 Okay, so uhhh see we saw in the previous lecture that uhhh using the implicit function theorem you know you can look at the locus of uhhh uhhh function of complex valid function of two complex variables uhhh which is smooth as a Riemann surface okay. Now uhhh what I am going to uhhh talk about in this and uhhh in a probably the next lecture is trying to understand uhhh what happens to uhhh function at what is called a critical point okay. So, basically uhhh what we have been looking at so far is uhhh the relationship between uhhh f an an an analytic mapping being a one to one mapping and for example uhhh the derivative non vanishing okay. So, uhhh what we have proved is uhhh if you have an analytic mapping uhhh if you have a analytic function which is uhhh non which is one to one then we know that it is an isomorphism on to its image maybe the inverse function is also uhhh analytic it is holomorphic okay. And uhhh on the other hand if the derivative of function does not vanish at a point then there is a of course a neighbourhood by continuity there is a whole neighbourhood surrounding that point where the derivative will not vanish. And what will happen is in that neighbourhood uhhh the function will be uhhh locally in uhhh bihologomorphic. So, in other words for given every point in that neighbourhood uhhh I I can find a smaller neighbourhood where the function becomes one to one okay. So, uhhh now the question that we turn to is what happens when uhhh the derivative uhhh vanishes okay. And uhhh just as we have in functions of one variable the set of points uhhh where the derivative vanishes uhhh is called the critical set such points are called critical points and the function values at these points are called critical values okay. And we want to study the function or the behaviour of the function uhhh at a critical point right. Now so, let me so, let me make the definitions uhhh uhhh let. So, uhhh behaviour at a critical point. So, here is uhhh so, here is a definition let uhhh f from D to C be uhhh analytic uhhh on a domain D D in the complex plane and uhhh a point z0 of D is called the critical point for f if f dash of uhhh z0 equal to 0 okay. So, uhhh it is simply a point where the derivative of the function vanishes and of course the derivative exists because the function is analytic. Of course you know if the function is analytic then it is infinitely differentiable okay. So, derivatives of all orders exist and the condition for the critical for a point being a critical point is that the derivative must vanish alright. And uhhh the value w0 which is f of z0 is called uhhh the critical value it is called the critical value corresponding to uhhh the critical point z0 okay. And uhhh so, if you look at uhhh z0 is it not is it not is a 0 of f dash okay. And uhhh of course f dash is an analytic function okay. And note that uhhh you know uhhh uhhh of course I am uhhh I am certainly not going to consider uhhh a constant analytic function okay. I am I am assuming that my analytic function f is non-constant because had it been constant then the derivative will be identically 0 then. So, what happens is that it will uhhh in that sense it will be you know uhhh every point will be a critical point. And uhhh there will be only one critical value uhhh namely uhhh uhhh uhhh the constant value of the function okay. I am not considering that case okay. So, I am considering a non-constant analytic function. So, if I take a non-constant analytic function then uhhh uhhh the uhhh then f then if you look at f dash what happens to f dash uhhh I am particularly interested in the case when f dash is also a non-constant analytic function okay. And uhhh uhhh of course you know uhhh uhhh if f has a critical point then f dash is uhhh 0 at that critical point. And uhhh if f dash is a constant then it will tell you that f dash is identically 0 and that will tell you that f is constant. Therefore uhhh you know f dash will be also a non-constant analytic function okay if f is a non-constant analytic function. And uhhh so if you look at this non-constant analytic function f dash uhhh z0 is a 0 of that and it has to be isolated you know the 0s of a non-constant analytic function are isolated okay. That means every 0 can be surrounded by a disc of finite radius where you cannot find any other 0s right. So, this uhhh this z0 is an isolated 0 of f dash and what this will tell you is that uhhh the critical points are isolated okay. So, uhhh the critical points of an analytic function are isolated points okay. So, that is the that is what I wanted to say and then uhhh you can also define the order of the critical point to be simply the order of uhhh the 0 uhhh uhhh z0 of f dash okay. Because after all z0 is a 0 uhhh namely the critical point is a 0 of the derivative and you look at the order of that 0 and call that the order of the critical point okay. So, uhhh so let me write that down uhhh if f is if f is non-constant so is f dash and hence uhhh uhhh of course if f is non-constant and uhhh f has a critical point then f f dash is also uhhh non-constant uhhh and hence uhhh and hence uhhh any critical point of f is isolated. So, the critical points are for an non-constant analytic function are isolated okay uhhh the order the order of the critical point of the critical point z0, f is defined to be the order of the 0 of uhhh uhhh f dash and z0 okay. So, this is the definition of what a critical point is what a critical value is and the what is the order of the order what is the order of the critical point okay. Now uhhh the uhhh of course you know if you take a point which is not a critical point alright if you take a point which is not a critical point then uhhh you are taking a point where the derivative does not vanish okay and that case we have already