 Okay so let us continue with whatever we were doing see we were trying to understand what removable singularity at infinity means okay and what we saw was that a function if it is entire that is it is analytic on the whole complex plane and it is also analytic at infinity which by definition means it has a removable singularity at the point at infinity and of course that is equivalent to it being bounded at infinity or it which is also equivalent to it having a limit at infinity okay then such a function must reduce to a constant. So an entire function which is analytic at infinity has to be a constant in other words a non constant entire function at infinity cannot have a removable singularity it has to be either a pole or a essential singularity so it has to be a real singularity it cannot be a non singularity which is what a removable singularity is a removable singularity is actually a non singularity in disguise okay so it is a fake singularity a removable singularity is a fake singularity it is not a real singularity okay so an entire function which is not constant cannot have a fake singularity at infinity it has to have a either a pole or an essential similarity. Now I told you that this is essentially the same as Lieuwell's theorem I mean this is just another avatar of Lieuwell's theorem and so you know the point is that somehow there is a generalization of Lieuwell's theorem okay which is exactly you know which is exactly like the Casorati Weierstrass theorem okay see so that is something that I want to say in this connection. See let me recall what is the big Picard theorem? The big Picard theorem is given a function analytic function which has an isolated essential singularity at a point then in every neighbourhood of that isolated essential singularity the function takes all complex values except with the exception of one value at most one value okay which means it might fail to take one value at the most or it might take all values okay and this it will do in every neighbourhood of an essential singularity and every value will be taken every value that it takes will be taken infinitely many times this is the big Picard theorem okay or the great Picard theorem and what is the first approximation to the big Picard theorem? So the big Picard theorem set theoretically what it says is that if you take the image of a deleted neighbourhood of an essential singularity you get either the whole complex plane or you get the complex plane minus a point so it is either the whole plane or a punctured plane the punctured being corresponding to removing a value which it will not take okay. Now what is the Casarati Weierstrass theorem? The Casarati Weierstrass theorem is a weaker version okay which we actually proved using Rayman's removable singularities theorem okay and what is the weaker version? The weaker version is that the image is dense you take any neighbourhood deleted neighbourhood of an isolated essential singularity and take its image under the analytic function the image is a set which is dense in the plane which means the closure of the image is the whole plane. The other way of saying that is that every complex value the function values come close to every complex value in other words you give me any complex value I can find a sequence of points such that the function values tend to that complex value okay so that is the same as saying that the image is dense that the image the closure of the image contains all the points in the complex plane which is the same as saying that every complex value can be approached by values of the function as close as you want okay. So this is the Casarati Weierstrass theorem which is relatively easy to prove because you can deduce it from the Rayman's removable singularity theorem that we did okay. Now in the same way there is a similar version of this for entire functions okay so you know so again let us go back to the Louisville's theorem what is Louisville's theorem Louisville's theorem says that a bounded entire function is a constant okay and so if you say it in a different way what it says is that an entire function which is not constant is unbounded okay what it means is that it will take values with bigger and bigger modulus because if all the values that it takes are bounded by a certain modulus that means it is a bounded function and if it is entire and bounded Louisville will say it is a constant. So if you take a non constant entire function what it will tell you is that the values of the function can become arbitrarily large in modulus okay now but then you can ask the question what is the image of an entire function and the answer to that is a little Picard theorem the little Picard theorem says that the image actually the image of an entire function is either the whole complex plane okay or it is a punctured plane namely it will omit one point at the most and that is a case for example if you take e power z it will omit the value 0. So and I told you that this is called the little Picard theorem or the small Picard theorem and this is supposed to be we would like to derive it as a corollary of the big Picard theorem okay but then the big Picard theorem has a weaker version which is the Casorati Weierstrass theorem and similarly the little Picard theorem also has a weaker version okay and that weaker version is the Louisville generalized Louisville theorem and what is the generalized Louisville theorem generalized Louisville theorem says that you take the image of an entire function then the image is dense okay that is the that is the weak version of the little Picard theorem which is called the generalized Louisville theorem okay and so the reason why I am trying to bring it at this point in the in our discussion is because it is to just to tell you that this is a circle of ideas with the generalized Louisville theorem is to the little Picard