 Okay so let us continue with the discussion of the monodromy theorem so what we have is so here is a monodromy theorem so the situation is like this you are given ahh 2 points z0 and z1 and you are given a path gamma ahh and you are given a function ahh so this path gamma is ahh ahh defined on close interval AB and takes values in ahh it traces out this geometric path okay it is given by this function on the defined on the close interval AB in R and ahh you are given a function ahh fa which which is analytic at this point okay ahh this point by the way is gamma of a initial point and you can continue it analytically along this path to ahh a function fb ahh analytic at this point and z1 is gamma of b okay and this analytic continuation along gamma is given by ahh is given by ahh even analytic continuation ahh so you know ahh analytic continuation are given by specifying a family of 1 parameter family of power series so you have ft of z is equal to ahh sigma n equal to 0 to infinity a n of t z-gamma t to the power of n defined in the disc of convergence of this power series which is given by mod z-gamma of t is less than r of t ahh ahh so I will put a subscript r sub f ahh t okay ahh r sub ft of t is ahh the radius of convergence ahh of this power series ft okay ahh and of course we always assume ahh all the radius of convergence of positive because they are all analytic functions when t equal to a you get fa the analytic function fa which is which is analytic at this point and when t is equal to fb you get fb which is analytic function at this point so fb is an indirect analytic continuation of fa along this path okay and suppose you are given another path like this which is say eta and suppose along this path also you have an analytic continuation of fa okay ahh and so again start with fa which is the same as ahh ahh g a okay so gt is another analytic continuation ahh it is another analytic continuation on another path the other path is what what is common between the other path and the first path is that both of them have the same starting point and ending point okay so ahh z0 is also eta of a and z1 is also eta of b so you have another path eta defined on ab ahh with values in c okay and ahh ahh with the same starting point and the same ending point and I again start for this for the analytic continuation along this path I am given an analytic continuation along this path of the same function fa but it is now given by ahh another one parameter family of power series which ahh which I denote by g sub t so this is sigma n equal to 0 to infinity ahh bn of t into z-neta of t to the power of n and this is this power series as disc of convergence z-neta of t is less than radius of convergence of gt ahh ahh as a function of t. So you are given given two analytic continuation like this okay and the question is I started with fa equal to ga at the end I am getting fb if I go if I analytically continue along this path and if I analytically continue along this path I am getting gb the question is are these two equal the question is are these two equal and the monodermy theorem says that it is they are equal under certain conditions so what are the conditions the conditions are first of all that ahh this path gamma ahh should be continuously deformable ahh ahh to ahh this path eta okay. So you should continuously be able to deform the path gamma to the path eta okay which in the language of topology or the language of homotopy is got but is is said as follows we say that gamma is fixed end point homotopic to eta okay ahh that is gamma and eta have the same end points and you can continuously deform gamma to eta okay. So ahh that is a condition and the other condition is of course that along any of these intermediate paths ahh through which you are deforming there is no obstruction to analytic continuation of fa okay. So so here so let me write it down ahh if ahh ahh if gamma is homotopic ahh that is continuously deformable deformable to eta and if there is no obstruction to analytic continuation of fa along any intermediate ahh gamma is ahh then ahh fb is the same as gb this is the monotermy theorem. So so an intermediate path is something here okay it is something it is one of those paths that occur in the continuous family of paths which start at gamma and end at eta okay. So this is the monotermy theorem okay ahh so the important thing the crucial thing is of course that ahh along ahh you you are you you at the at the starting point you have an analytic continuation at the ending at the starting ahh at the starting path you have an analytic continuation at the ending path you also you have an analytic continuation and in the intermediate along each of the intermediate paths also you have analytic continuation okay. Analytic continuation should exist the function the function fa should be analytically continueable in any along any intermediate path ok. We express that by saying that there is no obstruction to analytic continuation of this function along any intermediate path ok. In other words give me any intermediate path the function f a can be analytically continued along that path you make that hypothesis and then the monodromy theorem says that the final function that you get when you end when you go to the end point you are going to get the same function ok this is the monodromy theorem. Now so you know so how does so let me let me let me expand a little bit more on the statement I have to tell you what I mean by gamma is homotopic veneta and there is no obstruction to analytic continuation of f a along any intermediate path. So this is this is a definition this involves the definition of homotopy which you might have seen