studied where the derivative does not vanish uhhh if you take a point where the derivative does not vanish then there is a whole neighborhood surrounding that point where the derivative will not vanish because the derivative is continuous and then in that neighborhood you know uhhh sorry at he given any point in that neighborhood you can find a small smaller disc where the function is 1 to 1 and uhhh the function is invertible okay. So, you you can you can locally get an inverse but therefore what we want is we want to study the behavior of a function in neighborhood of uhhh a critical point in a neighborhood of the point where the derivative is 0. So, the first thing I wanted to say is that you know the the technique of uhhh studying this has always been uhhh trying to look at only zeros okay. The the whole theme of the lectures uhhh that so far has been to look study only zeros of analytic functions okay that is our theme and uhhh therefore you can see that uhhh uhhh even when I study a critical critical point I am I am studying the zero of analytic function namely the zero of the derivative which also is an analytic function okay. So, the so we want so let me write this down we want to study the behavior of f uhhh in a neighborhood of uhhh a critical point z0 of order uhhh m- m-1 okay. So, I am choosing m-1 for a particular reason uhhh with with critical value uhhh with critical value uhhh f of z0 is equal to w okay. So, uhhh so you see the what we do is why this m-1 is because we will instead of considering the function f of z if you consider f of z-w0 okay then what happens is the uhhh the a critical point for a function will be the same as uhhh for any other function which differs from the given function by a constant because when you take the derivative the constant is going to go away alright. So, if I consider the function f of z-w0 okay that function uhhh will still have z0 as a critical point because its derivative will be the same as f dash of z and which will vanish at z0 but the nice thing now is z0 is also a zero of the original function uhhh of this of this of this function okay. So, uhhh note that uhhh z0 is a zero of order m uhhh of f of z-w0 okay. So, uhhh so the method is that you uhhh you reduce everything to studying zeros of analytic functions alright and of course you know we have studied we have studied the uhhh uhhh in in all this uhhh I am assuming that uhhh m is at least uhhh you know uhhh 2 okay. Note because it is a critical point z0 is at least a simple zero of f dash okay. So, m should be at least 2 right uhhh. So, if you take f of z-w0 then z0 is at least a zero of order 2 okay. Now uhhh see the to to understand uhhh what happens to uhhh the mapping f in a neighbourhood of the point z0 okay uhhh given this uhhh condition. We will have to we will need to understand uhhh few other concepts uhhh uhhh probably you have come across them in a first course in complex analysis I am not sure if if you have but anyway I will recall them. So, these are to do with uhhh looking at the Riemann surface corresponding to various branches uhhh giving the inverse of a given function okay. The the so called uhhh Riemann surface of a multi valued function as it is called okay. Of course I should uhhh I should warn you that the uhhh the the terminology multi valued function is a is a misnomer in the sense that a function is not supposed to have uhhh take several values okay. A function by definition strictly is something that gives a well defined value for each value of the variable okay. So, if it is multi valued it is not a function okay but that is not what it means what it means is that you have several solutions to the inverse function. So, uhhh let me explain uhhh let me explain uhhh uhhh the point of view that uhhh when you try to write a function and write an inverse okay. The problem is that you may have several inverses okay and uhhh they will give you various so called branches of the function right and these branches will be different functions basically. But then if you want to think all of them as one and the same function okay then that is possible on what is called the Riemann surface of that inverse function okay. So, I will explain that so uhhh so so I will let let let me write this uhhh let me write the title as the Riemann surface of uhhh a multi valued function uhhh giving inverses inverses to a given function. So, uhhh so this is something that uhhh that needs to be understood. So, I will start with uhhh so I will uhhh start with the uhhh the most uhhh two most basic examples the first one is uhhh the logarithm which is the inverse to the exponential function okay and then I will uhhh look at the power function uhhh z going to z power m and uhhh why I am looking at the logarithm first is because uhhh that is this source uhhh if you understand uhhh uhhh that the logarithm has different branches and you understand how to define these branches. Then you can define branches for many other functions just using the logarithm for example for the power function. And then why am I interested in the uhhh power function I am interested in the power function because the final fact is that if you look at the mapping f of z in a neighbourhood of z0 up to uhhh change of coordinates it will behave exactly like the power function z going to z power m where uhhh you know uhhh where m is the same okay. So, if you look at this uhhh if you look at this function in a neighbourhood of z0 the behaviour of this function will look like z going to z power m. So, that is the reason I want to study z going to z power m and to study the inverse of z going to z power m okay which are the m throats of the variable they are there are uhhh you know m m throats they are the branches okay and you get those branches from the logarithm