 Alright, so what we are doing now is trying to understand what the spherical derivative of Meromorphic function is okay. So well you know the reason for all this is this idea of spherical derivative is important to study the topology of families of Meromorphic functions okay. See the reason is that normally you know there is a relationship between see basically we are interested in compactness okay and you know compactness is the same is strongly related to sequential compactness okay which is that you know given any sequence you have at least you are able to find a convergence of sequence okay and of course if you are worrying about Euclidean spaces then you know compactness is the same as closed and bounded but then you know for saying things like bounded you need a metric and so on and so forth but you know if you are working with spaces of holomorphic functions or analytic functions the point is that you know you will have to work only with normal convergence you will not get uniform convergence okay you will get uniform convergence only on compact subsets and then it is had it been only uniform convergence then you could have taken the soup norm okay and you could have used that to define a metric but the point is that you do not have uniform convergence you have only uniform convergence only restricted to compact subsets that is called normal convergence. So it is not so easy to think of a metric alright but then you still want to think of compactness and compactness is kind of related to sequential compactness and so then you know this is also connected with uniform boundedness it is connected with equicontinuity okay and it is also got to it is also connected with for example if you want boundedness of the derivatives okay so these are a bunch of interrelated results okay on the topological side this is the so called Arzela Ascoli theorem okay and on the holomorphic side or on the complex analytic side it is the so called Montel theorem okay which we need to prove okay and so you see the somehow we want to do it not only for analytic function we all do it for meromorphic functions because you see we have to worry about meromorphic functions because that is the these are the function that you need to study families of such functions to get to the proof of the Picard theorems which is what our primary aim is okay and since you are worried with meromorphic functions the problem is that you know they are not always different they are not differentiable everywhere I mean they are not analytic everywhere if you go to a pole at the pole of course the function goes to infinity so you cannot differentiate the function at the pole it is not differentiable because it is a singular point basically okay so your usual derivative will not work your usual derivative will not work at a pole so what will you do the method is that you introduce a spherical derivative because spherical derivative is something that will work even at a pole that is the whole point okay. So I want to tell you in general why we are getting so worried about why we are making so much noise about the spherical derivative because see that is what we need that is the thing that will work even for meromorphic functions it will work even at poles okay whereas ordinary derivative you cannot think of at a pole because it is a the moment you say pole it is a singular point and at a singular point derivative does not exist so you know you are in lot of trouble that is the reason we introduce a spherical derivative. So I was so let me continue with what I was telling you last time I was trying to tell you that the spherical derivative if f is a if small f is a meromorphic function as you can see here so let me use a different color for the moment say so if you have this f which is a meromorphic function on a domain D in the plane then there is this spherical derivative okay f hash of z and in fact this is spherical derivative in absolute value mind you I have put 2 times mod f dash of z by 1 plus mod f z the whole square so it is a non-negative real valued function okay normally a derivative should be derivative of a complex valued function should again be a complex valued function but this is not exactly a derivative which is complex valued it is actually positive real non-negative real valued and it is actually so you should think of it as absolute value of the derivative okay so we will when we say spherical derivative I mean absolute in absolute value okay and why is it that we are interested in this absolute value because it is a scaling factor you see so you see what is happening there you see the point is that you know as I was telling you last time see if you look at this if you take this function w equal to f of z which is a transformation from the z plane to the w plane okay then you know you assume it is an analytic function okay and assume it is not constant so then what happens is that you know if it is a non-constant analytic function then the image of any open set is open so if I start with this open set D here which is supposed to be for example in this diagram the interior of this dotted boundary okay then the image of that which is f of D is an open set okay and if I take a curve gamma inside D the image of gamma will be f of gamma okay and this f of gamma is now going to be a curve in the image which is f of D which is open and you know if I calculate the length of gamma it is given by the formula namely integrating mod D z okay because mod D z is the infinitesimal version of the Euclidean distance which is mod z1 minus z2 okay so you take if you take 2 points z1 and z2 on the complex plane then the distance between them is given by mod z1 minus z2 if these points are very close to each other you can call 