 So let us again recall what we are doing we were looking at the Casorati-Weierstass theorem as a first approximation to the great Picard theorem and we also wanted to get a good idea about dealing with the point at infinity okay. So the point the idea is you want to talk about when an analytic function is you know for example analytic at infinity okay or more generally infinity when can you talk about an analytic function having a singularity at infinity okay and then classify that kind of singularity and what does it mean to say that the singularities of a certain type let us say that it is the function is having an essential singularity or a removable singularity or a pole at infinity okay we need to understand this okay and so in order to study the behaviour at infinity you first must be able to think of infinity as a point okay because you always we are used to looking at things close sufficiently close to a point. So you basically when you want to study the behaviour of a function at a point you need an open neighbourhood of that point where the function is defined and for that matter even if the point is an isolated singularity you need a deleted neighbourhood of the point where the function is defined okay and then you want to study how the function behaves as you approach that point in various ways okay and you know that is how the classification of singularities goes. So for example if you take a point in the finite complex plane okay by that I mean the usual complex plane then if a function an analytic function is defined in a deleted neighbourhood of that point then you know that the way the function behaves as you approach that point can tell you what kind of singularity that is. So as you approach that point if the function goes to infinity that is a modulus of the function goes to infinity then that point has to be a pole as you approach that point if the function goes to a limit a finite limit namely you get a complex number in the limit then the function has a removal singularity at that point and on the other hand if neither of these happen namely that you do not get a limit then it is an essential singularity. So in the same way I want to be able to say I want to be able to treat the point at infinity okay so the question is how do you think of the point at infinity and the key to that is the Riemann stereographic projection okay which tells you that you can think of the point at infinity as the north pole on the Riemann sphere okay. So you let me draw attention to this diagram that I was drawing towards the end of last class the last lecture so here is the so you can see here is the 3 dimensional real space R3 and you have the xy plane which is being thought of as the complex plane okay and then instead of the usual z axis I am calling it the u axis which is the axis perpendicular to the xy plane and the reason is obvious because I want to use reserve the symbol z for x plus i y okay and then what I do is that I take this sphere centred at the origin radius 1 unit okay and that sphere minus the north pole okay which is the point with coordinates 0, 0, 1 if you throw that point out the rest of the sphere is mapped homeomorphically on to the complex plane okay by this map phi which is called the stereographic projection and then what you do is that I asked you to check that this map phi is a homeomorphism now what you want to do is that what is missing in the complex plane is a point that corresponds to infinity so basically you want to look at c union infinity you want to take this set which is c union infinity and c union infinity is called the extended complex plane okay and your idea is to send this infinity to the north pole because that is the c union infinity contains c as a subset okay and this c the complex plane is by the stereographic projection homeomorphic to the Riemann sphere minus the north pole okay so what is missing on the Riemann sphere side is a north pole what is missing on the extended complex plane side is a point at infinity so what you do is that you send the north pole to infinity under the stereographic projection okay this is a definition you make and once you do that you get a bijective map from the Riemann sphere to the extended complex plane okay and now the point is that this bijective map if you take this map outside the north pole it is a homeomorphism okay now what you do is you do the following thing you put a topology on the extended plane c union infinity in such a way that this extended map from the full Riemann sphere to the extended plane which is c union infinity that becomes a homeomorphism okay so you see what you must understand here is a technical point okay see I have a topological space and I am seeking to add a point at infinity to it okay this is done in topology what are called as locally compact house of spaces okay so if you have studied this in a first course in topology which you should have and if you have not it is not very difficult to read it up from a standard textbook the idea is that you want to compactify a space you want to make a space compact and what you should do is that you should start with a locally compact house of space okay and then you add a point at infinity okay and just adding a point at infinity will only give you a set okay but you have to topologize it and the method and the method you and the topology that you give is the following what you do is that for this for the set along with the point at infinity which will be the one point compactification okay the open sets are going to be not only the usual open sets of your topological space but to that you also add complements of compact subsets of your topological