Okay, so let us continue with whatever we were doing, so let me very briefly recall what we have been seeing, so you have been asking this question as to what will the size of the image of an analytic map be okay, so basically you have an analytic function, analytic is same as holomorphic function on the complex plane, it is defined on an open subset of the complex plane okay and you want to know what is the image, namely you want to know what is the set of values that the function takes and you want to know what this set is in the topological sense and also how big the set is and I told you that there is the to answer what the set is topologically, the image of a domain will again be a domain, the reason being that any non-constant analytic function will be an open map okay that is the so called open mapping theorem right, so the image of a domain which is by definition an open connected set will again be an open connected set and of course you know the image of a connected set will be connected because that is the property of a continuous function and analytic functions of course continuous, so at least you know that the image of a domain is a domain okay and the image is certainly an open set okay, now the question is the next question is that we asked was how big is the image okay, so the answer to that at least in the case of let us say analytic functions which are analytic on the whole plane which are called as entire functions, so in that case we have the so called little Picard theorem or the small Picard theorem which says that a function which is analytic on the whole plane okay, its image will be either the whole plane or it will be a punctured plane namely it will be the whole plane minus 1 value, so that is the only if at all it misses a value, if at all it misses values it will miss only 1 value okay or it will miss no values okay and the case when it misses 1 value is the in that case the image is a complex plane minus 1 value and that is called a punctured plane okay and the standard example is the exponential function is it going to e power z which misses the value 0 and takes every other value which you can verify because any non-zero complex number has a logarithm okay. So and of course you know if you take functions like polynomials okay then you will see that the image of the whole plane under a polynomial again be the whole plane and this is for example it is possible to deduce this using the fundamental theorem of algebra that any polynomial equation in one variable in one complex variable with complex coefficients always has all its roots as complex numbers okay. So well so see the idea is that you know in a first course in complex analysis the little Picard theorem is stated okay but since we are since it is advanced complex analysis we would like to see a proof of the little Picard theorem and interestingly I was telling you last time that the key to the proof of the little Picard theorem that we are going to see is actually having it deduced from the so called big Picard theorem or the great Picard theorem and that is interestingly a theorem which involves a singularity okay. So we have the so called great Picard theorem the great Picard theorem says that you know you take an analytic function and you take a point which is an essential singularity for the function okay and then you take a small disc about the essential singularity small open disc about the essential singularity such that in that disc the function is analytic of course leaving out the singular point and then the image of that disc no matter how small is going to be again the whole complex plane or the whole complex plane minus a single value namely the punctured plane and that is what the great Picard theorem says and that is an amazing fact so what I want to tell you is that we so this leads us to understand what singularities are okay and eventually let me tell you that you know when we try to prove the big Picard theorem we will have to study families of analytic functions with singularities and in particular singularities which are poles okay and such functions are called meromorphic functions and so we have to study families of meromorphic functions we have to do study the topology of the space consisting of elements which are actually meromorphic functions that is the generality in which we will have to go to understand the proof of the great Picard theorem okay but to begin with we need to worry about singularities so what I am going to do now is I am going to tell you I mean I am going to recall things about singularities which you have probably seen in the first course in complex analysis okay but it is anyway good to recall them so let me write here so singularities of an analytic function so singularities of an analytic function is what we are going to worry about so first of all let me recall so what is a singular point of an analytic function so you see by definition a singular point is defined only for a function is analytic okay if a function is not analytic then there is no question of talking about singularities okay so the idea behind defining a singular point is that the singular point should be approachable by points where the function is analytic a singular point should always be a limit of good points good points for the function means good points I mean points where the function is actually analytic okay so you know in a way when I say singularity of an analytic function it seems to be you know a misnomer or you know it is on the one hand an analytic function is supposed to be analytic at all the points where it is defined okay and then I say singularity of an analytic function it sometimes looks odd but that is not the point the point is that you see a function which is analytic is usually analyticity is defined on an open set okay there is an open set of points at each of which the function is analytic and of course you know for the definition of analyticity you need an open set you need every point to be an interior point of the domain of analyticity okay. So but the question is that if I move to the boundary of this open