1 point as z1 you can call the next point as z1 plus delta z then mod z1 minus z2 will become mod delta z and you replace delta by D to get the infinitesimal version so you get mod D z so mod D z is just the infinitesimal version okay it is called you may also call it the you know the element of arc length which you have to use to integrate okay and if you integrate the curve over the arc length you will get the length of the curve okay and of course it is very important that the curve needs to be rectifiable okay it should be a curve which has finite length okay and that is why we always put a condition that we work only with contours and contours are you know they are continuous images of a closed bounded interval which are in fact piecewise smooth and in fact piecewise continuously differentiable okay and for such curves for such contours length will always be a finite quantity and so you have this so the point is that you integrate mod D z you will get the length of this gamma alright now on the other hand what happens if you by the same philosophy you know if you integrate mod D w here over its image which is f of gamma you should get the length of f of gamma which is the f of gamma is the image of gamma and ref which is the image curve but you know but here the variable of integration w is f of z so if you integrate mod D w I mean if you substitute for w f of z then mod D f z will become mod f dash of z into mod D z okay and you see the difference between this formula here and this formula here is that there is this extra factor of mod f dash of z okay so what it means is that if you simply integrate over mod D z you will get the length of the source curve if you integrate if you multiply it by the modulus of the derivative you get the length of the target curve the image curve so the point is that the extra factor you have to put in the integrand to get the length of the image curve is the derivative the modulus of the derivative okay. Now in the same way what is happening going to happen is the following thing if you take f to be if you take the function f to be a meromorphic function on D okay now what you are doing is your see you have the complex plane okay and you have this domain D inside it and you have this function f and this is well I am now thinking of this function as a function into c and infinity mind you it is a meromorphic function so I am thinking of it as a continuous function into the extended complex plane because I define the value at a pole to be infinity okay and with that it is continuous and how do I actually think of this so you see that is this that is this this is identified via the stereographic projection to the Riemann sphere okay which I think I should put it as S2 S2 two sphere okay centered at the origin in the in three space real three space radius one unit okay and with the north pole being identified with the point at infinity okay. Now you see what you do is you essentially think of this look at this function okay if you look at this function think of the function as a map from the domain in the complex plane onto the sphere okay so geometrically whenever you are thinking of the Riemann whenever you are thinking of the extended plane think of the Riemann sphere geometrically that is how you should think of it okay so this function f think of the function f as going into the Riemann sphere so you know if you want you know its abuse of notation I will still call this as f okay in fact I should it is f followed by this stereographic projection so in principle I should call give it some other name but I will still call it as f because I am thinking of this as an identification I am thinking of the stereographic projection as an identification okay I am identifying the extended plane for all practical purposes with my Riemann sphere now what happens now what happens is that you see on the one hand you have the complex plane okay and you have this domain in the complex plane okay so this is my domain in the complex plane and the variable here is z and then on the other hand I have the sphere I have I have the Riemann sphere okay and what is happening is that you know if I now take if I now take a small if I take a if I take a curve gamma here okay in the complex plane in my domain and take its image under f okay then what will happen is that the image of this curve will also give me a curve here on the Riemann sphere okay now so this will become the curve f of gamma and how will I get the length of f of gamma okay now think about what I just told you some time ago to get the length of the image curve you have to integrate over you have to integrate over the original curve with the you with the original you know with the original infinitesimal element of arc length and then you have to scale it by the modulus of the derivative okay but now you see what I am doing is I am actually trying to get the I am trying to get the length of the image curve it is the length I am getting is actually the spherical length see it is the length on the Riemann sphere and the length on the Riemann sphere corresponds to the length on the on the extended complex plane given by the spherical metric see the only different with the only point you have to remember is in my target the metric is not the Euclidean metric it is the spherical metric okay I have to use the spherical metric I have to use the element of the spherical metric is what I have to integrate so what is the length of f of gamma under the spherical metric so I will put L sub s to just to indicate that this is length of spherical metric what is this is just I have to integrate over f of gamma okay I have to integrate over f of gamma I have to integrate over modulus of d z sub s I will keep putting