 Alright so we have been seeing about meromorphic functions and holomorphic functions converging normally you know with respect to the spherical metric. So I want to continue with the few examples so that you get a feel for what is going on. So let me write this down, examples of normal convergence under the spherical metric. So this is what I want to talk about. So the first question that we ask is we have already seen an example of a sequence of holomorphic functions which tends identically to infinity okay. So that is basically it is the sequence of functions z power n okay and the domain is the exterior of the unit circle okay and z power n converges as n tends to infinity to the function which is identically infinity okay point wise and this convergence is in fact normal okay we have seen this. Now you can ask this you can ask the following question can you have a sequence of holomorphic functions tend to a meromorphic function which is honestly meromorphic which means it has at least one pole and the answer is no okay. Sequence of holomorphic functions if it converges normally under the spherical metric either it converges to the constant function which is identically infinity or it converges to a good old holomorphic analytic function. You cannot get meromorphic functions proper meromorphic functions which are having some poles you cannot you cannot take a normal limit of analytic functions and expect a pole to pop up okay that will not happen that is what we proved okay but we think of other possibilities see you have sequence of meromorphic functions so can you think of a sequence of meromorphic functions which converges to say infinity and then of course normally under the spherical metric and can you think of a sequence of meromorphic functions which converge normally to a holomorphic function okay so we have already proved that if you take a sequence of meromorphic functions and suppose it converges normally under the spherical metric then the result is again a meromorphic function or it has to be the extreme case which is the function which is identically infinity okay and mind you let me again stress this is so important because you see in the limit the limiting function that you get could have developed bad singularities it could have developed a non-isolated singularity okay or for example it could have had singularities on a curve it could have happened you know and then it would have been terrible or it could have developed an isolated singularity which is essential and once you have an essential singularity that function is out of our discussion because our discussion we are trying to work only with meromorphic functions and for meromorphic functions we allow only poles as singularities okay so such bad things do not happen you do not have an essential singularity popping up sorry when you take the limit normal limit with respect to the spherical metric you do not have for example non-isolated singularities popping up when you take a normal limit of meromorphic functions okay so this is nice that this is what happens and that is what we proved in the last two classes okay now I just want to give you a couple of examples where you can have a sequence of I think you can have a sequence of meromorphic functions that goes to a holomorphic function okay and you can also have a sequence of so that example is an example where and of course you can also have a sequence of meromorphic functions honestly meromorphic functions which go to infinity okay so you know so let me write down those two examples so here is example 1 so this example 1 is well you know let me say I will just recall that example of holomorphic functions that goes to infinity okay and modify it carefully to make it into a sequence of meromorphic functions that go to infinity okay so this first example is just recalling you put the domain D to be the well the exterior of the unit disk a set of all z sets at mod z greater than 1 mod z greater than 1 this is exterior of the unit disk mind you this is a nice domain in the extended complex plane okay well the fact is that you know if you consider this as a domain in the complex plane it is not simply connected okay because it is a hole in between it has the whole unit disk closed unit disk out as a hole inside but if you consider it this as a domain in the extended complex plane mind you it becomes simply connected and how do you see that it is actually it is actually a neighbourhood of infinity okay and how do you see that if you just take its image under the stereographic projection you will get a small disk surrounding the north pole on the Riemann sphere and that is clearly a simply connected set okay so mind you including the point at infinity turns this from you know multiply connected domain to simply connected domain okay anyway that was just an observation so you take this domain D and then you take fn of z to be z power n n greater than or equal to 1 this is my sequence then you know that fn then fn converges the fn's are all holomorphic functions on D okay they are all in H of D these are all of course fn's are in fact entire functions they are just polynomials so they are holomorphic and well now but what happens fn converges to infinity normally so when I say normally this with