set if I go to a point in the boundary of this open set then how is it that the function is going to behave that point may be a point where the function might continue to be analytic or it may fail to be analytic okay and it is only about these boundary points in the boundary points of the domain or the open set where the analytic function is defined these are the points that we have to study for singularities okay. So the very first thing is that you know a singular point is defined only for an analytic function okay and by definition it is a point such that it is approachable or it is the limit of points where the function is analytic okay. So let me write that down a singular point z0 of an analytic function f of z is a limit of points where f is analytic but such that a priori f may not be defined at z0. So you see so look at this definition very carefully what it says is that you know a point z0 in the complex plane is a singular point for an analytic function if you can approach that point it is a limit of points where the function is analytic okay but at that point itself the function may not be defined okay and I of course the phrase a priori means is that to begin with okay or in advance you do not know whether f is defined at z0 or whether f can be defined at z0 these are things that you do not know okay. So it is a point which is outside so a singular point is a point which is outside the domain of definition it is outside the open set where the function is defined and your question is whether the function how does the function behave close to that point you see you what you must understand is the reason why we define a singularity like this is because you see I want to study a function at a singularity okay and I told you what is the motivation why should we at all worry about functions with singularities the answer is because singularities occur not all functions are going to be entire okay not all functions that you are going to study are going to be defined on the whole complex plane okay lots of functions occur they come up naturally with singularities so for example you take the identity function f of z equal to z that is the identity function that is of course entire okay but the movement you inverted if I take f of z equal to 1 by z okay then you see immediately at 0 it is not defined okay so the problem is that there are even you it is very easy you know to get hold of functions which you cannot define at a point and but that point is surrounded by points where the function is analytic okay therefore such a point is a singular point okay so singular points will come very naturally they are they are the most natural things that you have to come across okay you have to study okay and of course I told you the first motivation was that you know the we are trying to prove the big Picard theorem which is actually a theorem about the mapping properties of a function around an essential singularity okay so that is also motivation as to why worry about singularities okay. So let me come back to this definition of singularity see the point is that I have a point that is a point z0 where the function is not defined okay but I would like to study the function close to that point okay and why should I study the function close to that point because that is the only way I can study how the function behaves as I go closer and closer to that point okay so it means that no matter how close I get to z0 I should be able to study the function okay that means the function should be defined no matter how close I get to z0 that is the only if the function is defined can I study it okay so that is the reason why a singular point is always defined as a limit of good points okay so what I want to tell you is that there are functions for which you know singularities per say do not exist okay. So for example take the example of f of z equal to let us say mod z the whole square okay then if you take f of z equal to mod z the whole square this is of course defined on the whole complex plane okay and if you check mod z the whole square is z into z bar where z bar is a conjugate of z okay and you will see that the Cauchy Riemann equations are satisfied only at the origin okay so if at all this function is differentiable it will be only at the origin okay so certainly the function cannot be you cannot find a single point where this function is analytic okay because analyticity means that the function should not only be that point it should also be in a whole neighbourhood around that point but there is no such point the only point where this function f of z equal to z square mod z the whole square is differentiable is the origin and at that point it is not analytic because at no other point it is differentiable. So far if you take this function what is the set of singularities it is the empty set I mean in fact singularities are not even defined because it is not even analytic okay so what you must understand is that singularities are defined only for analytic functions okay you do we are not worried about functions are not analytic in the first place okay so that is one thing that is one point then the then the second thing is that you know singularities come in two in two categories if you want or two types okay and one is friendly one is less friendly the other is more friendly okay see the more friendly ones are called the so called isolated singularities okay what is an isolated singularity it is a point where the function has a singularity but there is a small open disk surrounding that point where the function has no other singularities so it means that there is a deleted neighbourhood of the point where the function is analytic okay such singularities are called isolated singularities okay and then there the less friendlier singularities are the so called non-isolated singularities okay and these are more difficult to study okay the standard example of non-isolated singularities is the is that of you know the log function okay if you take f of z equal to log z to be the principal branch of the logarithm okay you know that to make it analytic you