the sub s to just emphasize that I have to integrate over an element of the spherical arc length but what is the element of spherical arc length you see so in fact I think I should not put yeah so it is important that my variable let me call this variable as W okay and I should be careful inadvertently you make mistakes like this see I am integrating over f of gamma so my variable should be in f of gamma variable of integration and that has to be not z z is in the source z is on gamma whereas W is what is on f of gamma so you know if this should not have been z I should correct this this should be d w sub s okay it is an element of arc length spherical arc length with respect to the variable W on the sphere alright but W is f of z so this transformation is given by W equal to f of z okay and the only funny thing is that this W is now on the sphere mind you W can take the value of the north pole which is corresponding to W equal to infinity that is also allowed now okay because we have allowed values in C union infinity extended plane alright now you see but what is this what is this what is this element of spherical arc length I told you that last time that the element of spherical arc length is actually 2 times mod d W the usual Euclidean element of arc length divided by 1 plus mod W the whole square this is what the element of spherical arc length is and see the you know the reason why you got this is if you want as an aside let me write that down you know and use a different color so you see so what happened was that if you take the spherical distance between 2 points W1 and W2 okay then that turned out to be 2 times mod W1 minus W2 by square root of 1 plus mod W1 the whole square into square root of 1 plus mod W2 the whole square this is the spherical distance between 2 points W1 and W2 on the on the extended plane or the Riemann sphere okay this is the spherical distance it is actually the in fact I should not even say this is not spherical distance sorry this is actually the this is the chordal distance in fact okay yeah this is still not the infinitesimal version sorry so this is the d sub c this is the chordal distance so you take 2 points W1 here and W2 and you join them by this chord okay it is a line segment it is a line segment in 3 space joining those 2 points on the Riemann sphere but you see mind you those 2 points are actually points on the extended plane I am simply identifying the extended plane with the Riemann sphere so I am still writing W1 and W2 where actually I mean the stereographic projection of I mean the stereographic projection of W1 and the stereographic projection of W2 okay so see if so this is the chordal distance this is the chordal metric and what is this metric this metric is a metric in R3 mind you this sphere this Riemann sphere is sitting inside R3 this is this is inside R3 okay and in R3 I am simply measuring the distance and I asked you to check that this is a this is an exercise for you to check that oops it is an exercise for you to check that this is the distance formula okay I asked you to do that you should try you should do it if you have not done it so far. Now you see this is the chordal arc length if I want the spherical what is the spherical arc length is this this arc length and what is that arc length I take the great circle there is only one big circle on the on the Riemann on the sphere which passes through these 2 points and that circle with these 2 points 2 points in a circle determine a minor arc and a major arc okay and you take the length of the minor arc that is the definition of spherical distance. So if I want the if I want the spherical length okay and I want the infinitesimal spherical length that is that is what this quantity is this dW sub s is the infinitesimal element of spherical length and for that what I will have to do is I will have to bring W1 and W2 very very close okay and as I bring W1 and W2 very very close the chordal distance will approximate the it will come close to the spherical distance okay. So what I do is you know in this in this calculation in this formula what you do is you put W2 is equal to W1 plus delta W okay and then and then you write it in such a way that you only allow you only worry about delta W and do not worry about delta W whole square delta W whole cube higher order terms because you think of them as being very small and negligible and then you change the delta W to dw. So what will happen is that this will this thing this formula as W1 tends to W2 okay this formula will give you this formula. So you see in the numerator instead of W2 if you put W1 plus delta W the numerator will become 2 delta W and this both of these quantities will become equal to root of 1 plus W2 so you will get the square of that which is 1 plus mod W the whole square okay. So each of these quantities will become square root of so the first the second term will become square root of 1 plus W1 plus delta W mod the whole square okay and as delta W tends to 0 you will get this quantity. So this is the this is how you get this infinitesimal element of spherical arc length and integrating over that integrating that over the curve f of w f of omega should give you the length of f of omega the spherical length of f of omega which is what we are interested okay but then in this you plug in what W is see your W is fz so plug fz inside that if you plug fz inside that what will you get you will end up with well okay let me go back to the other color that I was using yeah so what will I get I will get well I will get integral since I have changed I have made a change of variable w equal to f of z now the variable becomes z and z now varies over gamma so I will put gamma here okay and if I now calculate this mod d fz