respect to this spherical metric okay and when I say fn converges to infinity it means by infinity I mean the function which is constantly equal to infinity at all points mind you we have put that extra function there okay and this is the extreme case alright and now what you can do is you can modify this so that you can get a sequence of meromorphic functions which go to infinity for example and you can also modify this to get a sequence of meromorphic functions which goes to a holomorphic function okay you can do things like that. So for example I can make each of these functions to have a pole that is say for example lying on the real line and it is approaching 1 okay and then if I take the limit the poles are moving towards 1 so what will happen is that the in the limit I will get a function which will have which will actually be holomorphic because I have included mod z equal I have excluded mod z equal to 1 from my domain okay so what you do is we set g of n of z to be well z power n plus let us add a pole at if you want well 1 plus 1 by n which is a little to the right of the point 1 okay so I put 1 by z minus 1 plus 1 by n and let me make this n greater than or equal to 1 okay so if you take these functions g n's they all have poles at the points the g n has a pole at has only one pole that is at 1 plus 1 by n okay and that is of course lying in mod z greater than 1 alright. So for example z 1 has a pole at 2 z sorry g 1 has a pole at 2 g 2 has a pole at 3 1 plus half 1 plus half which is 3 by 2 and then and so on okay and you can see that as n tends to infinity 1 plus 1 by n tends to 1 alright and now you can see that these g n's are actually meromorphic functions on D okay because they are not holomorphic because they have poles but and each has only one pole each function has only one pole and so it is a meromorphic function by definition these are all meromorphic functions and you know if you let n to tend to infinity okay then what will happen is that these z n's of course this z power n will go to identically to infinity for any fixed z with mod z greater than 1 mod z power n is going to go to infinity as n tends to infinity therefore point wise these g n's are going to go to infinity except you will have to worry about the poles okay mind you in a neighbourhood but the poles are all moving away okay so if you take any particular value of z even if it is one of these poles okay for example if you take 1 plus 1 by n that is a pole of g n but from g n plus 1 onwards it is not a pole and of course it is also not a pole for g n minus 1 and functions before that okay so but the mind you the value at a pole is by definition infinity for a meromorphic function we are allowing mind you we are allowing the value infinity okay so somehow what you will see is that if you now take the limit as n tends to infinity okay you will end up with you will end up with the function which is identically infinity okay so this is just a modification of the original example okay so g n tends to infinity right so here is one example so here is an example of honest meromorphic functions which go to infinity okay and well and in fact it would if I forget the z power n and I took only the function 1 by z minus 1 plus 1 by n okay now that function if you see if you look at it that will be a function which is again meromorphic with only one pole at 1 plus 1 by n but now the point is that this function 1 by z minus 1 plus 1 by n because z is in the denominator as you let z to infinity tend to infinity this will go to 0 so this function is bounded at infinity okay so as a result you know if you if you put h n of z to be just 1 by z minus 1 plus 1 by n okay then h n of z as they will all be meromorphic so that gives you you get you know an example of sequence of honest meromorphic functions which tends finally to a function which is holomorphic and it is also holomorphic at infinity it is analytic at infinity because at infinity it is bounded so that will give you the next example so this is this was example 2 which is a modification of example 1 and part of example and then we have example 3 so example 3 is put h n h of h n of z to be simply a z minus 1 plus 1 by n and then you put it put take the reciprocal so that you get a pole at 1 plus 1 by n for h n then these are all of course in they are all meromorphic functions on D so h n tends to 1 by z minus 1 okay as n tends to infinity and yeah and 1 by z minus 1 has a pole at 1 and that is not a problem because 1 is not in our domain alright it is in the it is in the boundary of our domain fine good. So yeah so now what I want to tell you is that what are we going towards next see we need to understand several technical concepts in order to get the proof of the Picard theorems which is our aim okay and one of these concepts one of the technical things that we have to worry about is the so called spherical derivative okay the absolute spherical derivative of a function alright now let me try to explain that and with reference to you know what we already know okay so the first thing is so I will tell I will put this heading as so let me change colour and put this heading as discussion of the spherical metric the spherical metric and spherical derivative because this is something that we