space along with that point at infinity okay and these are supposed to give you neighbourhoods of the point at infinity okay so this is how you do the one point compactification and that is exactly what is happening here okay so let me write here so topologize the extended plane C union infinity so that phi becomes a homeomorphs this is what you do okay so again let me tell you the idea the idea is you want to think of the point at infinity okay and of course so it has to be a new point no point on the plane can be thought of as a point at infinity okay you have to add a new point and we give it the symbol infinity okay and then now you have a set you have the complex plane which is a nice topological space it is even a metric space it is R2 basically okay but then you added one extra point and once you add the extra point you also have to tell me what is going to be a topology and the idea is that you make this topology in such a way so that this complex plane along with this extended point at infinity that is the same by same I mean homeomorphic to the Riemann sphere okay so topologically it looks like a sphere okay and at first sight this looks a little complicated because you know the complex plane is unbounded okay the complex plane is unbounded and what you have done is by adding this point at infinity or made it compact because you know the extended complex plane is now you are trying to put a topology on the extended plane which is a plane plus the point at infinity in such a way that it is homeomorphic to the Riemann sphere but you know the Riemann sphere is compact because it is a close and bounded subset of Euclidean space it is compact okay and therefore and you know if a topological space is isomorphic topologically isomorphic that is homeomorphic to another topological space which is compact then the original space is also compact anything isomorphic any space homeomorphic to a compact space is compact because the image of a compact set is compact under continuous map okay therefore what you have done is you have added this point at infinity and you have put a topology in such a way that the whole thing becomes compact just adding one point you are making it compact mind you if you remove that point then the space is even unbounded okay so the plane as it is unbounded okay so now what I want you to so let me write this down so I want you to check the topology above is the same as the unique one point compactification C is equal to R2 okay so this is something that I wanted to check okay alright and well in fact let me even illustrate it for a moment because I want you to understand what it is that is going on so you know so you have this you have this so let me again redraw this diagram so I have this 3 dimensional space R3 and I have this xy plane which is for me the complex plane okay and then I have this third axis which I do not call z I call it as u okay and this is the origin so I draw this I draw this sphere of radius 1 unit and look at the surface of the sphere so here is the circle this is the unit circle in the complex plane and that is the equator for the sphere that came out to be a little okay so this is the Riemann sphere okay and what the stereographic projection does is that there is this north pole and you give me any point on the sphere on the surface of the sphere it is mapped on to the point in the complex plane that is gotten by joining the north pole to that point and by a straight line and looking at the point where the line intersects the plane okay so this is the stereographic projection so it is it is P going to P of P so this is the this is the stereographic projection. Now what I want you to understand is that what is and this you see the point at infinity on the complex plane is not visible in the picture okay the point at infinity is really at infinity you cannot really see it okay but its inverse image under the stereographic projection is visible it is the north pole on the Riemann sphere okay and now you see since the topology on the extended complex plane is a homeomorphism with the Riemann sphere a neighbourhood of the point at infinity in the extended complex plane must correspond under this homeomorphism to a neighbourhood of the north pole on the Riemann sphere and what is the neighbourhood of the north pole on the Riemann sphere after all the topology on the Riemann sphere is just the induced topology okay the Riemann sphere is just a sphere in surface of a sphere in three dimensional space R3 and R3 has a standard topology and this inherits the subspace topology okay so what is the neighbourhood of the north pole going to look like it is going to be a set that contains a disk the intersection of a three dimensional open disk centre at the centre at the north pole which is what an open disk in three spaces with the surface of the sphere so what you are going to get is that you are going to get you are going to get something like this so let me again change colour to something else so you see basically a neighbourhood of the north pole is going to look like this this is how the neighbourhood of the north pole is going to look like on the Riemann sphere okay and this by our definition is its image should give you a neighbourhood of the point at infinity on the extended complex plane so that helps us to think of a neighbourhood of infinity on the complex plane okay on the extended complex plane so what is the image of this so you know if you draw the image of this see this boundary well of course you know I am I should not include this bounding circle okay because well then it will become a compact neighbourhood okay but I but throw that do not worry about the boundary