theorem what the Casorati Weierstrass theorem is to the big Picard theorem okay it is it is a similar thing both the Picard theorem tells tell you what exactly the image will be okay that the image will be the whole plane or the plane minus a point and the both the Casorati Weierstrass theorem and the generalized Louisville theorem will tell you that the images the image is dense that is the point okay and how do you prove the generalized Louisville theorem it is also it is actually the same proof as the same tactics that we use to prove the Casorati Weierstrass theorem except that you use Louisville theorem okay so let me let me write that down so here is the generalized Louisville theorem the image of an entire function is dense in the complex plane and of course it is very very important that all these theorems are valid only for non-constant functions okay so whenever you should not miss saying there it is a non-constant entire function because a constant function always the image is single point and it is entire okay so you should this non-constant should always be carried through if you want your statements to be accurate okay you you might you might tend to miss it but if you miss it it is a big miss okay so let me write that the image of well non-constant so this is very important the image of a non-constant entire function is dense in C it is dense in the complex plane what does it mean you can let us say it in 2 different ways it means that any complex value can be approached by function values that is one way and that if you want to say it more clearly you can say you can find a sequence of points such as the function values at those points approach the given value okay so let me write that in other words in other words given w not in the complex plane there exists a sequence is at n in there is a sequence in the complex plane such that the f of f of z n this sequence the image of the sequence under f this tends to w not where f is the given non-constant entire function so this is another way of saying what I said now so this is generalized level theorem and this is so let me write it this is the analog of the Casorati Weierstrass theorem okay this is the so I will write it as analog of the Casorati Weierstrass theorem okay and I am still putting this analog of Casorati Weierstrass theorem in quotes in some sense but it is actually Casorati Weierstrass theorem but you know the only thing is the case where it is actually a Casorati Weierstrass theorem for infinity as an essential singularity justifies this name actually and I will explain that to you later okay but what I want you to at this point understand is that the statement of the generalized level theorem is similar to this statement of the Casorati Weierstrass theorem and these are both weaker versions of the stronger theorems the stronger version of the Casorati Weierstrass theorem is a big Picard theorem stronger version of the generalised level theorem is the little Picard theorem okay that is the point. Now let us try to prove it the proof method is exactly the same as we have been doing for the Casorati Weierstrass theorem prove so you prove by contradiction okay you assume that there is a value which the function does not come close enough to and then get to prove that the function is a constant and you are done okay. So suppose there exists a W not which is not which is not in the closure of f of c f of c is actually the image of f the image of f is just all the set of values of f okay of course f is entire okay and of course let us assume f is entire non-constant okay. Suppose there is a point suppose there is a value W not which is not in the closure that means f u this W not cannot be approached by function values okay that means there is a neighbourhood surrounding W not which is which does not intersect the image okay so the W not is not in the closure of the image means W not is outside the closure of the image okay and the closure of the image is a closed set okay and anything outside a closed set is in an open set and anything which is in an open set is contained in a small disc which is also in that open set. So I can find a small disc surrounding W not which does not intersect the image okay. So let me write that down thus there exists an epsilon greater than 0 such that mod of f z minus W not is greater than or equal to f z okay I mean this is the same as saying that the disc mod W minus W not less than epsilon does not intersect f of c. So this disc does not intersect f of c means that its complement f of c can at most intersect its complement okay and well but now you know what to do you see the again the trick is that you know f of z is analytic everywhere its entire so f of z minus W not is also analytic it is also entire okay because it is just the constant minus W not added to an entire function and you know sum of analytic functions is analytic a constant function is analytic therefore f minus W not becomes a entire function but it never vanishes it is always greater than or equal to epsilon. So it means that its reciprocal which is defined because it does not vanish mind you the moment an analytic function does not vanish its reciprocal is defined and that also turns out to be analytic so 1 by f z minus W not also turns out to be analytic okay and because the denominator which is f z minus W not does not vanish and that is because it is always greater than or equal to epsilon in modulus okay and epsilon is a positive quality right. So I get this function 1 by f z minus W not which is on the one hand analytic and on the whole plane plus I also get that its modulus is bounded by 1 by epsilon so it is an entire bounded function and leave its theorem will tell you that it is a constant okay and so 1 by f z minus W not will become a constant that constant cannot be 0 okay because if that constant is 0 your f z minus W