need okay we need to worry about this so the first thing is let us go first Euclidean so that is on the complex plane okay. So you know if you give me 2 points Z1 and Z2 on the complex plane then the distance the Euclidean distance between these 2 points Z1 and Z2 is well mod Z1 minus Z2 okay this is the usual distance formula and the infinitesimal distance how is it given you get the infinitesimal distance by putting Z1 as say Z and Z2 as Z plus delta Z okay and then mod Z1 minus Z2 becomes mod delta Z okay and then you change the delta to D okay to get the infinitesimal distance so this is arc length is mod DS okay so in fact let me use the right notation mod DZ okay the arc length is mod DZ and roughly the way you think of it is you put Z1 equal to Z and you put Z2 equal to Z Z plus delta Z so the difference mod Z1 minus Z2 becomes mod delta Z and then change the delta to D the idea is you ignore if you do such an operation you have to ignore higher powers of delta Z than the first so if you get delta Z whole square delta Z whole cube okay all these terms have to be ignored and then you replace delta by D so this is a heuristic way of getting the infinitesimal distance okay and well and how do you for example check that this infinitesimal distance is correct so for example you know if I take integral from Z1 to Z2 of mod DZ if I do this I will get mod Z1 minus Z2 this is what I get okay and that tells you that the infinitesimal distance that you have computed is correct okay so you know how this calculation is done so you have so for example of course when I say this integral from Z1 to Z2 I am going along the this integral is along the straight line segment from Z1 to Z2 so whenever I write an mind you whenever you write an integral there is always a path involved without the path if you write it then it is very ambiguous so here when I say integral from Z1 to Z2 I mean the linear distance the equilibrium distance between Z1 Z2 that means I am taking the straight line segment from Z1 to Z2 okay so what you have to do is here is Z1 and here is Z2 and then you are taking this path the straight line segment okay and then you can parameterize this path you parameterize this path and then you evaluate that integral okay and if you evaluate it you will get mod Z1 minus Z2 so for example you know if you take a point T here how will you how will you write the parameter I mean if you take a point parameterized by T so I should write this Z of T so what will happen to Z of T, Z of T will be so you know Z of T should give me for example if T is the parameter varying from 0 to 1 you know Z of T has to be a convex combination of Z1 and Z2 so when T is 0 I am supposed to get Z1 and when T is 1 I am supposed to get Z2 okay so I will get Z1 plus I guess 1 minus T times Z2 right so in this if I put T equal to 1 I get Z1 sorry I should be the other way round let me change it Z2 plus 1 minus and I think I should put a T here also yeah so you put T equal to 0 I get Z1 and you put T equal to 1 I get Z2 this is a this is a parametric equation of this line segment okay and it is a function of T it is a differentiable function of T in fact it is a linear function of T and what is Z dash of T, Z dash of T is just Z2 minus Z1 okay it is a constant and what is the what does this integral come out to this integral comes out to if you evaluate it it is going to if you have to you know you have to transform the integral into terms of the parameter so you will put T equal to 0 to T equal to 1 and you will plug in this mod dz so it will become so dz is so you know the Z dash of T is dz of T by dt okay so you must think of mod dz as mod Z dash of T into mod dt okay so I am going to get mod Z dash of T into mod dt and you know well instead of mod Z dash of T I am going to put mod Z minus Z1 that is a constant that will come out and I will simply get integral from 0 to 1 mod dt and that is going to be 1 so I will get mod Z2 minus Z1 okay so that is how this formula is verified this is a very simple calculation but I am just trying to justify that mod dz is the correct infinitesimal version of the arc length okay and you know you would have seen this also in complex analysis how would you measure the length of an arc on the plane it is done by the same thing so you know suppose you have an interval suppose you have this say this interval or let me even put AB suppose is a closed bounded interval in the real line and suppose I have a continuous map gamma from that into the complex plane so you know it traces an arc sometimes the map is also called gamma the image is also called gamma like an abusive notation so this point is Z1 which is gamma of A and this point is Z2 which is gamma of B and well how do you get the length of the arc gamma you get it by simply integrating from A to B gamma dash of T mod gamma dash of T into dt okay and this is the same as integrating over gamma mod dz and basically the integral over gamma mod dz is transformed to the previous integral integral for me to be gamma dash of T dt mod mod gamma dash of T dt because you are just plugging in Z equal to Z of T because if you take a point on gamma the point Z there is given by Z of T it is a function of T where T is the variable here on the real line when T is A you get gamma