okay so if you look at the boundary circle okay so if you want you know we can I do not know if the whole of it will go away yes it does so let me put dotted lines like this okay tell you that it is actually an open set okay now this boundary dotted line which is a circle the image of that under the stereographic projection is going to give me a huge circle in the complex plane okay so if I if I draw this the image of this that circle I am going to get this I am going to get something like this this is what the image of that dotted circle on the Riemann sphere is after you do the stereographic projection on the plane okay and what about smaller and as a circles if you make this dotted circle on the Riemann sphere smaller this circle on the plane which is the image of that circle under the stereographic projection becomes larger okay so the image of this shaded region is going to be the exterior of this dotted circle on the complex plane so you know what I am going to get is I am going to get you know so you know so I am going to get something like this I am going to get all this everything outside I am going to get this is what I am going to get on the complex plane this is the image of that disk that open neighbourhood of the north pole on the Riemann sphere its image in the complex plane is the exterior of a circle okay so what this tells you is it tells you the following thing it tells you that the neighbourhood of a neighbourhood of the pointed infinity on the extended plane should be thought of as the exterior of a circle in the complex plane okay so that is the reason that when you try to write a neighbourhood of infinity in the complex plane you write it in the form mod z greater than 1 by delta okay you have to write it the exterior of a circle centre at the origin okay equation for a circle centre at the origin is mod z equal to R where R is the radius and you want R to be sufficiently large and you want to look at points outside the circle so that means mod z should be greater than R okay and instead of taking R large you can replace R by 1 by delta where delta is small okay therefore what this tells you the Riemann stereographic projection tells you that in the extended complex plane a neighbourhood of infinity is just the exterior of a circle okay and what is it that corresponds to smaller and smaller neighbourhoods of infinity it corresponds to the exteriors of larger and larger circles the larger your circle is and you take the exterior of that that means that you are getting closer to infinity okay a larger circle means if you take the exterior of a larger circle it means you are going to a smaller neighbourhood of infinity and how do you actually visualize this you visualize this by looking at what it translates to in terms of this stereographic projection on the Riemann sphere on the Riemann sphere you get smaller and smaller and smaller discs close to the north pole so it does make sense okay so the moral of the story is that so let me write it here it is very very important neighbourhood of infinity in the extended complex plane C union infinity is mod z greater than R if you if you want you can write it as 1 by delta R sufficiently large delta sufficiently small okay and the larger you take R you are going to a smaller neighbourhood of infinity okay of the point at infinity okay and because of this stereographic projection this is very easy to now visualize okay okay now given this we can now we are in a good position to define the limit of the limit as z tends to the point at infinity okay so I think let me see whether I have done this no I have not done it before so let me do this so let me anyway recall you see so let us go back to this definition what is the meaning of limit as z tends to z0 for z0 an ordinary complex number it means that your mind you z0 is fixed it is a fixed complex number z is a variable and you are letting the variable z tend to z0 and this is happening on the plane okay and what does that mean it means that you are making the distance between z and z0 which is mod z minus z0 you are letting that to go to 0 okay and which in terms of delta notation is that you are looking at this mod z minus z0 less than delta which is a sufficiently small open disk centered at z0 radius delta the interior of that disk okay and you are making delta small enough okay that is what it means now what I want to do is that I want to translate this definition and or rather give a version of this definition for which z0 is actually the pointed infinity okay because I want to make sense of limit as z goes to infinity okay and how does one do that so one uses this one uses this the intuition that comes in the last line of the definition namely you letting z to tend to z0 is supposed to mean that you are looking at a you are allowing z to vary in a small neighbourhood of z0 okay that is what limit z tends to z0 means alright in the same way if you now think letting z tend to infinity should mean that you should allow z to vary in a small neighbourhood of infinity okay so that is what it should mean so but now you know how to think of a small neighbourhood of infinity a small neighbourhood of infinity is the exterior of a large circle on the complex plane centered at the origin. So now you have a very easy way to give a definition as to what it means when z tends to infinity so let me write that down so let me write it here we may define we may thus define limit z tends to infinity for z in mind you now z should be now z is in the for z in the extended complex plane which is or let me just keep for z in c union infinity as z as mod z greater than r for r sufficiently large r tending to infinity