not cannot be a finite quantity okay so the constant is nonzero and once it is nonzero reciprocal of that constant will become f z minus W not so it will tell you f z minus W not is a constant so f itself will be a constant and that will contradict with the assumption that f is non constant okay and that ends the proof and if you really see this is exactly the same proof as the same technique of proof as we proved the you know Casarati wire stress theorem okay. So let me write this down now f z minus W not is entire and non vanishing so 1 by f z minus W not is also entire and non vanishing but has modulus bounded by 1 by epsilon so is constant by Lieu's theorem and of course the fact that 1 by f minus W not does not vanish will tell you that this constant cannot be 0 okay because after all this constant is the value is the function itself okay so this constant is nonzero because 1 by f z minus W not is equal to that constant and 1 by f z minus W not never vanishes and so f becomes constant and that is the end of the proof. So I am not so the way you should read this proof is that either you assume f is a constant and get to the end of this proof which will say that you have got a contradiction or you assume that f is a function an entire function which misses a value I mean which stays away from a value then it has to reduce to a constant that is what this proof says okay. So this is the generalized Lieu's theorem okay and I just wanted to say that this is an analogy with the Casorati Weierstrass theorem okay fine now you see so what we have now at the moment we have at the moment the idea of a function having a removable singularity at infinity okay and then of course you know you can ask all you can ask regular questions that you would ask usual questions that you can ask for a function which is analytic at a point okay. So for example so to begin with suppose you have a function which is analytic at a point in the complex plane the fine in the usual complex plane then what do you know about what are all the things that what are the basic things that you know about such a function. So the first thing is that you know that it is infinitely differentiable okay in fact all its derivatives exist and they are also analytic at that point okay this is one thing and the second thing is you have Cauchy's theorem that you take a neighborhood sufficiently small neighborhood of the point in fact you take any disc surrounding that point or even a domain surrounding that point containing that point where the function is analytic and the integral over of the function over a simple closed curve such that the function is analytic on the curve and in the interior of the curve also will always be 0 that is Cauchy's theorem essentially and then of course you have that the function it can be expanded as a Taylor series this is Taylor's theorem okay and you have a converse criterion for analyticity which is given by Morera's theorem which says that if you have a continuous function and if the integral over every simple closed curve is 0 then the function is analytic okay. So these are all the things that you know about analytic functions in this is all you know for a function analytic in a domain in the complex plane but now we are interested in analyticity at infinity okay so you can ask all these questions of a function which is analytic at infinity okay you should see what happens. So the first thing is there is one point we have to be very careful about see you cannot define the derivative of function at infinity it is a it is troublesome because you know I cannot really write limit z tends to infinity f of z minus f of infinity by z minus infinity which it really does not make any sense okay but there is a tricky way of doing it see the tricky way is well you know how we defined a function to be analytic at infinity we cleverly said that it has to have a removable singularity at infinity and for having it making it have a removable singularity at infinity we found that either you say it is bounded at infinity or we it has a limit at infinity or it is continuous at infinity you just put one of these weak conditions okay and all that all these three are good enough and they are they are powerful enough is because the inspiration comes from the Riemann's removable singularity theorem. So you can ask this question suppose a function f is analytic at infinity then are all its derivatives also analytic at infinity okay so and the answer is yes the answer is the answer is yes. So it is it is funny you are saying that the function is analytic at infinity all the derivatives are analytic at infinity but you really do not go and define the derivative at infinity in fact the truth is that the derivative at infinity will always be 0 okay except at the worst it can be a constant okay I mean at the best I mean it can be a constant all the time most of the time it is 0 okay. So you know so let me let so let me first answer that question so remarks number 1 if f of z is analytic at infinity then so is every derivative of f okay so if f is analytic at infinity then all its derivatives are also analytic at infinity that is the that is the first that is the first statement okay and the truth is that the moment you say something is analytic at infinity its value at infinity has to be defined the fact is that all the derivatives vanish okay in fact in fact in fact all derivatives of f at infinity vanish okay except possibly f itself not vanishing at infinity it may be a constant and f is thought if you want to think of f as the 0th derivative of itself okay then the 0th derivative can at worst be can be a constant which is non-zero but all other derivatives have to be 0 okay so why is this true? So this is true because of well basically the many ways to see the easiest way to see it is by looking at the by looking at this philosophy that if a