of A which is Z1 the starting point when T is B you get gamma of B which is Z to the ending point okay so basically if you want to mod dz is infinitesimal distance and you integrate an arc integrate that over an arc you will get the arc length and of course it is very important that your arc should be reasonably good so that you get a finite number okay so arcs for which the arc length turns out to be finite they are called rectifiable arcs okay they are called rectifiable arcs and one has to worry about these things because there are if you just of course you know whenever you say arc it is always this gamma is always continuous alright but then you know even to write that integral you see that integral actually means this mod gamma dash of T dt so you know gamma dash should exist so gamma is not only continuous but actually it is differentiable and you can relax it to being piecewise differentiable piecewise continuously differentiable mind you whenever I integrate something it has to be continuous so first of all gamma dash has to exist and gamma dash has to be continuous only then I can write that integral okay if I am being very naive of course if I use measure theory I can do much more I can allow discontinuity on a measure a measure 0 set but we are not going to that level of you know complication where we assume the naivest point of view and even for the naivest point of view you will have to assume that gamma dash exists and it is continuous and of course all Riemann integral can be evaluated even if there are a few finite finitely many discontinuities which are with finite jumps okay so that is the reason we use the word contour we say that gamma should be a contour which means gamma should be piecewise continuously differentiable function okay fine so this is the length of gamma and of course there is this important restriction that gamma should be rectifiable arc you have to worry about such thing because there are strange things like which are called as space filling curves and these are curves that can fill out a whole region of space okay and obviously the length of such a curve will be infinite okay so you do not want to end up with such horrible things so that is the reason you have to worry about computing this integral only for rectifiable arcs okay. Now what is it how is this connected with analytic functions so you see the point is suppose you have the complex plane and suppose you have a domain here some domain okay so here is some domain D in the complex plane and suppose there is an analytic function f analytic or holomorphic function f which is defined on this domain and suppose I have a suppose I have a path I have an arc in the domain okay so it means that you know I have this continuous piecewise continuously differentiable function gamma from an open I mean a close bounded interval in the real line and it is giving me this arc with this as a starting point this as the ending point okay and this is also I label the arc also as gamma where actually it is image of gamma but I will use abuse notation also call that gamma okay so we use gamma for the image of the arc we also use gamma for the function that parameterizes the arc okay. So well you know what is length of gamma length of gamma is as I told you just now it is integral over gamma mod ds this is the length of gamma okay now what will you get if you integrate over f circle gamma mod f dash of mod f dash of z so let me write this so let me write something here mod dw okay so you see so you know I am writing this mapping as w equal to f of z okay which is similar to y equal to f of x if you are looking at functions for a real variable x is the dependent independent variable y is the dependent variable but now z is the independent variable and w is the dependent variable w depends on z by means of the function f so this z varies over this copy which is a source complex plane and in fact z is varying in d that is where your function is defined and then you have well here is a target complex plane okay and w is in the target complex plane and mind you that the image of f I am of course assuming that f is not a constant function I am assuming f is a non-constant analytic function we have this deep theorem that a non-constant analytic map is an open map okay therefore the image of f is an open set okay that is a very deep fact so this is f of d if you want that is open actually and of course I am not looking at the case when f is a constant function okay f is not constant I do not want to worry about that case okay because in the image is only one point okay so f of d is open and well what happens to what is f of what is the image of gamma well I will get another curve here so I get this oops I get this is f of gamma okay this is just the image of the curve gamma under f okay and you know if I now take the variable in the target if I take the variable in the target plane which is w and if I integrate f circle gamma okay f circle gamma is just mind you f of gamma okay it is just what is f circle gamma you first apply gamma then apply f which is the same as taking the image of gamma under f okay so f circle gamma is in principle f of gamma okay and if you integrate f of gamma over f of gamma mod dw then you should get the length of f of