 You see we have been looking at the point at infinity and what we have seen is the situation when infinity is an isolated singularity okay and I have already told you what happens when infinity is a removable singularity or infinity is a pole and even more generally we have also seen the so called residue theorem for the external complex plane okay. So the point that one has to remember is that the residue theorem for the external complex plane which we discussed in the last lecture in that the contribution of the residue at infinity will always be there okay irrespective of whether the point at infinity is a removable singularity or a pole or an essential singularity. So the issue is that you know even if infinity is a removable singularity okay for example for the function f of w equal to 1 by w okay infinity is w equal to infinity is a removable singularity okay because the function tends to 0 as w tends to infinity but still the residue at infinity is not 0 okay which is which does not happen for a point in the usual plane. In the complex plane at a point at which the function has a removable singularity is a point where the function can be redefined so that it becomes analytic and therefore if you take the if you calculate the residue at that point you will get 0 okay whereas this does not happen for the point at infinity that is one point that you should always keep in mind. Now we are going to talk about the situation when infinity is an essential singularity okay so what does it mean to say that infinity is an essential singularity so basically you are looking at a function for which the point at infinity is a singular point an isolated singular point okay that means it is defined in a neighbourhood of infinity in a deleted neighbourhood of infinity and it is analytic in a deleted neighbourhood of infinity which means that basically you are looking at an analytic function which is analytic outside a circle of sufficiently large radius okay and because that is what a neighbourhood of infinity is okay you know that while stereographic projection. Now what does it mean to say that the function has infinity as an essential singularity this is the same as saying there are the 2 ways of saying it of course in fact 3 ways of saying it and all the 3 ways of saying it are correct so there is one way which is using the Laurent expansion at infinity so one way of saying that infinity is an essential singularity is by saying that if you take the Laurent expansion at infinity then the you know the singular part has infinitely many terms okay and there what you must remember is that the singular part is actually the polynomial part I mean it is the if you expand it in powers of the variable okay if W is the variable okay then you are going to expand the function in positive and negative powers of W okay and you should look at an expansion which is valid outside a circle of sufficiently large radius mind you that is the Laurent expansion in a neighbourhood of infinity okay and because you are looking at the point that infinity the positive powers the terms involving positive powers of W that is a singular part okay that is a singular part and the terms involving the constant and the negative powers of W is the analytic part because that because negative powers of W behave well at infinity okay. So the condition for the function to have infinity as an essential singular point is that if you write out its Laurent expansion in a sufficient which is valid outside at all points outside a sufficiently circle of sufficiently large radius centered at the origin okay then you must get you must get the positive powers of the variable that occur they must be there must be infinitely many terms okay which is the same as saying that the singular part at infinity has infinitely many terms and so what you what should not happen is that the singular part has only finitely many terms at infinity which means that it is the singular part is actually a polynomial okay. So in other words what we are saying is that you know how do you recognise that function has infinity as an essential singular point you write out the Laurent expansion at infinity by the neighbourhood of infinity then what happens is that the singular part is not just a polynomial okay in particular what this means is that if you take polynomial functions they are certainly not going to have infinity as an essential singularity and you know that we have seen it last time polynomial of degree polynomials in fact have infinity as a pole and the order of the pole is actually the degree of the polynomial okay. So that is so this is one way of defining infinity as an you know essential singular point of course the other way is that the limit of the function as you approach infinity does not exist okay that is also correct okay. So usually there are 3 characterizations one of any singular point there are 3 characterizations one is you look at the Laurent series and look at the condition of the Laurent series whether it has no terms at all in the singular part or whether it has finitely many terms in the singular part or whether it has infinitely many terms in the singular part and then the second condition is on the limit of the function as you approach that point okay. Of course the third condition is about the boundedness of the function okay or the unboundedness of the function but of course the boundedness is equivalent to the point being a removable singularity okay and the unboundedness means that it is either it could either be a pole or an essential singularity okay. So let me write these points down infinity as an essential singularity of f of w. So you see so f of w is assumed to be analytic in a deleted neighborhood of infinity okay that is a standard assumption blanket assumption and well so what are the conditions well you write the Laurent expansion of f at infinity that is f of w equal to sigma n equal to minus infinity to infinity an w power n valid in a deleted neighborhood of infinity has infinitely term infinitely many terms in its singular part and what is the singular part the singular part is sigma the singular part is the positive powers of the variable so it is sigma n equal to 1 to infinity an w power n and that is the same as saying that an not equal to 0 for many infinitely many n for infinitely many n okay. So and you see this definition is just the same as usual definition that to in order to say that a certain point for example in the complex