 Okay so let us continue with what we are trying to do, so you know we are trying to prove the final aim is to prove the Picard theorems okay and the little Picard theorem will be deduced as a corollary of the great Picard theorem okay and but the great Picard theorem the little Picard theorem is about the image of an entire function, the great Picard theorem is about the image of a deleted neighbourhood of an isolated essential singularity okay and what we are going to now first prove which is the easiest thing to prove to begin with to give to get you a feel of things is the Casorative Asterium which says that if you take any neighbourhood of an isolated essential singularity the image of that neighbourhood of course with the singularity omitted of course the image of that neighbourhood will always be a dense subset of the complex plane okay namely it is closure will be the whole complex plane okay what this means is that take any complex number you can always find a sequence of points in any neighbourhood of an isolated essential singularity such as the function values at those points approaches that complex number okay. So let me write that down let us prove it the key to proving the Casorative Asterium is the Riemann removable singularity theorem which we proved last time okay so let me write that down so Casorative V Astras theorem so let z not be an isolated essential singularity analytic function f of z given any complex number w not we can find a sequence of points in a neighbourhood of z not such that if you take the function values at those points f of z n that tends to w not okay so this is the this is the Casorative Asterium you take an isolated essential singularity z not of an analytic function f of z and then you can always find a sequence of points in a neighbourhood of z not such that the function values at those points tends to the limit w not okay so in this way so what does this mean it means see w not can be f of z n tends to w not means that you can no matter how close you go to w not you can find function values okay so that means that w not is in the closure of the set of function values that is what it means so w not is a limit point of the image set of f okay the image set of f is just a set of values of f okay and what this says is that every complex number is in the closure of the image set the set of values that f takes okay and this is this is a very deep theorem because what it says is that it tells you that therefore the if you take the image of a deleted neighbourhood of the isolated essential singularity then the image is going to be huge because a set which is dense is huge okay the image is going to be dense in the complex plane so that means the image closure is the whole complex plane so it is the image is huge and this is the this is kind of much weaker when compared to the statement of the great Picard theorem which says that the image is actually the whole complex plane or at most a punctured plane namely it can at most omit a point okay but you must remember that that omitted point is also in the closure because it can be approached by other points okay so so this Casarati Weierstra's theorem is a weaker version of you know the great Picard theorem but it tells you it answers this question that we have been worried about namely what is the image of an analytic function okay so let us prove this the proof of this is just going to involve it is just going to involve Riemann's removal similarity theorem okay the use of Riemann's removal similarity theorem so so let me so let me write this also in other words the image of f is dense in the complex plane okay this is another way of seeing it okay so let us go on to the proof of the theorem so you know so you know the idea of proof is very very simple what you do in the proof is that you prove by contradiction okay so you assume that there is a complex value which is not in the closure of the image that means there is a complex value which is not in the limit of values in the image okay that means that the image is bounded away from a certain complex value okay you assume that and then you show that if this is the case then your function cannot have an isolated essential similarity at z0 but the similarity has to either be removable or it has to be a pole and this is where you will be using Riemann's removal similarity theorem so this is the technique of the proof the technique of the proof is just by contradiction okay you assume that a certain value is not in the limit okay and then you show that this will imply that either z0 is a removable similarity or it is a pole and both of these are not possible they will give contradiction because you have assumed that z0 is an essential similarity which by definition is something that is neither a removable similarity nor a pole okay so that is how the proof works. So let me write down the proof so prove so let me go to a different colour by contradiction assume that there exists a W0 complex value such that the conclusion of the theorem does not hold so you assume this okay the conclusion of the theorem is that given any complex value at W0 okay you can find a sequence of points Zn such that the function values at Zn approaches W0 now you assume that that is not the case assume that there is at least one W0 for which this does not happen okay and we will try to get a contradiction. So so what does this mean what it means is that it means that W0 cannot be approached by image points okay this the other way of saying this is that you are trying to say that there is a neighbourhood of W0 okay which has nothing to do with the image okay which does not intersect the image so that means that there is an epsilon greater than 0 such that this open disk centre at W0 radius epsilon is disjoint from the image of f okay that is the that is what it means okay so let me write that down so thus there exists epsilon greater than 0 such that mod W minus W W0 less than epsilon does not meet the image of f image of f IM f is just the set of values of f and the set of values of f includes all values that f takes in a also in the deleted neighbourhood of the isolated singularity Zn okay which we have