in a course in algebraic topology but nevertheless it is very easy to understand. So you see the idea is like this so what is happening is that you know you have you have this on the on R2 ok you have you have the axis and you have this interval it need not be positive real line it could be some ab and you could have some cd. So ab is a close interval on the real line where where the paths all the paths are defined ok and cd is a parameter is a parameter s so you know this is the parameter t this is the parameter s ok. So what you get is you get a you get a you get a square like this I mean or rather a rectangle like this you will get something like this ok. So the x coordinates vary from x equal to a to x equal to b which I am calling as t. So t coordinates vary from t equal to a to t equal to b and the y coordinates which in this case is I am labeling by the variable s the s coordinates vary from c to d ok and what is happening is that you have you have a f is a continuous function from this into the complex plane ok and what is this f doing it is doing the following thing you see you know when s is equal to c and t varies from a to b alright you are you are getting the path gamma sub c which is the same as gamma ok. So you start with so the diagram is like this so you have gamma sub c which is gamma it starts at z0 ends at z1 ok and then and you know as you as as as t moves along this line the path that is traced is gamma which is gamma sub c ok and then if you take any value of s in between ok any value of s of the of the pi coordinate in between then what you get is an intermediate path gamma sub s ok which is what I wrote there and when s becomes d s equal to d and t varies from a to b you get the path gamma sub d which is beta so you get this path this is gamma sub d which is the path beta ok. So this so this function f is a continuous function it is a continuous function it is a continuous function of two variables two real variables so we write f is as f of s, t ok and we write gamma s to be the path f of s gamma s of t we call gamma s of t to be f of s, t for fixed s is also a path which starts at z0 and ends at z1 ok and the so what you what is what you are seeing is you know see try to imagine it like this if you have a square here or if you have rectangle like this if you take a continuous image of this rectangle on the complex plane you will get something like this you should you should expect to get something like this with you know if I take a continuous image of this on the complex plane I should I should get these these four should correspond to these four ends of a distorted rectangle alright and and and the image of this line segment will be this path, the image of that line segment will be that path the image of this line segment will be a path like this and the image of this line segment will be a path because the images of all these line segments are going to be continuous images of an interval. So they are going to be paths so you are going to get something like this But then you know if I put the condition that uhhh at the point t equal to a all the uhhh all these uhhh each of the uhhh each of the paths gamma s they all started the same point. So then it will it will amount to actually you know collapsing all this into a point okay and at the point t equal to b if I insist that all the points end at the same point all this will get collapse. So you know all this gets collapse to uhhh z0 and all these all these points on this arc uhhh on this path they all collapse to z1 and then you know if you collapse it this is the resulting diagram this is what is happening and this is this is an intermediate path the intermediate path is here okay. So if you collapse this diagram this is what you will get okay and that is the and we say that capital F is a fixed end point homotopy from uhhh uhhh gamma equal to gamma c to gamma neta equal to gamma d okay this is what uhhh this is what I mean by saying that gamma is homotopic that is continuously deformable to neta okay. So this is this explains this statement in a very precise way alright. So that is that is one thing and the second thing is I should tell you that there is no so the second hypothesis is that there is no obstruction to analytic continuation of a along any intermediate path. So you know I know that F a can be analytically continued to uhhh along along the path gamma which is gamma c and I know uhhh that F a can also be analytically continued along the path neta which is gamma sub d but I need also that F a can be analytically continued along any intermediate path gamma sub s okay. I need that condition that is part that is part of the hypothesis and then the monodromy theorem says that uhhh in fact for any along any of these intermediate path you continue F a the final function you are going to get will have to be the same. So it has to be F b which is what it was for the first path okay that is what the monodromy theorem says alright. Now uhhh now the point is uhhh uhhh uhhh let us uhhh uhhh let us postpone the proof of this theorem but let us try to see how uhhh why it is important and why it is so uhhh you know so I have I have told you that uhhh given uhhh so so applications one of the application is the following. See suppose you are given uhhh u,f which consists of a pair uhhh uhhh f and analytic function on the domain u suppose you have given a given a pair like this. We have defined two types of uhhh domains connected with uhhh uhhh this we can define two we have defined and we can uhhh we have done that earlier there are two types of domains that you can uhhh define with respect to u,f. One domain is called the domain of maximal analyticity