plane which is an isolated singularity is an essential singularity all you have to do is write out the Laurent expansion at that point and then you find that the Laurent expansion has infinite singular part okay and there it will be infinitely many negative powers okay so of course I should tell you that this is equivalent to f of g of z is equal to f of 1 by z has an essential singularity at 0 okay so g of z is f of 1 by z has an essential singularity at z is equal to 0 okay and this is of course you know this is of course based on this philosophy that you know the behavior of f of w at infinity is the same as the behavior of f of 1 by z at 0 okay where you put w equal to 1 by z that is you already know that this is a homeomorphism okay and it is a holomorphic isomorphism of the punctured plane the plane minus the origin onto itself okay and okay so this is one thing then the other thing is so this is one condition the other condition is limit w tends to infinity f of w does not exist this is the other condition and of course this is the same as saying that limit z tends to 0 g of z does not exist so this is also something that you know and so well and of course the usually we will have another condition which will be on the behavior of the function in the neighborhood of that point and well certainly you cannot expect the function to be bounded in a neighborhood of that point okay because that is equivalent to the function being actually analytic at that point and that is Rayman's removable singularity theorem right anyway so let me go ahead and say some other things so maybe I will put some I will change color and put a few boxes here so this is equivalent to this and this part is the same as this okay alright so now what I want to say next what I want to say next is that you know let us look at let us analyze this condition that infinity is an essential singularity okay now the so for example what are the functions what are the entire functions which have infinity as an essential singularity you can ask this question okay so we have already answered a similar question for poles okay and for removable singularities see you take an entire function mind you an entire function means a function which is analytic on the whole plane okay and the moment it is analytic on the whole plane the whole plane is also mind you a deleted neighborhood of the point at infinity you must not forget that therefore a function which is analytic on the whole plane is also having infinity as an isolated singular point okay automatically okay if you think of the Rayman's stereographic projection you see that the infinity corresponds to the north pole and the whole plane corresponds to the remaining part on the Rayman's sphere which is the Rayman's sphere minus the north pole okay by the stereographic projection so the plane itself is an neighbourhood of the point at infinity and therefore an entire function will always have the point at infinity as an isolated singularity. Now what happens if that isolated singularity is a removable singularity well then you are saying that the entire function has infinity as a removable singularity okay which is the same as saying that infinity at infinity it is bounded or it has a limit okay and then by Liouville's theorem it will reduce to a constant okay so the moral of the story is that a non constant entire function cannot have infinity as a removable singularity we already seen this okay and then you can ask the question when will an entire function have infinity as a pole okay and we have seen that will happen if and only if the entire function is a well is a polynomial okay and the degree of the polynomial will be the order of the pole okay. So now we ask the question when will an entire function have infinity as an essential singularity and the answer to that will be that if you it should not be a polynomial okay basically if you write out the if you write out its well Taylor expansion at any point the Taylor expansion of course you will get if you have to write only a Taylor expansion because it is an entire function okay there is no question of Laurent expansion okay. So at any point in the you choose any point in the plane it is analytic everywhere the plane so you choose any point and write the Taylor expansion at that point that Taylor series will have infinite radius of convergence because this function is entire okay and the point is that the Taylor series should be a power series it should not be a polynomial okay. So the moral of the story is an entire function has infinity as an essential singularity if and only if its Taylor series is not a polynomial okay so and the Taylor series of an entire function being a polynomial is the same as the entire function itself being a polynomial okay. Therefore all your sayings that you know an entire function if you want it to have an essential singularity at infinity okay then it should not be a polynomial alright and for this reason we call such entire functions as transcendental okay so the usually the polynomial functions and the meromorphic functions which are given by quotients of polynomials okay they are called algebraic and everything that is not algebraic is called transcendental so such entire functions which have infinity as an essential singular point are called transcendental entire functions okay. So let me write that down an entire function that has infinity as an essential singularity is one that is not a polynomial okay so and you know what will happen if it is a polynomial infinity is a, infinity is a pole okay if it is a polynomial of positive degree of course if you we also consider constants as polynomials polynomials of degree 0 and that is the case of a constant function. So if you are looking at a non-constant entire function okay then it is the only way infinity is an essential singularity is that it is not a polynomial of positive degree okay. So for a polynomial if constant has infinity as a removable singularity if non-constant has infinity as a pole of order equal to its degree okay so a polynomial is certainly not an entire function that has an essential singularity at infinity and conversely if an entire function has an essential singularity at infinity because you take any point and you write out the Taylor expansion at that point you will see that if infinity is a pole then the Taylor expansion should terminate and it has to be a polynomial. So if infinity is not a pole then your Taylor series will have