assumed is essential okay so what does this mean so how do I do now? Now you see so let us rewrite that thus for all Z not equal to Z not of course okay we have mod f of Z minus W0 is greater than or equal to epsilon this is what it means okay so let me again tell you what I wrote down before what I wrote down before is that the disk centre at W0 radius epsilon does not meet the image okay it means that there is no point in the image whose distance from W0 is less than epsilon so it means that if you take any point in the image the distance from that point to W0 is at least epsilon or greater than epsilon that is what I have written down so when you write mod f when I write mod f Z minus W0 is greater than or equal to epsilon I am actually saying that the distance between f of Z which is the point in the image of f the value of f at Z and W0 is at least epsilon okay. Now so you know the advantage with having this kind of a thing is that see whenever you have a function which is bounded away from 0 okay what is the advantage of having a function bounded away from 0 the advantage of having a function bounded away from 0 is that you can invert it okay it is reciprocal makes sense okay. So here f Z minus W0 mind you is also a function okay f of Z is a function f Z minus W0 is just the function f of Z added to the constant minus W0 and adding this does not change the analyticity of the function except at the point Z0 where which of course we are not going to worry about okay mind you when I write f of Z I am of course assuming Z is in the domain of analyticity of f okay it is implicit and of course Z is not Z0 because Z0 is not a point where f is analytic to begin with I have assumed that Z0 is an essential singularity it is a singular point okay so the point is that since I have mod f minus W0 is greater than epsilon 1 by mod f 1 by f minus W0 make sense as a function okay. So and what it tells you is that that function in a deleted neighborhood of Z0 is bounded by 1 by epsilon okay that is what it says. So now look at G of Z defined to be 1 by f of Z minus W0 okay in a deleted neighborhood of Z0 okay now look at this function now you see this function mod G Z is greater than or equal to is less than or equal to 1 by epsilon you have that that is just inverting mod f Z minus W0 greater than or equal to epsilon. So what you have is now you have two things see G of Z make sense as an analytic function okay it is because it is a reciprocal of an analytic function and it is defined where the denominator does not vanish okay. So f Z minus W0 never vanishes because if it vanishes then its modulus will be 0 and vice versa but the modulus is always bounded away from 0 it is always greater than or equal to epsilon so f Z minus W0 never vanishes okay and mind you wherever f is analytic f Z minus W0 is also analytic okay because it is just f of Z added to the constant minus W0 adding a constant does not change the analyticity of a function because a constant function is also analytic and some of analytic functions is analytic okay. So f Z minus W0 is also analytic in a deleted neighborhood of Z0 okay and it is non-zero so it is reciprocal 1 by f Z minus W0 is as well analytic in a deleted neighborhood of Z0 okay. So what I have now is I have this function GZ it is analytic in a deleted neighborhood of Z0 okay Z0 is a singular point but look at the last inequality this function is bounded in a neighborhood of Z0. Now Riemann's removable singularity theorem will tell you that Z0 has to be a removable singularity for G okay so since so let me write that down since G of Z is analytic in a deleted neighborhood of Z0, Z0 is also a singular point of G of G and further is a removable singularity by Riemann's theorem on removable singularities since it is bounded okay so here is where we are using the Riemann's removable singularity theorem okay. So what it means is that so it means that G can be redefined at Z0 so that you get a function which is analytic at Z0 as well okay and the fact that you can define G redefine G at Z0 okay should tell you that f has to have at Z0 either a removable singularity or a pole it cannot have an essential singularity that is the conclusion of all this and that is the contradiction to the hypothesis that Z0 is actually an essential singularity and that is how we get the proof of the theorem of the Casorati Weierstrass theorem okay. So let me write that down thus limit Z tends to Z0 G of Z exists call it G of Z0 and then G extends to an analytic function at Z0 okay so this is what removable singularity means you can take the you can so this is probably the right time for me to recall the Riemann's removable singularity theorem what does it say it gives you an equivalence of 4 statements the first statement is that the point in concern the isolated singularity in concern is removable okay which is equivalent to saying by definition that the function can be extended to an analytic function at that point the second condition is slightly weaker that you can extend the function continuously to that point okay namely the condition is that the limit of the function as you approach that point exists okay as a finite complex number and then the third condition was the condition equivalent condition that involve the Laurent expansion and that condition was that if you write out the Laurent expansion about removable singularity you actually get a Taylor expansion namely there are no negative powers there is no principal part there is no singular part okay and then the fourth condition which was the most amazing was the condition that the function is bounded in a neighbourhood of the singularity okay bounded of course bounded means bounded in models okay and I told you that that is the weakest of all the four conditions and that boundedness in the neighbourhood of a singularity can only happen if the singularity is a removable singularity that is essentially Riemann's removable singularity theorem so you know so because of that the limit as Z tends to Z not G of Z exist let us call it as G of Z not so G extends to an analytic function at Z not so here I am using several equivalent versions of the Riemann's removable