function is I mean the behaviour of function at infinity f of w at infinity is the same as the behaviour of g of z which is f of 1 by z at 0 okay so if you use that you will see that the then you use the theorem that for a point in the complex plane if a function is analytic at a point in the complex plane then it is infinitely differentiable there all the derivatives are also analytic at that point okay and that gives you this result okay so let me explain that so proof f is f w so I will change z to w f w is analytic at infinity if and only if g of z is equal to f of 1 by w 1 by z is analytic at 0 at 0 okay and the point is that you know see so g of z is analytic at 0 so g is given by a Taylor expansion okay so g has a Taylor expansion at the origin so this is the otherwise called the Maclaurin expansion the Maclaurin expansion is actually the Taylor expansion at the origin at 0 okay and what is the Taylor expansion is just g of z is equal to this is f of 1 by z and that is sigma n equal to 0 to infinity An z power n this is what the Taylor expansion is and you know that the An's are the derivatives the An's are the derivatives of well you know An An is just the nth derivative the nth derivative of g divided by factorial n the nth derivative of g at 0 divided by factorial n this is what the An's are okay so you know that okay and this g upper n and let me put a bracket the bracket is supposed to be in derivative g round bracket upper round bracket n okay. Now you see now see the point is that this is where is this valid this is valid in mod z less than r okay where r is the radius of convergence radius of r is the radius of convergence of this power series in z okay you know whenever a power series converges it converges in a disk with centre the centre of the power series and the radius is called the radius of convergence okay and this radius of convergence is actually the distance from the centre of the series to the nearest singularity of the function that is what the radius of convergence is for example if the function has no singularities then the radius of convergence is infinite that is what happens when you write out a Taylor series for an entire function okay the fact that you write out a Taylor series and you get infinite radius of convergence is a proof it is equivalent to saying that you have the function that you are really dealing with is actually an entire function okay. So you have this now you know let us put w equal to 1 by z and you will get a similar expansion for f of w in a neighbourhood of infinity so what you will get is you will get f of w equal to sigma well n equal to 0 to infinity a n w to the minus n valid in mod z mod w greater than 1 by r okay this is what you will get because I just put z equal to 1 by w that is the relationship between z and w okay and if I put that I will get mod w greater than 1 by r and you know you will easily recognise that mod w greater than 1 by r is a neighbourhood of infinity okay in the extended complex plane alright and there you have this function and this is kind of this is very nice for f because you see when you write I told you that this is how you should treat this as an expansion for a function at infinity okay which is good see if you want to expand a function at infinity you use the powers of the variable just as you will when you want to expand it at 0 the only thing is that at infinity it is a negative powers that behave well okay at 0 it is a positive powers that behave well okay. So you see that this expression has only negative powers of w it does not have any positive power of w and that is the proof that it behaves well at infinity because you know if I if I let w tend to infinity then this expression is every term is expression is going to go to 0 and this and it is going to go to 0 uniformly because you know whenever these things converge whenever series converge they always converge whenever power series converge or Lorentz series converge they always converge normally they converge uniformly on compact sets okay therefore see the fact is that this is very well behaved okay the moment you see only negative powers of the variable in a series you must really understand that you are looking at a function which is entire I mean which is actually analytic at infinity okay. So for example the simplest case is if you are looking at 1 by z okay which is the same as 1 by w if you want if you take a think of w as a variable 1 by w is good at infinity it is bad at 0 okay it has a pole at 0 whereas at infinity it is very good similarly if you take 1 by w square that is bad at 0 but it is good at infinity all the negative powers are good at infinity okay. So the fact that your f has an expansion in a series of negative powers of w tells you that it is good at infinity okay and the fact I want to say is that if you well you know if you take this expansion for f okay and you differentiate you can differentiate this expression for f this series for f of w term by term that is because of normal convergence okay and if you do that what you will get is again expressions of the same type okay because when you differentiate of course you know when I put n equal to 0 I will get a not and what is a not? a not is the value at the centre see if you write a if you write a series power series centre at z not okay and you plug in z equal to z not what will you get you are supposed to get the constant okay if you write out a power series in z minus z not it will be of the form a not a not plus a 1 times z minus z not plus a 2 times z minus z not squared and so on when I plug in z equal to z not which is the centre of the series all the terms except the first vanishing I get the constant term so the constant term is the function value at z not at the centre okay and in the same way if you look at this expression for f of w at infinity in neighbourhood of