have to make you have to throw out the negative real access along with the origin of course origin will never come into the picture because you cannot define log 0 okay and then you will have to cut out the negative real access okay and then you get the so called slit plane it is a plane minus the real access from the origin to the going to minus infinity though that whole line segment that whole ray is cut off okay so this is the function on way this is the domain the slit plane is the domain where the principal branch of the logarithm can be defined and it is analytic there and every point on the negative real access is a singularity by definition because the function is not defined there and it is not analytic at those points okay. So well in fact the truth is that the function can be defined at each of those points but you cannot define it in such a way that it becomes analytic okay on the whole puncture plane okay so the negative real access in the case of along with the origin is all the points on this ray they are all examples of non-isolated singularities okay so let me write that down so singularities are of two types isolated and non-isolated well so let me write that down z0 is an isolated singularity of f of z if there exists an open disk 0 less than mod z minus z0 less than epsilon for some epsilon greater than 0 where f is analytic this is just another way of saying that there is a small neighbourhood around z0 where the function f is analytic okay and so and what about a non-isolated singularity well singularities which are not isolated are non-isolated singularities okay as the name says. So singularities that not isolated non-isolated okay and of course the examples are well you take f of z equal to 1 by z then z equal to 0 is of course an isolated singularity then I can take f of z equal to if you want sin z over z again z equal to 0 is an isolated singularity okay but you will recognize immediately that the limit as z tends to 0 sin z by z is 1 so it is a singularity that can be really removed okay we will see about that very soon I will let me give you the principle branch of the logarithm f of z equal to principle branch of log z which is ln mod z plus i times principle argument of z with principle argument of z varying from minus pi to pi minus pi included pi not included plus pi not included and this is so let me write here the negative real axis including 0 or let me say points on the negative real axis including 0 are non-isolated singularities okay. So the logarithm of course you know you have to also keep in mind the domain of definition so in the case f of z the first example f of z equal to 1 by z the domain of definition is the punctured plane the complex plane minus the origin and origin is the isolated singularity in the second case also it is the punctured plane it is the complex plane minus the origin and the third case of course the principle branch of the logarithm it is a slit plane so it is the plane minus the negative real axis along with the origin removed okay fine. So well that is that now what are we going to do so let me tell you that we are worried only about isolated singularities we will not be worried about the non-isolated singularities but then let me also tell you that what is the way to study non-isolated singularities the one of the theories that helps in the study of non-isolated singularities is the theory of Riemann surfaces okay. So the point is that when you have non-isolated singularities then you basically have usually you have a curve where which is full of points where there are singularities okay. So in the case of the principle branch of the logarithm this curve is actually the negative real axis okay and such a curve is called a branching curve or a branch locus okay of your function and the way to study that is to do what is called to go to what is called the Riemann surface of the corresponding function okay so there is the key to study studying simple non-isolated singularities which lie on a curve is a study of we will lead you to study of Riemann surfaces okay but anyway we are not going to do that but this is just to tell you that non-isolated singularities can also be studied alright and then you can also have a very strange situation like there may be a function which has only one singularity and that one singularity alone may be non-isolated and all the other singularities may be isolated you can have all kinds of examples so here is let me give you one more example so here is another example you take f of z to be 1 by sine of 1 by z okay look at this function 1 by sine 1 by z you see the point is that whenever you take the reciprocal of a function your reciprocal is always in trouble whenever the function vanishes okay so when I write 1 by sine 1 by z this is cosecant of 1 by z okay and the problem with this function is whenever the denominator which is sine 1 by z vanishes and you know sine 1 by z vanishes when whenever 1 by z is n pi so the problem is that problem is at the point z is equal to 1 by n pi where n is where n is an integer okay so this is the these are the points and among these you know you can see that if you take the function sine 1 by z that is already that already involves 1 by z and 1 by z is not defined as 0 okay so 0 is already a problem for the function sine for the function 1 by z so it is also a problem for the function sine 1 by z okay therefore you see of course when I write 1 by n pi I must make sure that n cannot be 0 okay because it does not make sense so n cannot be 0 but then I should also include z equal to 0 because this is a point where the function even the function the denominator is not defined namely sine 1 by z is not defined okay now if you look at it carefully this as n becomes larger in size okay 1 by n pi comes closer and closer to the origin okay and therefore you see but all these 1 by n pi for various n not 0 they are all isolated singularities in fact they will be simple poles as we will see later okay but the origin will be a non-isolated singularity so here is an example of a function which has 1 singularity which