the various branches of the logarithm okay. So, uhhh that is the uhhh outline of what I am going to do. So, start with start with uhhh f of z is so let me not use f probably I will use uhhh g of z is equal to e power z the exponential function. So, uhhh this is uhhh this is an entire function. Entire function uhhh means uhhh if you recall it is a function that is analytic on the whole plane okay it is defined on the whole plane and it is analytic on the whole plane. And you know pretty well that uhhh uhhh the image of this function is the whole plane minus uhhh the origin the only value that the exponential function does not take is the is is the value 0 right. And you know also pretty well that uhhh uhhh the inverse of this function is given by uhhh a logarithm okay. So, uhhh you know how to write the logarithm if uhhh z0 uhhh if if uhhh z is x plus i y then you know that uhhh we have you can define log z okay which is uhhh you define it as uhhh uhhh half uhhh natural logarithm of uhhh x square plus y square uhhh plus i into uhhh argument of z uhhh plus 2n pi okay. This is how you define the uhhh logarithm of uhhh uhhh z if provided of course provided uhhh z is not 0 okay. So, probably I I should uhhh uhhh maybe change notation uhhh let let let me do the following thing uhhh. So, I want to think of an inverse function okay. So, I I would like to solve for e power z equal to omega okay uhhh in that equation I would like to solve for z. So, I am thinking of e power z equal to w. So, let me call it w not omega. So, I have e power z equal to w. I am trying to solve it for z as a function of w and what I will get is z equal to log w. So, I will change let me change everything to w. So, I will uhhh the source complex variable z real part are real and imaginary parts are x and y. The target complex variable is w I will put the real parts as u and v okay w is u plus i v. So, if I do that I will get something like this. So, log w uhhh. So, I will get u squared plus v squared uhhh and i argument of w plus 2 and 5 uhhh provided w is not 0 okay. So, uhhh the point is that uhhh here the uhhh uhhh this argument of w is something that uhhh uhhh has many values okay. The argument has many values and you get all the values by fixing one value of the argument and adding all possible uhhh you know integral uhhh all possible uhhh integer multiples of 2 pi okay. And this is the reason why you are uhhh why you getting uhhh many inverses okay. So, if you solve e power z equal to w then the solution is z equal to log w that is the inverse function alright and it has many values log w has many values. The reason for so many values is because the argument of w is not uhhh is not well defined it is defined only up to uhhh a multiple of 2 pi okay uhhh. Now, so in some sense it is not correct to call this as the inverse function okay because the inverse function should be unique but we can make this into a function. So, uhhh so what do you do? So, this is the uhhh this is the whole point we uhhh we uhhh define what is called a branch of the logarithm okay. We define what is called a branch of the logarithm that gives you a single valued inverse function okay to the exponential function which is analytic okay. And the fact is that this single valued inverse function will not be defined on the uhhh target plane of course you know when I take the inverse function you see this the situation is like this I have the I have the complex plane here which is the variable is z and then I have the mapping uhhh g of uhhh z is equal to e power z equal to w and uhhh well the uhhh the images in the w plane okay. And uhhh I am missing out uhhh I will have I will miss out the origin because uhhh e power z will not take the uhhh value 0. So, if at all I try to define uhhh an inverse function uhhh g inverse is log and you know let me let me put this in quotes let me put this in quotes why I am putting that in quotes is because you know there is no if you start with the non-zero uhhh value of w you do not get a unique value of the logarithm you get so many values of the logarithm okay you get several values of the logarithm. So, uhhh so you know the way it is is if I start with uhhh if I start with the value say w uhhh one let us say okay. Then I will get if I look at log w one alright I will get uhhh lot of values all the values will have the same real part which will be half log mod w okay which is uhhh which is uhhh say some uhhh some value here. So, this is this line this line is given by uhhh real part of z is equal to half log mod w okay uhhh maybe I should not write it is just log mod w you should not write half because it is already uhhh line of mod w right and uhhh and of course uhhh what are the uhhh what is what are the imaginary parts the imaginary the the imaginary uhhh parts of this logarithm they all differ by uhhh multiples of uhhh uhhh 2 pi. So, what is going to happen is well you are going to get you are going to get various you are going to get various points uhhh uhhh you know uhhh so you know if if if you if you have a point here then you will have you will have another point here and you will have another point here and so on and all these distances will be 2 pi you will get so many values okay. So, you will get uhhh you will get so many values uhhh all having the same real part alright namely log mod w alright and uhhh the but the imaginary parts will differ by uhhh every every successive point uhhh 2 nearby points if you take uhhh 2 logarithms uhhh the imaginary uhhh part will differ by 2 pi alright. So, in some sense you know the inverse mapping is trying to send w1 to uhhh uhhh of course this is this should be w1 non mod w1 okay. So, the inverse mapping is trying to send w1 to this uhhh uhhh all these points. So, it is not a function okay because