infinitely many terms okay which is the same as saying that there are infinitely many terms in the Lorentz expansion at infinity okay. Mind you if you take a polynomial the polynomial itself is expansion is the Taylor expansion of the function that it represents at the origin okay and since that expansion is valid on the whole plane it is also a Lorentz expansion at infinity the expression for the polynomial itself is a Lorentz expansion at infinity and it is except for the constant part the positive part the terms involving positive powers of the variable that is automatically the singular part of infinity and that it has only finitely many terms tells you that infinity is a pole okay. So let me write that down conversely if f of w has infinity as an essential singularity then its Taylor expansion at any point of c has to have at any point of c has to have infinitely many terms so it is not a polynomial okay. So to say things in short a transcendental entire function is something that is different from a polynomial now I want to make the following statement that take any entire function okay take any entire function exponentiate it okay so you take f of w to be an entire function okay f of w may have infinity as a you know either a pole or it may be essential but if you take e power f of w okay if you take e power f of w I claim that it will always be transcendental okay. So that means I am saying that for e power f of w w equal to infinity will always be an essential singularity okay that is also very easy to see and how do you see it let me tell you the argument in words you see so I know that f of w is entire and I am looking at e power f of w okay of course e power f of w is also entire. Because it is a composition of entire functions exponential function is of course entire okay and e power f of w is f of w followed by the exponential function it is a composition of entire functions so it is entire okay and what are the possibilities for e power f of w at infinity infinity can either be a removal singularity or it can be a pole or it can be an essential singularity. If infinity is a removable singularity you know then e power f of w must be a constant because of Lieu's theorem and if e power f of w is a constant then f has to be a constant because f is if e power f w is a constant that constant is a complex number which cannot be 0 because e the exponential function never takes the value 0 and f has to be log of that okay f can be one of the logarithms of that constant non-zero constant okay and so in that case e power f so what I am trying to say is that if you are looking at an entire function a non-constant entire function okay e power f has to be transcendent okay the only case you will have to worry about is the constant function when of course e power a constant is also constant okay so if you have a non-constant entire function okay e power f also will be non-constant alright so infinity for e power f if infinity is a removable singularity then f is a constant so if you are looking at a non-constant f then e power f will not have infinity as a removable singularity so it can be a pole now if e power f is a pole has a pole at infinity is a pole for e power f then e power f has to be a polynomial because we have seen that the only entire functions which have a pole at infinity are the polynomials so e power f has to be a polynomial but that is not possible because you see if you take a polynomial of constant non-constant polynomial of positive degree the you know fundamental theorem of algebra says that it will have zeros so there are it will assume the value 0 but that is equal to e power f and e power anything can never be 0 okay therefore e power f cannot be a polynomial alright that means that infinity cannot be a pole for e power f so therefore the only thing that is left is that e power f should have infinity as an essential singularity so the moral of the story is that you take any non-constant entire function and you take e power that the resulting function will certainly have infinity as an essential singularity in other words I am saying that the resulting function it will always be transcendent okay so let me write that down okay so let me use a different colour an entire function that has infinity as an essential singularity is called okay an entire function that has infinity as an essential singularity is called transcendental and so all entire functions which are not polynomials are transcendental okay thus the entire functions the transcendental entire functions are precisely the non-polynomials precisely those that are not polynomials okay if f of w is entire and non-constant then e power f of w is always transcendent for if e power f w has w equal to infinity as a removable singularity then e power f w reduces to a constant by lewill and contradicts f being non-constant also e power f also infinity cannot be a pole as then e power f would be a polynomial which must have zeros okay so that also cannot happen so as a result you take any entire function and you which is not constant and you exponentiate it what you get is a transcendental entire function okay so in this so in this context let me also say that you know the entire function the functions which are either polynomials or quotients of polynomials okay they are called meromorphic functions they are all called algebraic okay and the non-algebraic functions are the transcendental ones okay so let me write that in general the polynomials and more generally the meromorphic functions in the extended plane are called algebraic okay is at least in the sense of algebraic geometry okay in the sense of complex algebraic the non-algebraic ones are called transcendental okay examples of transcendental functions this is something that one should look at so you take for example e power z okay this is transcendental because z equal to infinity is an essential single point and why is that so because basically if you write the expansion for e power z Taylor expansion or McLaren expansion which is Taylor expansion of the origin you write the usual expansion that we all know and you know it has infinitely many terms okay so e power z and then similarly you can take the trigonometric functions you can take sin z you can take cos z and so on okay so these are all transcendental functions and the way of course you know you can also see that infinity is an essential singular point because if you invert the variable you will get origin as the