will become mod f dash of z into dz so what I will get is 2 mod f dash of z mod dz by 1 plus mod f of z the whole square and what is this if you look at it carefully what is this this is just 2 times sorry this is just integral over gamma oops this is just integral over gamma 2 mod f dash of z divided by 1 plus mod fz the whole square this whole thing multiplied by mod dz so what are you getting you are getting length of the image curve and in the spherical metric of gamma under f is this formula and go you go back to what I was telling you some time ago if you want the length of the image curve you have to multiply by the modulus of the derivative okay if you simply integrate you will get the length of the source curve but if you multiply by the modulus of the derivative you will get the length of the image curve so you see what this tells you this tells you that if you want the length spherical length of the image curve f gamma under f you will have to multiply by this quantity here and that quantity therefore has to be the absolute value of the spherical derivative therefore the spherical derivative so this is what we call as the so this is what is called as f dash of z this is called the spherical derivative of f okay and mind you this is I should write it in bracket it is in absolute value this is an absolute value alright because it is a so you must remember you know go back to your first course in complex analysis when you take f dash of z okay if you take a point z0 if you take f dash of z0 where suppose z0 is a point where function is analytic so the derivative exists you take f dash of z0 what is the what is the geometric significance of f dash of z0 see the modulus of f dash of z0 is the scaling factor it is locally the factor by which an image is scaled if you take some if you take a small square containing z0 a very small square containing z0 and you take its image under f you will get something that looks like a square okay okay and its area will be you know its length will be scaled by mod f dash so the modulus of the derivative is a scaling factor the argument of the derivative is a rotating factor it is a factor of rotation okay but the argument of f dash of z0 is the angle by which the tangent rotates okay if you have a source point and you have a curve passing through the source point z0 and you have tangent at that point now you take the image curve which will pass through f of z0 the image point and you take the tangent there the difference in the angles that the tangents make with the x axis is precisely argument of f dash of z0 okay so the geometric meaning of f dash of z0 is that the modulus of f dash of z0 gives locally at z0 the magnification factor and the argument of f dash of z0 gives locally the rotation factor this is how geometrically the map f behaves locally and you know if the derivative f dash is not 0 then you know it is conformal you would have studied this in a first course conformal means you know it will preserve angles between curves okay so for example if you take something like a something like a square okay its image will be something like a distorted square alright if you take something like a circle its image will be something like a distorted circle you can expect it to be like for example something like an ellipse or something like that okay and this is of course if you consider it sufficiently small okay and at a point where the derivative does not vanish okay and this is why it has got so many applications to engineering because of conformality so you see the why I am saying trying to tell you all this is that the modulus of the derivative is the magnification factor and therefore multiplying by multiplying the infinitesimal arc length by the modulus of the derivative always gives you the length of the arc length of the image curve and that is what is happening here you see this is the multiplication factor this multiplication factor is therefore called the spherical derivative okay now I want to tell you a few things few very very interesting things in this integral see first and foremost the amazing thing about this is that you know I told you f is a meromorphic function okay f is a meromorphic function so you see f could have poles okay f could have poles of course they are isolated but f can have poles and the beautiful thing is your curve gamma you see your curve gamma can pass through those poles okay now that is the amazing thing see normally when you do integration you never tried the integrand is always supposed to be continuous okay when you integrate an analytic function or for example whenever you do Cauchy's theorem or you know you do argument principle in all these things when you want to apply you always make sure that the contour does not pass through any singular points it cannot pass through poles it cannot there are cases when you are doing the logarithmic integral in the argument principle you will assume that the contour does not pass through any poles and also through any zeros okay you do not allow the contour even to pass through zeros because you are integrating the logarithmic derivative which is f dash by f and if there is a zero then denominator f will have a zero then you cannot integrate it so in all these things that you have been that we have been doing so far we always make sure that the contour on which you are integrating does not pass through any zeros or poles it should never pass through poles of course but also not through zeros if you are trying to apply the argument principle but now mind you we are not we are dealing with meromorphic functions they have poles and my point is now the contour gamma can pass through as many poles as you want it will not create any problems okay it will not