it is called the domain of maximal analytic continuation of F okay. So uhhh uhhh there is the domain uhhh p,g of maximal uhhh v1,g1 of maximal analytic uhhh maximal direct analytic extension continuation or extension or extension of u,f. There is there is this is the domain of maximal analyticity I mean so in in other words what this means is that v1 is the largest open set in the complex plane uhhh which is uhhh this is the largest domain in the complex plane uhhh uhhh which contains u okay which will contain u and to which uhhh f extends to an uhhh uhhh uhhh single analytic function on all of v1 and I am calling that as g1 okay. So let me recall that is uhhh v1 in c is the largest domain v1 containing uhhh u such that uhhh v v1,f1 is a direct analytic extension extension or continuation of u,f which means which is just trying to say that you know if you take g1 and restricted it to you will get f okay. So this is the largest uhhh uhhh this is the domain of uhhh maximal analytic continuation okay this is the largest domain to which you can continue analytic function okay. So uhhh you know uhhh we have seen uhhh we have seen examples I just recall uhhh if you take uhhh uhhh f of z uhhh to be 1 plus z plus z squared plus so on geometric series and u to be the domain uhhh mod z less than 1 unit disc then uhhh uhhh then we have seen that uhhh v1, g1 is simply uhhh is simply the pair that consists of the whole complex plane minus the point 1 and the function is 1 by 1 minus z. This is the the domain of analytic maximal analytic continuation is the complex plane punctured at z equal to 1 and the corresponding analytic function is 1 by 1 minus z okay and of course uhhh uhhh this analytic function cannot be continued to z equal to 1 because uhhh at z equal to 1 it has a uhhh it is a it has a singularity it has a simple pole it cannot it is not a removable singularity it cannot continue it okay uhhh if because if you continue it you if you if you can continue it means that uhhh that singularity is removable at at least locally okay that is not possible because it is a pole pole is not a removable singularity uhhh so so this is what happens and then uhhh second example is that of the zeta function zeta of z is equal to sigma uhhh n equal to 1 to infinity uhhh 1 by n power z. So, this is defined to be sigma n equal to 1 to infinity uhhh 1 uhhh 1 by uhhh e power uhhh z lawn n and this is the Riemann zeta function and you know the domain on which uhhh uhhh uhhh so here I am taking f to be uhhh f to be zeta and I am taking u to be the right half plane uhhh to the right of the vertical line uhhh uhhh uhhh real part of z is equal to 1 okay. So, this is set of all z in in in the complex plane such that real part of z is greater than 1. So, this is the zeta function we we all know that it represents an analytic function in this uhhh in this right half plane we proved that earlier and then I told you that it was it is a theorem it is a theorem it is not trivial it is a theorem that in this case uhhh v1 uhhh is actually again like the geometric case of geometric series the domain of maximal analytic continuation is just the complex plane punctured at the point 1. That means this zeta function extends to the whole complex plane except at the point 1 and what happens and of course the extended function uhhh is called g1. So, g1 g1 is equal to extension or it is called the extended zeta function called the extended Riemann zeta function uhhh which is just an extension of this function you extend it from the right half plane to the whole complex plane but the only point you cannot extend it to is the point z equal to 1 where uhhh z equal to 1 it cannot be extended because z equal to 1 will be a simple. So, uhhh z1 z z equal to 1 is a simple pole pole for g1. So, this is something that we uhhh prove later on in the course okay and this is a non-trivial example and then uhhh that is of course uhhh there are of course uhhh uhhh uhhh there is of course another important example you take f of z to be the principle logarithm of z principle branch and in this case you take the uhhh uhhh the open set to be the slit uhhh plane. So, this is just complex plane minus you remove the you remove the negative real axis including the origin okay. Then you know the principle branch of logarithm is analytic here alright and in this case what happens is that uhhh the uhhh if you take the maximum uhhh if you can if you take the domain of maximal uhhh uhhh direct analytic continuation it will simply be the same in this case uhhh v1 will be equal to u and g1 uhhh will be simply equal to f. So, in this case uhhh uhhh uhhh uhhh you you you you simply cannot directly analytically continue it across the uhhh across the slit which you have made uhhh by deleting the negative real axis okay. So, this is the this is the situation with respect to so these are 3 examples the situation with respect to uhhh uhhh the domain of maximal analytic continuation okay the domain of maximal direct analytic continuation or this is called the domain of maximal analyticity of the function okay. Then then what we have defined is we have defined another thing the given of function analytic function we also define another domain that domain is called the domain of uhhh uhhh regularity it is called the domain of indirect analytic continuation okay. So, uhhh we also have the domain uhhh uhhh uhhh v2, g2 of indirect analytic continuation analytic continuation of uhhh u, f okay we also called the domain also called the domain of regularity of regularity of u, f you also have this now what is what is this domain see this is the domain which consists of all those points to which the original function