okay or as mod z greater than 1 by del 1 by delta for delta sufficiently small and delta tending to 0 okay so this is how you can define what limit z tends to infinity means okay you are allowing z to come closer and closer to infinity that means you are restricting z to be in a neighbourhood of infinity and you know what a neighbourhood of infinity looks like it is the exterior mod z greater than r of a circle of sufficiently large radius and if you want to get it to get more closer to infinity then you must increase r you must take the exterior of larger and larger circles okay. Now well let me let me go back to something that I completely forgot which I said I would do and I did not so here it is about this check here see this check was that the topology that you put on the extended complex plane so that there is stereographic projection so homeomorphism is the same topology that you would give to the 1 point compactification and that is correct because in the 1 point compactification the topology is such that the open sets are the usual open sets of the space plus you also take complement of compact subsets of the space and add the point at infinity and if you look at the stereographic projection okay what is an open what is an open neighbourhood of infinity an open neighbourhood of infinity in the extended complex plane corresponds to an open neighbourhood of the north pole on the Riemann sphere okay and if you take an open neighbourhood of the north pole on the Riemann sphere mind you it is a complement of a compact set because it is complement on the Riemann sphere is a closed set and it is a closed subset of the Riemann sphere is already compacts okay so it is already compact so it is a complement of a compact set okay So this argument should tell you that this check is correct I mean this is how you check it okay so that is what I forgot so I just wanted to go back to it. Now let us come back to this business of defining limit as z tends to infinity now that you have this you can now for example define the limit of for example say you can take the limit of a function as z tends to infinity okay. So for example limit z tends to infinity f of z is equal to lambda which is a complex number means that given epsilon greater than 0 there exists a delta greater than 0 such that the distance between f of z and lambda can be made smaller than epsilon whenever the distance between z and infinity can be made small enough which is the same as saying that mod z can be made larger than 1 by delta okay. So this is how you define the limit as z tends to infinity of a function being a finite value finite complex number okay and you see so you must understand limit as z tends to infinity f of z is lambda means that the function values f z the distance from lambda which is mod f z minus lambda can be made as small as you want how small as small as epsilon which is already given to you that epsilon could have been any small okay you can make it as small smaller than that epsilon provided you choose z sufficiently close to infinity and what is choosing z sufficiently close to infinity you have to choose z outside a sufficiently large circle in the complex plane centre at the origin and that is exactly this condition okay that is this condition that mod z minus mod z is greater than 1 by delta for delta sufficiently small okay. So mind you in order to say this it is implicit that the function should be defined in a neighbourhood of infinity that means the function should be defined outside a circle okay the function should be defined in a neighbourhood of infinity of course if a function is not defined in the neighbourhood of a point there is you cannot talk about the limit of the function as you approach that point okay because to take a limit you must be able to approach the point and if this is to be a well defined limit then you should be able to approach that point in all directions basically you need to allow that allow the variable to approach that point from a neighbourhood of the limiting point. In this case the limiting point is a point at infinity a neighbourhood of that is exterior of circle in the complex plane and so what you must understand is it is very very important so let me write it in a different colour f needs to be defined in mod z greater than r for some r greater than 0 otherwise this definition does not make sense okay. At least this is you need to require this of f if you want to study properties of f such as analyticity okay so now what I want to say is that so suppose you are looking at a function f which is analytic and which is defined outside in a neighbourhood of infinity namely outside a circle in a complex plane okay then you see that infinity becomes a singular point it becomes an isolated singularity okay it is because the function has been assumed to be analytic in a neighbourhood of infinity okay therefore infinity if you take a deleted neighbourhood of infinity in the extended complex plane you will simply get the exterior of the exterior of a sufficiently large circle in the complex plane and then the function is analytic therefore infinity becomes a singular point and then now you can talk about whether the singular point is removable or essential or a pole okay and the standard way this is done in a first course in complex analysis is that you study the behaviour of the function at infinity you study the behaviour of f of z at infinity by looking at the behaviour of f of 1 by z at 0 this is what you do but what I want to tell you is that that is the same as what we are doing now okay so there are 2 definitions and we need to reconcile them okay and we will try to do that next okay.