singularity theorem you must realise that. Now the question is it all depends on what G of Z what is the value of G of Z not is the point is that if G of Z not is 0 okay then f has a pole at Z not okay and if G of Z not is not 0 then f has a removable singularity at Z not okay and then thus we have manifest a contradiction to what we have assumed okay so let me write that down now if G of Z not is 0 we see that limit Z tends to Z not f of Z has to be infinity this has to happen because you know the limit as Z tends to Z not G of Z is 0 so the limit is Z tends to Z not 1 by f Z minus W not is 0 that means the denominator has to become unbounded so that means and if f Z minus W not has to become unbounded in modulus then f has because W not is just a constant f has to become unbounded in modulus and that is the condition for a pole okay so if G of Z not is 0 then limit Z tends to Z not f of Z is infinity and Z not is a pole of f and you see Z not is a pole of f which is not possible so that is ruled out so we have ruled out the case that G of Z not is 0 the only other case is when G of Z not is not 0 okay if G of Z not is not equal to 0 then limit as Z tends to Z not f of Z is actually W not plus 1 by G Z not you will get this okay which means again by Raymond's removable singularity theorem that f has a removable singularity at Z not okay which is again not possible so in both cases you get a contradiction and we are done okay so that is the that brings you to the end of the proof okay so you see we are applying the you must see that we have applied the Raymond's removable singularity theorem twice we have applied it once to G and then we have applied it in one of the cases to f itself okay fine so this theorem is a very nice theorem and so what it tells you is that you take neighbourhood of an isolated essential singularity and take its image you are going to more or less fill up the whole complex plane you are going to get the image is at least dense and we have to move towards the proof the great Picard theorem okay. Now what I am going to do next is I am going to deal with the point at infinity okay so you see the you see in this proof itself for example when I wrote down limit Z tends to Z not f of Z is infinity okay I am using the point at infinity so you would have seen the point at infinity as the extra point that is added to get a one point compactification of the complex plane and you would also have seen it as a Riemann sphere in a first course and but anyway I want to recall these things because you see it is very important from now on to be able to think of infinity both in the domain of definition of the function as well as in the range of values of the function so you want to have a situation where you can talk of a function a variable an independent variable going to infinity and its value at for example the value of a function at infinity you want to say that and you also want a function to take the value infinity okay you want to include infinity into your set of values of the independent variable and the set of values of the dependent variable so you have to deal with infinity carefully and usually in a first course probably this is sometimes not covered very thoroughly so I want to just revise these things so that you are comfortable about thinking about limits at infinity and infinite limits okay so that is what I am going to do next so and I need that because of the following reason see I want to be able to think of infinity as one of the values of a function okay and I also want to be able to think of infinity as a singularity okay see for example if you take an entire function okay then the point at infinity is of course approachable by infinity is of course approachable by any curve on the complex plane which is non bounded okay so you can always approach the point at infinity and then the question is whether the function is analytic at infinity or it is not analytic at infinity so you want to think of infinity as a singular point okay and then the question is what kind of singularity is it because you know we have already classified singularities as either removable or pole or essential so the question is I want to be able to think of infinity the point at infinity as a singularity and question or study when this singularity is either removable singularity or a pole or an essential singularity okay so the point at infinity is very very important so let me go to that so I will take another colour the point at infinity infinite limits infinite values so this is what I am going to I am going to tell you about okay so well so the idea is as follows so what we do is let us so first of all let me you know the approach to everything is since we are doing calculus approach is always through limits so let me recall what finite limit is okay as a way as an independent variable approaches a finite value okay so you know see limit so when I write limit z tends to z0 okay when I write this what does it mean for z0 in C for let z0 be a complex number okay what does limit z tends to z0 mean see it means that you are going closer and closer to the point z0 okay so basically what it means is that and what does it mean to say that you are going closer and closer to z0 if you think of z as a moving point a variable point then you are saying that the distance from z to z0 is becoming smaller and smaller okay so this limit as z tends to z0 can be interpreted as limit as mod z minus z0 tends to 0 okay that is how you can interpret it okay so this means so let me write that here means mod z minus z0 tends to 0 okay means we let so let me write that we are letting mod z minus z0 tend to 0 and mod z minus z0 mind you is the distance between z and z0 and what it means is that if you go to definitions what is the business of letting something go to 0 in analysis trying to let something go to 0 is the same as making it as small as possible so that is where your epsilon comes in so usually we use epsilon for the values of the function and we use delta for the values of variable so let me use delta so the point is that you are putting your choosing deltas as small as you want and you are letting mod z minus z0 less than