gamma so this is going to give you length of f of gamma okay by what we have seen alright but what is now you plug in what dw is fz so mod dw will become mod f dash of z into mod dz okay so what will happen so let me write that down I need a little space let me rub this off and write the open on this side so this will be integral over f circle gamma of mod f dash of z into mod dz this is what I get okay so if you look at it very carefully you see that if you normally integrate over dz okay and of course you know since I change the variable from w to z I should change the curve from f circle gamma to gamma itself because now my z is in the source and what is in the source is gamma not f circle gamma f circle gamma is in the target okay so I should change this to gamma so now look at this now compare the first equation which is that L of gamma is gotten by integrating over gamma just mod dz and look at the second equation what you have done is instead of simply integrating mod dz over gamma you have multiplied it by this mod f dash of z okay and what does it give you it gives you the it gives you the length of the image of gamma under f okay so try to understand this if you simply integrate over mod dz you get the length but if you integrate over mod dz the derivative of your function okay then you get the length of the image of the curve under the function okay so this is what I want to tell you whatever you put whenever you integrate over gamma a curve gamma mod dz and you put something in the integrand okay that something is a scaling factor okay it is a factor of scaling alright so if you for example you know if I put 2 if I instead of integrating over gamma integrating over gamma mod dz suppose I integrate 2 mod dz I will get twice the length of the curve okay so and similarly if I integrate it with some constant times mod dz I will get constant I will get of course constant should be positive constant because you are worried about length then you will get positive constant times that length okay if I integrate let us say k mod dz I will get k times the length of the curve that is because mod integrating just over mod dz is going to give me the length of the curve as it is and it is multiplied by k the k will come up but then this is alright if I am putting a constant but instead of a constant I can actually put a function which varies as the point z varies on the curve okay and then what will happen is that you are scaling the you are just scaling the you are getting a scaled version of the length of the curve and the fact here is that multiplying by mod f dash of z gives you the image the length of the image curve and the image curve is thought of as the original curve scaled by the map f the map f maps gamma the original curve onto the image f gamma it scales it and that scaling factor is mod f dash of z and that is what appears in this integral okay so why I am saying all this is I am trying to say that you know you look at the coefficient of mod dz in the integral that should give you the modulus of the derivative okay the modulus of the derivative mod f dash of z is the coefficient in the this is the coefficient of mod dz in the expression for the length of the curve okay. Now what you have to do is if you try to mimic this with the following modification namely you take again a function f defined on a domain in the complex plane but now assume that the function is taking values on the Riemann sphere I mean which is essentially taking values on the extended complex plane okay and on the extended complex plane you use the spherical metric okay then you know you would like to have an idea of what is the derivative of the function with respect to the spherical metric okay what we are looking at is trying to get an idea we are trying to guess at what is the derivative of a function with respect to the spherical metric okay and what does it mean you are taking a function which is defined on a domain in the complex plane it is taking values in the extended complex plane because that is where the spherical metric makes sense and then you want to take the derivative of the function with respect to the spherical metric okay it will be ideal if we get a form we can guess a formula for that but then more importantly what this argument tells you is that if you take a curve in the domain you take its image under the function in the extended plane that means you are actually looking at its image on the Riemann sphere okay and if you find the length of that curve in the image in the Riemann sphere with respect to the spherical metric and in that if you take out the coefficient of mod dz okay or if you want mod d of whatever variable you are using then that coefficient should give you the modulus of the spherical derivative okay so I am just trying to tell you how this argument tells you how to guess what the modulus of the spherical derivatives okay. The only difference is you are not looking at an analytic function you are looking at a function which is having values in the not on the complex plane but in the extended complex plane and mind you the advantage is that because you are allowing extended complex plane your function can take the value infinity okay and mind you once you do that you can also deal with meromorphic