is non-isolated and all other singularities are isolated okay so let me write this down z equal to 1 by n pi n an integer which is different from 0 these are all isolated singularities and z equal to 0 is non-isolated so you see you can have so this is another interesting example okay fine so what we will do is that we will start worrying about only isolated singularities okay and so we will leave out the case of non-isolated singularities and go to the case of isolated singularities so how do you classify isolated singularities so this is again something that you should have done in a first course in complex analysis the isolated singularities are classified as removable singularities poles and essential singularities okay so let me recall what these things are so let me first say in words what is a removable singularity a removable singularity is essentially a singularity that can be removed namely that is an isolated singularity okay but the function can be extended to the singularity in a way that it becomes analytic okay it is like it is the analog of removable discontinuity that you study in first grade analysis okay so well then you have the so called poles what kind of singularities are poles these are going to be poles are supposed to be thought of as zeros of the denominator okay so the point is that you cannot divide by 0 so whenever the denominator becomes 0 the function is not defined so all the places where the denominator becomes 0 these are the poles okay and they should be and when I say 0 it should be 0 of certain order okay and in general you think of poles as zeros of the denominator the other way of saying it is that 0 the poles are actually zeros of the reciprocal of the function okay so a 0 of the reciprocal of the function okay is exactly what a pole of the functions okay and then what are essential singularities by definition these are the singularities which are neither zeros nor neither poles nor removable okay that is a clever way of defining them because then you do not have to you do not have to define them separately okay so let me write down these definitions so isolated singularities are of 3 types so the first one is they are called removable singularities the second ones are called poles and the third ones are the essential and by definition they so if you go to definitions essential is defined as not removable not pole okay that is how you define essential singularity and you may be wondering why the name essential singularity well the let me tell you they are really essential because they kind of completely the function the behaviour of the function the neighbourhood of the essential singularity can distinguish the function from other functions. So it is that though it is a singularity of the function it is like it is an it is very essential for this function it can it can distinguish the function it holds all the information more or less about the function okay that is why it is called essential the behaviour function in a neighbourhood of the essential singularity completely holds the holds the full information about the function okay that is why it is called essential okay we will see more about this later and of course pole is let me say this is 0 of the reciprocal this is what a pole is and removable is well to say it in the simplest words it can be removed. So this is as simple as it goes but then you know so there are many ways of characterising the so called removable singularities poles and essential singularities and one of the keys to that the doing these things or for that matter one of the keys to studying the function around an isolated singularity is the so called Laurent expansion of the function okay. So this is how you would have you would have all gone through a first course in complex analysis where you have used Laurent you have come across Laurent series and then you have used the residue theorem often trying to find the residue at a pole and so on and so forth. So the Laurent series is one concrete way of trying to get a formula for the function as a series of some of powers both positive and negative around a isolated singularity. So let me state the following thing so this is let me just recall this is Laurent's theorem which is kind of very helpful to study functions around isolated singularity okay so here is Laurent's theorem if z not is an isolated singularity of f of z then f of z is equal to sigma a n z minus z not to the power of n n equal to minus infinity to plus infinity okay this is the Laurent series of f about z not okay valid in 0 less than mod z minus z not less than r okay where r is the distance is the distance from z not to the next singularity of f. So what do I mean by next singularity of f what I mean by that is the next nearest singularity of f so maybe so let me write next nearest okay so this is the Laurent's theorem okay I have stated Laurent's theorem for an isolated singularity okay but Laurent's theorem is also valid in an annulus actually okay so and you know a deleted a punctured disk is a special case of an annulus with the inner radius 0 you know an annulus about a point is the open region between 2 circles centered at that point of different radii okay and if you make the inner radius 0 okay then you get a punctured disk which is also a special case of the annulus so and in that if you make the outer radius infinity then you get the punctured plane so a punctured plane is also a special case of an annulus okay. So for example if you take the function e power 1 by z okay and you write out the you simply take the what is the Laurent expansion the Laurent expansion is you know e power z has a Taylor expansion which is valid for all z and in that Taylor expansion you simply replace z by 1 by z and that continues to be valid on the whole plane except the origin okay and the whole plane except the origin is again an annulus with inner radius 0 outer radius infinity okay punctured plane is also a special case of an annulus okay. So this is Laurent's theorem and when I write f of z is equal to sigma n equal to minus infinity infinity a n z minus z not