create any problems for this integral because you see that is a matter of calculation that you have to understand I will tell you roughly suppose your contour gamma passes through some poles mind you that can be only finitely many such poles on the contour because you see the set of poles is anyway an isolated set by definition of pole is isolated okay it is an isolated similarity so the set of poles is an isolated set and if you take the set of poles lying on gamma it is an isolated set it is an isolated subset of a compact set mind you gamma is compact any contour is compact it is closed and bounded okay because it is actually continuous image of an interval closed and bounded interval so it is closed and bounded it is compact and you know any isolated subset of a compact set is finite because had it if it were infinite it will have a limit point okay and that limit point will not be isolated okay therefore what will happen is that you have if gamma passes through poles of f mind you f is air meromorphic it could have poles if gamma passes through poles of f it can pass through only finitely many poles because gamma is compact and what happens at a pole see nothing happens to the integrand at a pole it is bounded that is the beautiful thing that is why this integral is valid even if gamma passes through a pole or several poles that is because you just imagine suppose f has a pole at z0 okay then in fact you know we can write this down suppose so let me write this down suppose f of z so let me rub this and probably go down a little bit so that I have more space suppose oops suppose f has a pole at z0 which is lying on gamma this will create this will not create any problems this will not create any problems why because you see in a small disc centered at gamma sorry centered at z0 you see f of z will look like g of z by z-z0 to the power of n where n is the order of the pole okay now you calculate you calculate this quantity okay notice that and of course you know g of z0 is not 0 okay g of z0 is not 0 and of course this is how a pole locally function looks locally at a pole of ordering alright and g is of course analytic okay g analytic at z0 okay you have this now you see just look at this expression that I have written about if I take f dash if I take the derivative the derivative will continue to have a pole of one more order if I differentiate this gz by z-z0 power n if you want using quotient rule then I will get a z-z0 to the n plus 1 of the denominator so what I will get is I will get a pole of higher order okay of order greater order and if you go to the denominator the denominator will have a 1 by it will have a mod gz the whole squared by z-z0 to the power of 2n and as z tends to z0 you will see that the numerator tends to I mean this whole quantity will go to either 0 or to a finite value okay because what is actually happening is that as z tends to z0 z-z0 is going to go to 0 so f is going to infinity alright but the fact is that the denominator will go to infinity faster than the numerator because the denominator contains f squared f squared has a pole of order 2n whereas the numerator has a pole of order only n plus 1 okay therefore the denominator will go to infinity faster than the numerator as a result the integrand is bounded so the point is that this integral is valid even at a pole that is the big deal there so this integral is your gamma can pass through poles of f there is no problem and geometrically also you should believe this because if gamma passes through a pole of f its image will pass through the north pole on the Riemann sphere after all at a pole of f f is taking the value infinity and that corresponds to the north pole on the Riemann sphere so after all what you are saying is that the image curve is passing through the north pole of the sphere how does it matter it is not going to affect it is no the north pole of the sphere is in no way different from any other point on the sphere okay so the moral of the story is that shows that the L s of f gamma is well defined even if gamma passes through poles okay so this formula is always works it works even for a meromorphic function it even works if gamma passes through poles no that is a very very important thing okay and then there is so this is something that you need to know and there is another fact that I expand into in the next lecture the fact is that if you take the reciprocal of f you see f is a meromorphic function then 1 by f is also meromorphic function and the beautiful thing is that if you take each spherical derivative you will get exactly the same as a spherical derivative of f. The reason is because of the fact that the spherical distance is invariant under inversion I told you the inversion on the complex plane translates to a rotation of the Riemann sphere about the x axis and it leaves spherical distance is invariant therefore the spherical derivative is also invariant if you invert the function okay so this is another important fact that we use and the advantage that you can replace f by 1 by f is that whenever f has a pole 1 by f has a 0 so you can reduce from the meromorphic case to the analytic case so you can still work with analytic functions you see this is the advantage of having spherical derivative okay so spherical derivative does not distinguish between f and 1 by f and the advantage of moving from f to 1 by f is that you can read what is a pole for f becomes a 0 for 1 by f and 0s are very friendly more friendlier than poles okay in the neighbourhood of a 0 you can usually you apply usual analytic function theory because the function is after all analytic okay so that is the advantage so that is another motivation for having a spherical derivative okay so I will stop here.