functions the whole purpose of allowing the value at infinity is to deal with meromorphic functions so you can now generalize all this to taking the image under a meromorphic function of a curve in a domain and taking its image in the extended complex plane thought of as a Riemann sphere and trying to measure the spherical distance. So whatever we did with the Euclidean distance you have to do it with the spherical distance so let me try to quickly tell you how that goes. So you see if you take the so now with respect to spherical distance so you see I wanted to first check that well so if you take the Riemann sphere okay which we draw like this it is just the unit sphere in 3 space 3 real 3 space and where the xy plane is thought of as a complex plane the third axis is called the u axis and this is the north pole which corresponds to the point at infinity okay. Now you know you give me 2 points on the Riemann sphere suppose these are 2 points that correspond to 2 points on the plane okay so well so there is a point z1 and then there is a point well well if I go by the diagram then I will have to be careful so let me let me redraw this a little bit so let me have 2 points so if this is z1 let us say this is z2 and you know by the stereographic projection z1 will correspond to a point here which is p1 and z2 will correspond to say another point here which is p2 okay and well and you know now on the Riemann sphere you know if I join this if I join this chord so let me use a different color if I join this chord from p1 to p2 okay so the chordal distance between p1 and p2 which you can define to be as the chordal distance between z1 and z2 okay you can define it like that okay because you are just transporting distances on the Riemann sphere to the extended complex plane because of stereographic projection and what is the chordal distance between p1 and p2 I want you to check this is a very simple exercise it is actually this is actually equal to 2 times modulus of z1 minus z2 by root of 1 plus mod z1 squared into root of 1 plus mod z2 squared okay this is what it is I want this is just analytic geometry okay I want you to calculate this okay and then what happens is that you see as this is a chordal distance between 2 points and what it is is actually this is actually the distance of those 2 points p1 and p2 in R3 okay mind you this whole diagrams in R3 the xy plane is the complex plane okay so you can check that this is the distance in R3 okay it is a very simple analytic geometry calculation right. Now what is the what is the spherical distance between p1 and p2 it is the length of the arc minor arc from p1 to p2 on the major circle that passes through p1 and p2 on the Riemann sphere so let me use a different color for that and that is going to be oops let me use something else let me use green okay so here is the so here is this so I also have this d sub so there is this d sub s this is spherical distance between z1 and z2 okay and how do you get this spherical distance okay mind you it is a length of an arc it is a length of this arc this circular arc along the great circle passing through p1 and p2 and how do you get the length of the arc how do you get the length of an arc you get the length of the arc by integrating an element of the infinitesimal distance okay. Now the chordal distance as p1 and p2 come closer the chordal distance infinitesimally becomes the spherical distance okay therefore and you know as you make z1 and z2 closer okay if you put z1 equal to z and put z2 equal to z plus delta z you know what you will get is that you will get the element of spherical distance you will get this expression you will simply get mod dz by 1 plus mod z the whole square okay because you see in this expression here okay in this expression you put z2 is equal to z1 plus delta z okay and you look at only delta z terms and you change the delta to d you will get this alright and of course of course there is a 2 I forgot the 2 there is a 2 here okay and therefore this spherical distance between z1 and z2 is simply given by integrating okay along that arc of the great circle from p1 to p2 that is the same as integrating along this line along this line segment L because image of this line segment L on the Riemann sphere is exactly that great that minor arc of the great circle passing through p1 and p2 so this integral over L 2 times mod dz by 1 plus mod z the whole square okay. So this is the spherical distance between 2 points on the complex plane and this works even if you assume one of the points to be the point at infinity mind you because your points can vary on the external complex plane so both points could be the point the north pole itself okay fine so now what we can do is that we can use this to guess what the spherical derivative should be you can use this and the previous argument to guess what the spherical derivative should be and I will just state it and stop the spherical derivative is of f is f hash of z it is 2 into mod f dash of z 2 by 1 plus f mod f z the whole square this is what it will be this is the absolute value of the spherical derivative this spherical derivative so I should say in absolute value this is what you will get if you think about it and I will explain this in the next lecture okay.