I do not have a unique uhhh value I cannot pick out a unique value from this alright. Now uhhh what we do is we do the following thing we try to define a map in this direction and of course on this side you will have to take c- the origin okay you have to throw out the origin. So, which I will I will draw it like this I will I will I will put it put it circle here saying that uhhh uhhh uhhh the origin is thrown out okay and I will try to I can try to think of defining uhhh a branch of the logarithm. So, let me write this uhhh. So, let me not even say branch now for the moment let me say uhhh try to define uhhh uhhh log in this direction from the w plane to the z plane right. Now uhhh the fact is that I there is no of course I I am trying to define a inverse function to an analytic function and I would like that also to be analytic okay uhhh I would like to define uhhh any function that I would like to study is uhhh something that I would like it uhhh like to uhhh something that certainly I would like of that function is that it should be analytic alright. Now which means in particular it has to be continuous alright. Now the continuity itself forces that you cannot define it on this okay and uhhh uhhh so you explain that what we will do is we will we will let us uhhh cut out uhhh the uhhh the negative real axis you it we if you cut out the negative real axis uhhh it is called this slit plane okay that means you throw out the line segment from minus infinity to uhhh 0 including 0 okay. So, it is not possible to define it on uhhh the punctured plane C-origin but it is possible to define it on a subset of that namely the slit plane and what is the slit plane the slit plane is uhhh plane- the complex plane- uhhh the line segment from minus infinity to 0. So, this is the uhhh interval on the real line thought of as a real axis extending from minus infinity to 0. So, that is the that is the shaded piece I am I am just taking out this okay. So, uhhh the the way to picture it is uhhh that you know uhhh you must think of uhhh uhhh this you must think of this line being cut okay. And then what you do is you do the following thing you define uhhh so let me define this. So, I will define for any W here alright I will define what is called the principle branch of the logarithm alright and the so called principle branch of the logarithm uhhh uhhh actually uhhh depends on what is called as choosing a principle branch of the argument of W okay. So, what you do is you define log W principle branch of the logarithm of W to be uhhh ln mod W plus i times principle argument uhhh uhhh of W and what is this principle argument of W the principle argument of W uhhh is the argument that I will take from uhhh it is it is the angle from minus pi to plus pi okay. But then uhhh I throw out one of uhhh these end points okay. So, probably I throw out uhhh I mean the uhhh I wonder what the convention is uhhh maybe I throw out uhhh a plus pi okay. So, what you will do is uhhh so you put this condition where uhhh minus pi less than or equal to argument of W strictly less than pi okay. So, you see see the uhhh uhhh what you must understand now is that uhhh when I do this I I have when I when I write this I have I have not thrown out uhhh the uhhh the negative real axis okay. Mind you the argument W the argument of a complex number is always defined for a non-zero complex number. So, this this argument is always defined on the punctured plane for any complex number any non-zero complex number you can define the argument. The only thing is the argument is ambiguous in the sense that you can add any multiple of 2 pi to it okay that is because as far as the angle is concerned adding 2 pi does not change the point on the plane alright. So, uhhh well uhhh so what you must understand is that this this is uhhh defined on the punctured plane okay. But the problem is it will fail to be continuous on the negative real axis because you know if you take a point uhhh below the uhhh negative real axis the argument will be close to minus pi okay. But if you just push that point above the negative real axis and in fact it will it will exactly be equal to minus pi if it is a point on the negative real axis alright. Because I have taken minus pi to be the argument uhhh uhhh also I have taken minus pi also to be one of the values okay. So, every point on the negative real axis has uhhh argument minus pi and points below that will have arguments close to minus pi. But if you just go a little bit above the negative real axis the argument will be close to plus pi. So, you see there is a jump in the argument there is a jump in the argument which is uhhh a jump of uhhh uhhh uhhh nearly 2 pi okay. And it is this jump in the argument that prevents or this function from being a continuous function on the negative real axis. And you know uhhh if you think of log w as a function okay uhhh and of course ambiguity was what argument use suppose I use the principle argument okay to remove any ambiguity. If I want this uhhh I want this log to be a an analytic function. So, I want it to be a continuous function alright. And mind you if it if it is continuous then both the real and imaginary parts should also be continuous. A complex valued function is continuous if and only if it is real and imaginary parts considered as real valued functions are continuous alright. Therefore, if you want this to be continuous then ln mod w and r argument of w should both be continuous functions. Of course, ln mod w will always be continuous okay because it is a natural it is the it is a natural uhhh logarithm okay it is a real valued non negative real valued function. There is no problem with this okay. Whereas the problem is with the argument as you can see because the argument is not going to be continuous on