can be indirectly analytically continued along a path okay. So, uhhh so how what is the definition uhhh v2, g2 is so v2 uhhh so whenever I say domain I am also uhhh when I say domain but I include both the domain and the analytic function you must always remember that. So, uhhh uhhh that is why uhhh a pair like this is called a function element which namely it consists of a domain and analytic function defined on the domain holomorphic function defined on the domain it is called a it is the pair is called as a function element okay. So, uhhh so what is the domain of regularity v1 is the set of all z belonging to c such that there exist an indirect analytic continuation continuation of f uhhh from a path uhhh uhhh f along a path uhhh starting at the at a point of uhhh u to z okay. So, so the diagram is something like this you have uhhh so you have this is my complex plane and here is my domain u and here is my analytic function f with values in c and what am I doing uhhh I am I am I am collecting all those points z with the property that uhhh you know the whenever uhhh there is a so there is a point z0 in the domain and there is a path there is a path gamma starting at uhhh this point z0 in the domain u and such that along this path you can analytic analytically continue indirectly analytically continue f to get a new function new analytic function at the point z and you put together all these points okay you put together all such z's okay. So, you see you put you put together all these points and the result is uhhh the result will be an open connected set because the point is you know if you can if you uhhh after all you know if you can analytically continue it along this point then you know uhhh uhhh I am getting an analytic function at that point. So, it is analytic in a small disc surrounding that point. So, for every other point in that small disc I can I can extend that that analytic function itself extends along a smaller uhhh radial path okay. So, if I can extend from z0 to z uhhh analytically indirectly analytically uhhh f okay then I can do so for every point in its small disc surrounding z okay uhhh and mind you an analytic function on a domain is of course analytically extendable along any path trivially on that domain. So, the moment I say that I can extend analytically f to obtain a new analytic function here at this point it means that it is analytic in a small uhhh disc surrounding that point and it means that along uhhh for every point in that disc it is analytically continueable for a path starting from the centre of the disc to any other point. So, uhhh so the moral of the story is the set of all such z is open okay and it is by definition path connected okay. Therefore, it is a domain. So, this v1 is a domain by definition okay and what is happening is what is happening is uhhh uhhh you are getting a you are getting a new you are getting a new set but the only problem is that in this case uhhh at various z you will get various analytic functions okay. You started out with an analytic function f on a domain u but then uhhh uhhh if you go to different points you do not know whether the final functions that you get uhhh whether they are indirect analytic continuation or whether they are direct analytic continuation you do not know okay. And so the the so the fact is that uhhh uhhh uhhh so you know you you can look at these three examples uhhh uhhh with respect to uhhh example 1 uhhh you see that uhhh uhhh uhhh if you take the domain of indirect analytic continuation uhhh uhhh that is the domain of regularity of of of uhhh you will still get the same domain because you cannot continue it to the point z equal to 1 because that is a simple point that is the only point that is left out. The same thing will happen uhhh to example 2 okay. So, in both examples 1 and 2 the domain of uhhh direct analytic continuation the domain maximal domain of analyticity is the same as the domain of regularity for 1 and 2 because the only uhhh point that is left out is the point z equal to 1 but at z equal to 1 you cannot even indirectly analytically continue the function because of the fact that uhhh uhhh any indirect analytic continuation we will amount to a direct analytic continuation because every point allow every point every a whole uhhh any deleted neighbourhood about one is already the function is already analytic there okay. So, any indirect analytic continuation will amount to actually a direct analytic continuation you cannot have a direct analytical analytic continuation you cannot continue these 2 functions directly to the point z equal to 1 because z equal to 1 is a pole ok. So in example 1 and example 2 domain so you have domain of domain of maximal analytic continuation is the same as the domain of regularity that is of maximal indirect analytic continuation. So this is what happens in these 2 examples but something striking happens in this case. So in this case what happens is that the domain of maximal direct analytic continuation so of course here I should say maximal direct analytic continuation will be strictly smaller than the domain of maximal indirect analytic continuation. So in example 3 namely if you take the fundamental branch of the logarithm what will happen is that the domain of maximal direct analytic continuation will be of course the slit plane which is a complex plane minus the negative real axis along with the origin removed and this will be properly contained in what is going to be the domain of maximal indirect analytic continuation it will be just the function plane ok. So the whole negative real axis except the point 0 the whole negative real axis will also come in so it will be this is