delta okay so let me write that down which in turn means means let delta be small consider mod z minus z0 less than delta and let delta tend to 0 this is what it means you are making something small means you are actually taking values of that which are getting closer and closer to 0 okay and you know therefore you know the of course all this conveys very clearly what is happening topologically mod z minus z0 less than delta is actually a disc it is an open disc centred at z0 radius delta so you are it means that you are going to a infinitesimally small very very small neighbourhood of z0 and the smaller the delta is the smaller is the neighbourhood is okay so you are basically looking at you are just concentrating attention at a very small neighbourhood of z0 that is what it means okay and now you know in the same way you can so now this can be used to also define what limit z tends to infinity means okay so limit z tends to infinity what do you make of this what do you make of limit z tends to infinity okay so now this is the point where you will have to have a little bit of imagination okay so the idea is that first of all you should be able to think of infinity as a point okay as a concrete point and the second thing is that once you think of it as a concrete point in a space then you can think of limit z going to infinity just as you thought of limit z going to z0 you know limit z going to z0 meant that you were going to a small neighbourhood of z0 now if you can think of infinity as a point in the same way limit z tends to infinity means that you are going to a small neighbourhood of infinity okay so all you need is a way of thinking of the point of infinity as a point on a space where you can think of a small neighbourhood around that point okay and the key to this is as you would have seen in a first course in complex analysis the key to this is the so called Riemann's scenographic projection okay so let me explain that so this is so let me write this down as go to a small neighbourhood of infinity now what does that mean what does that mean so the key is the Riemann's scenographic projection that is the key so let me recall what that is so basically so the idea is you should have seen this so what you do is the following you so let me draw a diagram so here is well this is the this is a three dimensional space so this is my okay so let me draw it like this so this is my usual xy plane this is the origin okay and this is the x axis this is the y axis okay or rather this is if you want to be right handed then that is the y axis is the negative y axis right so if you want the point 1 is here point 1 on the y axis is here so this point is 1, 0 this point is 0, 1 so if you think of this xy plane as usual complex plane then you have 2 coordinates x and y and of course 1, 0 is the point 1 complex number 1 0, 1 is a complex number i okay but I the point is I want to put in a third axis which normally in three dimensions you would call the z axis but I do not want to use z because z is already supposed to be x plus i y so you use let us use something else for it you use you if you want okay some books use you let me also use it so what you do is now you take the three dimensional space now it is not xyz but it is xy u this is a positive u axis then what you do is that you draw this you draw this circle I mean you draw this sphere centered at the origin and radius 1 okay so let me rub these coordinates off because it will make things easier for me to draw so you know I draw this so I have this I have this circle this is the unit circle on the complex plane and then I have this point 1 with u coordinate 1 okay and x and y coordinate 0 okay so it will be the north pole of a sphere of the sphere centered at the origin and radius 1 so what I am going to get is I am going to get something oops so I am going to get something like this so here is my sphere so this is this sphere is this is the sphere it is called the Riemann sphere it is a sphere centered at the origin radius 1 unit and this point here which has coordinates 0, 0, 1 for x, y and u it is called the north pole so I will use the word I will put the symbol n okay and of course you know if you project it all the way down you are going to get the point with coordinate 0, 0, minus 1 which is the south pole so called south pole okay you can think of the earth as a sphere and you have the north pole and the south pole it is just like that and so what is the stereographic projection so what it does is that so let me call this sphere let me give you a name for this sphere so let me call this as I will put this S like a dollar symbol and put S2 and I will put a subscript R so this is the this is standard topological notation okay the 2 on top S is supposed to denote a sphere okay the 2 on top is supposed to denote the dimension okay it is the surface of a sphere okay mind you I am not taking the solid sphere I am taking only the surface of the sphere which is surface okay so it is 2 dimensional and by dimension I mean real dimension so it is real 2 dimensional okay and the subscript R is to remind you that this is being done in real space okay this is being done in real space real 3 space so the ambient space here is R3 oops so the ambient space here is R3 it is in 3 space the only thing is that I am treating the X Y plane as the complex plane and instead of the usual Z axis I am calling it the U axis because Z is already reserved for X plus I Y okay. Now what you do is well this is the stereographic projection it is a very very simple projection what it does is that it goes from the sphere I will call the sphere the Riemann sphere okay so it is called the Riemann sphere okay it goes from the Riemann sphere minus the north pole to the complex plane to the complex plane okay and what is the map so what you do is you take any point on the on the sphere okay mind you you are taking a point on the surface of the sphere okay and you are not taking the point at infinity I mean you are not taking the point n okay you are not taking the point n so as I just said inadvertently the point n will be the missing point at infinity okay so that is the that will be the analogy so we