infinity you see a not is what you get when you plug in w equal to infinity okay you plug in w equal to infinity which means you take the limit as w tends to infinity what happens is only a not survives and this is in perfect analogy that when you as you go to the centre of the expansion what you get is a constant term okay and so in some sense what I want you to understand is that this expansion in negative powers of z is like the good expansion that is the reason why it is called analytic part at infinity okay all the negative powers along with the constant they form the analytic part at infinity and the positive powers of the variable they form the singular part at infinity okay this is exactly upside down for what happens at a finite complex number okay. So well now what I want to tell you is that this f can be differentiated okay you can differentiate f and if you look at all those derivatives the derivatives the moment you differentiate it even once the constant will go away okay and then you further differentiate it the constant is not going to remain in the first derivative the constant will go away and mind you when you when you and you can do the differentiation term by term okay if you do it term by term you are only differentiating negative powers of z of w and if you differentiate negative powers of w you will get further negative powers of w you are not going to end up with the positive power. So the moral of the story is that you can keep on differentiating this as many times as you want and you are going to get a functions which are analytic at infinity because you are going to get analytic functions in a neighbourhood of infinity and they will all go to 0 at infinity because they involve only negative powers therefore you see that all derivatives exist and they are all 0 and that is the remark okay so I am just using the fact I am just using the simple fact that you know whenever a functional series converges normally that is it converges uniformly on compact subsets of a domain and if you know that every term in the series is analytic then the series converges to an analytic function and the derivatives can be computed by doing term wise differentiation okay that is all I am using okay so that is the first remark. So you know it is very very funny you must understand that we do not define what derivative at infinity is okay but we indirectly define a function being analytic at infinity as being continuous at infinity or being bounded at infinity or having a limited infinity and then we get that all the derivatives also are analytic at infinity and we get in fact also that all the derivatives is 0 okay so you see it is very very funny you are not able to define derivative at infinity okay but for the derivatives at infinity you are getting an expression they are all 0 okay so well then let us go to the second remark okay so let me so in this regard let me actually tell you that in some sense Cauchy's theorem fails okay Cauchy's theorem will fail for function analytic at infinity okay and the idea is very simple you see take the function f of w equal to 1 by w okay 1 by w is one of is the best I mean all negative powers of w are the best functions at infinity okay. Now if I integrate 1 by w over a curve which contains 0 then I am certainly not going to get 0 because 0 is a pole but 1 by w is analytic at infinity okay so Cauchy's theorem will fail so the moral of the story is that you have to be careful when you try to apply integration theorems when you want to work with the point at infinity you have to be a little careful okay so let me write this here Cauchy's theorem fails which is in a way sad because but then you cannot you should see this as inevitable because you cannot define the derivative at infinity okay directly right so what is the example? The example is you take f of w equal to 1 by w you take this function in you take the extended complex plane C union infinity okay mind you it is also analytic at infinity its value at infinity is 0 that is you make it continuous at infinity by putting f of infinity equal to 0 okay and it is defined in the punctured extended plane okay which is the exterior you throw out 0 but you include infinity okay this is a mind you this is an open set in the extended complex plane given the you know 1 point compactification topology okay as we have seen earlier now you take this function now it is not true that integral over every closed curve is 0 okay so well so you know see take mod w equal to 1 okay you take mod w equal to 1 for that matter this is a unit circle okay and see the what is Cauchy's theorem the usual Cauchy's theorem is you take a function which is analytic on and within a simple closed contour okay and you integrate over the contour you should get 0 okay now the point is that if you take mod w equal to 1 if I if you give me the if you give me the positive orientation if you give me the positive orientation then the interior of the unit circle will be the interior of the curve as usual and the exterior of the unit circle which is the neighborhood of infinity will be the exterior of the curve okay. So and if you give it the positive orientation okay then is the function analytic in the interior no because if you give mod w equal to 1 the positive orientation then the interior is the interior of the unit circle which is mod w less than 1 it contains w equal to 0 where it is not analytic so you should not give it the positive orientation if you want it to be analytic so what you do is you take mod w equal to 1 but put the negative orientation okay that is a curve for which the interior of the curve which will be the exterior of the unit disc will be a domain where the function is analytic but still if you calculate the integral over that I am going to get minus 2 pi i and that is not 0 so Cauchy's theorem fails okay. So so let me write