power n notice first of all that what this is supposed to mean is that the series on the right converges and it converges to f that is what it means okay and technically what are these a n's the Laurent coefficients they are they are given by integrals okay they are given by integrals so where so a so here a n is 1 by 2 pi i integral over gamma f w d w by w minus z not to the power of n plus 1 this is the this is the these are the values of these a n's and what is this what is this gamma see gamma is a simple so this is so here is z not and gamma is some simple closed curve gamma is some simple closed curve going once around z not okay and of course in the region enclosed by gamma z not is the only singularity and there are no other singularities on gamma for the function okay. So so this is the Laurent's theorem okay and of course you know you must keep in mind that whenever you write an integral like this when you write integral over gamma you know it is very important that for such an integral to be defined the curve should be contour it should be piecewise smooth which means piecewise continuously differentiable curve okay and the integrand should be piecewise continuous at least on the contour for the integral to be defined okay so of course I can deform gamma a little bit and the integral will not change that is because of Cauchy's theorem okay so in particular if you want to make calculations you can take this gamma to be a small circle centered at z not okay with sufficiently small radius okay where really the shape of gamma does not matter okay it is only the fact that gamma should be a simple closed curve simple means that it does not cross itself okay and it goes around once exactly once around z not okay and this is Laurent's theorem and the point the important thing about Laurent's theorem is that as you would have learnt in the first course in complex analysis the most important thing about Laurent's theorem is the coefficient a minus 1 okay when I put n equal to minus 1 what I get is a minus 1 is 1 by 2 pi i integral over gamma fw dw okay and that is important that is the residue of f at z not it is the a minus 1 and it is important because it gives you it tells you what the integral of the function is around a singularity okay if I put n equal to minus 1 I get a minus 1 is equal to 1 by 2 pi i integral over gamma fw dw so integral of fw dw over gamma where gamma is going around a singularity that is a very important thing okay Cauchy's theorem tells you that if you go around a point where the function is analytic okay if you try to integrate a function around a closed along a closed curve okay such as the function is analytic inside and on the curve then you are going to get 0 that is what Cauchy's theorem says it says you will not get anything so but then you can ask a question what will happen if you integrate around a singularity if you take a function and you integrate it along a singularity around a singularity what will you get the answer is a residue so that is why residues are important they help you to calculate the integral of a function around a singularity okay so and that is so essentially it is the that the residue is the residue is 2 pi i so a minus 1 is the residue and 2 pi i times a minus 1 is equal to integral over gamma fw dw that is exactly the residue theorem okay residue theorem actually says that the integral around if you go once around if there is a curve which goes once around a singular point then the integral of the function along that curve is going to give you 2 pi i times the residue that is if you are going around one singularity and then the residue theorem in general says that if you have several singularities then you have to take some of all those residues it will be 2 pi i times some of all the residues this is the residue this is the residue theorem okay which helps us to compute lot of integrals even real integrals which you would have seen in a first course in complex analysis okay so very well so this is law of theorem now what I am going to do is I am going to you know go back to our our study which is the study of singularities and I am going to tell you you know we saw first that there were that there are 3 types of singularities there are the rest there are the removable singularities there are the poles and then there are the essential singularities now let me say something about poles which is something that you would have which you would have come across but you should try to now do this as an exercise that will help you to revise your basic knowledge of complex analysis so here is a theorem so here is a theorem so this theorem about poles let z0 be an isolated singularity of f of z then the following conditions are equivalent number 1 f of z has a pole of order n greater than 0 at z0 okay so and the that is 1 by f of z has a 0 of order n greater than 0 at z0 so this is the definition of a pole this is one of the definitions of the pole of a pole in fact a pole can be defined in many ways and what this theorem says is that it gives you various you know equivalent conditions so the first thing is the definition of a pole which is the which is as the 0 of the reciprocal okay so f has a pole of order n if 1 by f which is the reciprocal of f has a 0 of order n okay and so and what is a what is a second one the second one is limit z tends to z0 f of z is infinity okay so this is this is another condition for a pole the function becomes arbitrarily large in modulus as you approach a pole okay and here is a third one the Laurent expansion of f at z0 has only finitely many negative powers of z minus z so this is the this is the way you define this is the way you define a pole using the Laurent expansion see the Laurent expansion helps you to also classifying lattice so what I want you to do now is that you should you need to I want you to go back and as an exercise prove that all the 3 statements are equal okay at least you should have seen this in a first course but I want you to recall the proof that is an exercise okay and so let me stop here and we will continue in the next lecture.