properly contained in C minus 0 punctured plane which will be the domain of maximal indirect analytic continuation. So this is properly so this properly contains this ok so in other words if you take the principle branch of the logarithm what is happening is that you know it is analytic to make it analytic I have to throw out the negative real axis and the origin ok. But if I want to analytically continue it indirectly along paths I can always analytically continue it across any point on the negative real axis so long as the path does not go through the origin ok and therefore the domain of maximal regularity ok the domain of regularity the domain of maximal indirect analytic continuation will also include the whole negative real axis ok and it therefore it will be just C minus 0 of course 0 can never be remedied because at the point 0 logarithm is not defined logarithm is not defined for the point 0 ok. So you see there is a big difference so you know all these examples so there are two definitions there is one definition of maximal direct analytic continuation there is another definition of maximal indirect analytic continuation which is regularity and the question is how are these two related and we have seen that this could be bigger than that ok. Now so what the monotromy theorem says the monotromy theorem says the following thing so you can ask the question when are these two when can you say these two are equal ok so the monotromy theorem in another version ok which is actually equivalent to this version says that suppose your domain of maximal indirect analytic continuation is simply connected ok suppose your domain of maximal indirect analytic continuation is simply connected then it is the same as the domain of maximal direct analytic continuation ok so that is a reformulation of the of the monotromy theorem. So monotromy theorem says that if you start with a function element namely a pair consisting of a domain and a polymorphic function on the domain analytic function on the domain then and if the domain of regularity of that element of that function namely the domain of maximal indirect analytic continuation of that function if that is simply connected then that has to coincide with the domain of maximal direct analytic continuation in other words both these domains will be equal and on the domain of maximal indirect analytic continuation the domain of maximal indirect analytic continuation will actually become a domain of maximal direct analytic continuation which means that the function will extend to a single valued function on the whole domain of regularity ok whereas you know you do not expect it in this case for example log cannot you cannot extend log to a single valued analytic function on the on the punctured disk you cannot get a single valued analytic function of the logarithm on the punctured disk ok that is a if you want that is a basic exercise simple exercise in the first course in complex analysis right you can never find a single valued analytic branch of the logarithm on the punctured disk ok. So the moral of the story is that in this case the domain of regularity is bigger than the domain of maximal direct analytic continuation and the problem is that this domain of regularity is a punctured plane it is not simply connected ok. So the problem is that you are not able to get a single valued function you are not able to get a single direct analytic continuation of the function to this domain of regularity the reason is that this domain of regularity is not simply connected it has a whole ok and what the monetary theorem says is that whenever you have a domain of regularity which is simply connected such a thing cannot happen ok. So let me so you know let me state that version of the theorem let me keep this as it is so and let me state that version of the theorem and try to convince you why the monotromy theorem implies that so this is a very important question that the monotromy theorem answers it says that whenever you are in a situation where you can you know that the domain of regularity is simply connected then you can for sure say that that domain of regularity is the same as the domain of direct analytic continuation ok that is what the monotromy theorem says. So here is monotromy theorem version 2 so you know I therefore let me call this as version 1 this is monotromy theorem version 1 this is monotromy theorem version 2 so what does it says it says given a function element u,f which means that is a holomorphic or analytic function f on the domain u ok if the domain of regularity of f namely the domain of maximal indirect analytic continuation of f v1 is simply connected then it is equal then v1 so v2 is what I have used for so let me use v2 so let me read this for a moment given a function element u,f namely a holomorphic or analytic function f on the domain u if the domain of regularity of f namely the domain of maximal indirect analytic continuation of f v which is which we call v2 is simply connected then v1 is equal to v2 that is the domain of maximal analytic extension maximal direct analytic extension extension is the same v2 v1 is the same as the domain of regularity. In other words what we are saying is that wherever you can analytically extend f indirectly no matter your extension may be indirect analytic extension you are only going to get a direct analytic extension so what it says is every indirect analytic extension of f has to be a direct analytic extension ok so it is also saying that is a very indirect analytic extension of f is direct that is what it says ok this is the conclusion you can get you can come to if you assume the domain of regularity simply connected ok.