this down take mod w equal to 1 give it the negative orientation orientation so that its interior lies in C union infinity minus 0 its interior is actually yeah mod mod w greater than 1 this set along with the point at infinity this is the interior okay it is a kind of it is actually exterior because if you change the orientation you know the interior and exterior will get interchanged and again now at this point let me recall from first course in complex analysis what is this interior and exterior so the rule is the following you say the interior of the region is actually the region that lies to your left as you walk along the curve okay so that is the rule. So you know if I take the unit circle and if I walk along and give it the positive orientation and if I walk along it which means I am going to walk in the anticlockwise sense then what lies to the left is the inside of the unit circle okay which is the interior okay and if I give it the clockwise orientation which means I am going to walk clockwise around it then the interior what lies to my left will be the exterior of the unit circle and that is the reason what lies towards the left will be the interior and that will be the exterior of the unit circle okay it will be so let me again say this just to relieve you of some confusion the exterior of mod W equal to 1 with the clockwise orientation is the exterior of the unit circle okay because it lies to the left okay so fine. So now but so now you know if you take but integral over mod W equal to 1 FWDW is going to be minus 2 pi okay mind you I am getting this minus because this mod W equal to 1 is given the clockwise orientation you know you have done this computation with anticlockwise orientation in a first course in complex analysis and you always get 2 pi I but since I have changed the orientation to you know clockwise I will get a minus sign because you know after all if you change the if you change the orientation of the path of the integration then the integral will change by a minus sign you know that okay. So the point is the Cauchy's theorem fails okay so you should not expect anything out of Cauchy's theorem here okay so for a function which is analytic at infinity Cauchy's theorem fails then the next thing I want to talk about is about Morera's theorem okay so thankfully Morera's theorem works and Morera's theorem works just for the case just because of the fact that Riemann's removable similarity theorem works and basically Morera's theorem works because we cheated by saying that analytic at infinity is same as continuity at infinity okay so let me say that Morera's theorem works so what is so you know so let me recall what is the usual Morera's theorem for a domain in the complex plane mind you Morera's theorem is supposed to be a it is supposed to be a converse to Cauchy's theorem okay now the only beauty about Morera's theorem is that whereas in Cauchy's theorem you always try to apply it to a simply connected domain okay you do not allow any holes and that is because Cauchy's theorem says integral over a curve is 0 provided the function is also analytic inside the curve okay there should be no point inside the curve where the function has a singularity or is not analytic there should be no holes in the domain of analytic with inside that curve okay whereas Morera's theorem is valid even for non-simply connected domains that is the beauty of Morera's theorem it is slightly stronger in that sense but it is a converse to Cauchy's theorem so what is Morera's theorem for a domain in the complex plane it is just that suppose I know I have a continuous function on a domain in the complex plane and suppose I know the integral over every simple closed curve is 0 then the function is analytic okay and the proof actually is very very easy what it does the proof is actually that because integral is 0 okay you can fix a point and then you can define an antiderivative the antiderivative will be independent of the path it can be defined as an integral the integral is independent of the path because the hypothesis of Morera's theorem is that the integral over any closed curve is 0 the integral over any closed curve is 0 mind you is equivalent to integral being independent of the path okay and therefore you can define an antiderivative and this antiderivative will be a function whose derivative is the given function okay but the moment the fact that the antiderivative has the derivative as the given function tells you that the antiderivative is analytic and you then use the theorem that the derivative of an analytic function is analytic therefore the original function which you assume to be continuous is also analytic so this is how Morera's theorem works okay now you will have to modify this this more or less works also for the for a domain in the extended complex plane okay so suppose you are in the external you are you have a domain in the extended complex plane and suppose integral over any closed curve is 0 and suppose your function is continuous okay then you forget the point at infinity what for whatever is left Morera's theorem still works and tells you that the function is analytic so infinity is becomes a singular point but then it is continuous at infinity and we have cheated by saying that continuity at infinity is as good as analytic at infinity so it becomes continuous everywhere I mean it becomes analytic everywhere and you are done okay so Morera's theorem works for a domain in the extended complex plane also so it works at infinity Morera's theorem works at infinity okay so only thing you should be careful about is that you should not try to integrate over a curve which passes through infinity which does not make sense okay by a curve we always mean a curve in the finite complex plane okay not involving the point at infinity okay so I will write this down in more detail in the next talk