is g, okay and by pullback what you mean is that you compose f with the isomorphism so that you get a function on the source and that is why it is called pullback because you are taking a function from the target a function defined on the target and from that you are cooking up a function on the source and all you have to do is compose with the isomorphism from the source to the target and for that matter you do not even need an isomorphism even if you have a morphism this will work, okay. So let me write that down suppose phi from D1 to D2 is an analytic isomorphism of domains in the complex plane so mind you phi it means of course you know I am assuming D1 and D2 are non-empty they are open connected sets their domains and they are isomorphic by a map phi which is an analytic or holomorphic isomorphism what it means is that it is basically it is an analytic map which is 1 to 1, okay. You have this deep theorem which says the open mapping theorem which says the image of a non-constant analytic map is always open and you also have this theorem that the an injective analytic map is a holomorphic isomorphism namely its inverse is also going to be the inverse is also going to be holomorphic, okay. So saying that phi is an analytic isomorphism is the same as saying that phi is analytic and phi is injective, okay. So let me write that down that is phi is analytic and injective now you can now I can say the same thing if f is a complex valued holomorphic function complex valued function on D2 then f is holomorphic or analytic if and only if f circle phi is holomorphic or analytic, okay the same argument applies, okay. So clearly if f from D2 to C is a function f is analytic if and only if G is equal to f circle phi is analytic, okay. So the philosophy is the same, okay and now I want you to now I want to give you the justification as to why this studying f at infinity is same as studying f of 1 by z at 0 that is because the map is it going to 1 by z that is the and that is what phi stands for in our particular case that is an isomorphism it is a homeomorphism and from the punctured plane to the punctured plane it is a holomorphic isomorphism, okay and that gives you the justification that studying f at infinity is same as f of z at infinity is same as studying f of 1 by z at 0 and the z going to 1 by z is the takes the role of phi, okay. So I will stop here.