will see that so you take any point P here on the sphere other than the north pole and then what you do is you join this take the straight line passing through n and in P okay that straight line will go down and hit the plane at some point and that point will give you a complex number because for me any point on the plane is a complex number I have thought of the X Y plane as a complex plane and that is the complex number to which I am going to send P to okay and you can see clearly it is a projection map okay it is a projection map and that is why it is called the stereographic projection okay so so basically what I do is I take this line from n passing through P and then it will go and hit the let me call this as P of P so this is the map P which sends P to P of P and P of P is a complex number P of P is a complex number and this is a stereographic projection so you have if you want to think of it as a projection it is like this okay now the beautiful thing is that so the beautiful thing is that this map P is actually bijective map okay you can very well see that if P changes then P will change so it is an injective map okay and conversely give me any point on the complex plane it is of the form P P for a unique point on this on the Riemann sphere because I can get that point by simply joining that point to the north pole that will hit the sphere at a certain point and that will be the point which will be mapped to the given point under P okay so it is very clear that this map is bijective okay it is very clear that this map is bijective and in fact you can try it out as an exercise this map is actually a homeomorphism it is a topological isomorphism see the complex plane has a topology okay and this topology is also this topology that it is the same as the topology that the plane inherits as a subspace of 3 dimensional space okay you take the xy plane take the complex plane xy plane and treat it as plane in 3 space and you take the natural topology on R3 you restrict that topology to the subset that is the same as the topology on the complex plane okay so the topology on the complex plane is same as the topology that it inherits as a subspace from 3 space and the sphere is also living in 3 space so it is also a subset of the 3 space so it also has inherits the topology okay and the fact is that this map from the sphere to the plane is in fact a continuous bijective map which is open and therefore you know its inverse is also continuous so it is a homeomorphism it is a topological isomorphism it is a map which is continuous whose inverse is also continuous okay of course the inverse being defined because the map is bijective okay so the beautiful thing is that Phi is actually a homeomorphism okay so the fact that Phi is a homeomorphism so let me write that down but before that let me tell you something the fact that Phi is a homeomorphism tells you that therefore you can think of the whole complex plane as a punctured sphere okay see S2 minus n is a punctured sphere it is a sphere minus the north pole okay and what this homeomorphism tells you what does a homeomorphism tell you it tells you that 2 spaces are topologically the same means of isomorphism so when you say Phi is an isomorphism topological isomorphism namely homeomorphism you are actually saying that you can this is just another way of saying that the complex plane can be thought of topologically as a punctured sphere okay that is the significance of this statement okay so let me write that down so here let me take another colour Phi is check so this check is something that is more or less obvious but you should do it Phi is a homeomorphism okay I am certain many of you would have done this in a first course in complex analysis but it is not very difficult to do if you have not done it so Phi is a homeomorphism which tells you that the complex plane can be thought of as a punctured sphere okay now the punctured sphere misses only one point namely the north pole and now you know you have you wanted a point at infinity you wanted to attach to the complex plane a point at infinity but the point is where do you attach it I mean you cannot see it okay but then if you look at this picture you think of the complex plane like a punctured sphere now what is the extra point that you will have to add to make it the whole sphere and mind you when you make it the whole sphere only then it becomes compact okay if you remove a point from a sphere it loses compactness okay because it will not be closed since we are in our in Euclidean space a subset is compact if and only if it is closed and bounded so the sphere is of course any subset of the sphere is of course bounded but the problem is that unless it is closed it is not compact so the only way to make it compact is add that missing point that in this case which is a north pole okay so here comes the nice upshot of this what you do is you think of the complex plane plus the point at infinity you denote the point at infinity as with the symbol of infinity and think of it as an extra point that you add to the set of complex numbers and then what is the topology you give you give the topology which makes the natural extension of this homeomorphism phi into a homeomorphism from the sphere to the extended complex plane which is a complex plane with that extra point at infinity added okay so define phi from the Riemann sphere to C union infinity by sending n to the point at infinity okay so the north pole maps to the point at infinity and once you do this what you have done is you have given a bijection between this sphere and C union infinity mind you C union infinity will be called the extended complex plane it is a complex plane plus the point at infinity and therefore the complex plane plus the point at infinity is now nicely identified as a sphere okay and the advantage now is that you have therefore the point at infinity being thought of as a north pole on the sphere and it is a point on a topological space you can do topology in a neighbourhood of the north pole and think of it as doing as working in a neighbourhood of infinity okay that is how you think of infinity the point at infinity okay so I will stop here.