essential singular point so if you take e power 1 by z origin will be an essential singular point if you take sin 1 by z origin will be an essential singular point if you take cos 1 by z origin will be an essential singular point okay so these are examples of transcendental functions and the other thing that I want to tell you is that I want to also recall the big Picard theorem in this connection okay take now you see you know take an entire function which is not constant okay of course if infinity cannot be removable okay because if infinity is a removable singularity then it will reduce to a constant since I have taken a non constant entire function infinity is not a removable singularity the next possibility is infinity is a pole in which case the entire function is a polynomial okay and you know the image of a polynomial map is the whole plane okay because a polynomial can assume all we will assume all values okay you take any value you can equate to the polynomial and you can solve for it and you will get solutions that is because of the fundamental theorem of algebra so if you take a polynomial mapping it will be the image of the whole plane under the polynomial mapping will be the whole plane then you look at the third case namely when infinity is an essential singular point okay now if infinity is an essential singular point okay you see what does the big Picard theorem say the big Picard theorem says that you take any neighbourhood of an essential singular point no matter how small the image will be the whole plane and or it may be a punctured plane it might miss at one point okay now you see this is a so what you are saying is that if I take a transcendental entire function okay for example e power z or sin z or cos z okay then the big Picard theorem will tell you that even if you take the image of not the whole plane but the exterior of a circle no matter of how large radius you will still get the whole plane or the punctured plane and you can compare it to the little Picard theorem which tells you that the image of the whole plane is either the whole plane or the plane minus a point so I want you to understand the significance see if I take for example take e power z okay the little Picard theorem will tell you that the image of e power z is either the whole plane or it is the whole plane minus a point because you know it is e power z we know it is the plane minus the origin okay but the little Picard theorem never tells you what is the image of anything other than the whole plane okay but if you apply the big Picard theorem to e power z okay you are applying mind you when I apply the big Picard theorem to e power z I am trying to apply I am trying I have to apply it only to an essential singularity and where does e power z have an essential singularity at infinity so if I apply the big Picard theorem to e power z at infinity okay then you see that you take any neighbourhood of infinity okay which is you take the outside the region exterior to a circle of any radius no matter how large the image of that itself will be the whole plane or a punctured plane okay and in fact every value is taken infinitely many times so you see you see in that sense how the big Picard theorem is far, far stronger than the little Picard theorem in the case of entire functions you can see that of course for polynomials and there is nothing special because you have fundamental theorem of algebra which will tell you things very clearly but if you are looking at non-pollinomials if you are looking at transcendental entire functions you see you can really see the amount of difference in the conclusion and you can see the strength of the theorem the see the great Picard theorem tells you much more than the little Picard theorem okay. So if you take e power z and you take the image of the exterior of a circle no matter how large radius it will still be the punctured plane okay that is what the big Picard theorem applied to e power z will tell you okay whereas if you try to apply the little Picard theorem you will not get anything it will give you only the image of the whole plane but it will not tell you anything about the image of the exterior of a circle okay so in that sense you see how powerful the big Picard theorems okay. Now what I want to do next is tell you about Meromorphic functions okay so you see so I want to concentrate on Meromorphic functions and you know we need to study Meromorphic functions in order to get to our main aim which is the proof of the Picard theorems alright. So the first fact about Meromorphic functions is that you see if you take a domain either the domain may be in the plane or it may be a domain in the extended plane which means it could include the point at infinity as an interior point on a domain if you look at the set of Meromorphic functions on that domain that is automatically a field okay so you get an algebraic structure called a field and it will be an extension field of the field of complex numbers okay so and this field extension its algebraic properties are deeply connected with the geometric properties of the domain you are studying okay. So let me make that statement so let me go to the next thing the field of Meromorphic functions so let me recall what is the Meromorphic function so it is basically a function which is analytic and has only isolated singularities which can be at most poles okay of course they could be removable singularities but if they are removable singularities you really do not consider them because they are actually points where the function is analytic okay and otherwise the only honest singularities that it has are poles okay so the moment I am talking about a Meromorphic function I have to remember that first of all what is allowed is what kind of singularities are allowed are only poles which means that there are only isolated singularities there are no non-isolated singularities okay there are only isolated singularities and they these isolated singularities are actually only poles okay and the point is that you see if you look at a Meromorphic function in the extended plane okay then you see that since an isolated set in the extended plane has to have only finitely many points okay because the extended plane is compact okay therefore there will be only finitely many poles okay so the moral of the story is that if you are looking at a Meromorphic function on the extended plane it will have only finitely many poles I think probably I will stop here