the negative real axis. And that is the reason why why if I even though this function is defined on c-0 okay this principle logarithm as it is called which is defined using the principle branch of the argument. The principle logarithm defined using principle argument uhhh where uhhh we usually use capital L and to show that it is the principle branch of the logarithm and capital A to say that it is the principle branch of the argument or principle argument. The point is that uhhh this function even though it is defined on the punctured plane it is not continuous on the punctured plane. To make it continuous you will have to cut out uhhh the uhhh you will have to cut out the uhhh the negative real axis okay you have to cut it out. What that means is that uhhh what is the advantage of cutting it out I mean cutting it out does not mean remove it in in in a strict sense. Cutting it out means that you separate the portion of the uhhh uhhh uhhh you separate the portion above the negative real axis from the negative real axis and the portion below. You you you you separate them and why do you separate them you separate them because once you separate them you cannot move from here to there. You cannot think of see what is the problem in this being continuous the problem in this being continuous is that I can take a point here and I can I can take a point which is uhhh uhhh on the negative real axis or slightly below the negative real axis uhhh and then very easily push it to go above the negative real axis. That is because the region above the negative real axis is close to the negative real axis and the region below it okay. Now by slitting the plane what I am doing is I am just making them far away okay. Therefore I would not have this I I simply cannot push a point on the negative real axis or below the negative real axis to above the negative real axis that is because I have cut it. I have I have purposely created a disconnection okay and once you look at this slit plane like this then you see that this becomes a continuous function because the only problem was continuity on the negative real axis. So it becomes a continuous function and the truth is it even becomes an analytic function okay. This becomes an analytic function and uhhh of course uhhh uhhh this function called a branch of the logarithm with the principle branch of the logarithm becomes an inverse function to the function e power z okay. But mind you that the inverse function uhhh as a function is defined on c-0 but as a continuous function can be defined only after you after you make the slit okay and then of course if you make the slit then it becomes not only continuous it is actually analytic alright. And so you get uhhh so the moral of the story is if I take this function exponential function uhhh even though the exponential function is not uhhh 1 to 1 you know uhhh because if I change if I to z if I add any multiple of 2n pi i I will still get the same value and apply the exponential. So it is a many to 1 function. So obviously the if I take if I think of an inverse function the inverse function will be multiple valued in general we call this multiple valued function as log. We call it as a logarithmic function but to make it a single valued function you have to take a branch of the logarithm. That branch of the logarithm for example in this case is a principle branch is defined on the punctured plane but even though it is defined on the punctured plane it will not be continuous. If you want to think of it as a continuous function then you have to slit the plane okay. You will have to separate the the the portion above the negative real axis from the negative real axis and the portion below it by making a slit okay. And then if you consider the slit plane then this function becomes an analytic function and it becomes an inverse to this function. In the sense that you apply this function then apply the exponential function you will get the identity map on the slit plane. And this becomes a single valued function and it becomes analytic and it becomes really an inverse function to this. So the moral of the story is you are able to get an inverse function only after making a slit in the plane okay. In fact this slit need not have been made here you can even make it along a radial line okay. Only thing is on whichever line you make it you make the argument to you know vary from there to I mean you you just it is just you rotate this slit by whatever angle you want. So you can slit along any radial line okay that does not matter that will also give you a branch this is the principle branch okay. Now so so what are the other branches see the other branches other branches are given by the principle branch plus a fixed multiple of 2 pi i okay other branches is equal to or are given by of log or given by the principle branch plus 2 n pi i for fixed n okay. So in particular you know here I have taken the principle argument from minus pi to pi alright but then I can add 2 pi to it if I add 2 pi to it my principle argument will vary from pi to 3 pi that is another branch of the argument that will give you another branch of the logarithm that branch of the logarithm will be this principle branch of the logarithm plus 2 pi i so I have just added 2 pi okay and this is how you get every branch alright from this from one branch. Now the point is where does the idea of Raymond surface coming the idea of Raymond surface comes in the following nice way we want to think of all the branches as one function you want to think of all the branches as giving one and only one function okay the way to do it is of course they are not one and the same function they are different functions so the idea is that you modify the domain of the function okay you change the domain of the function instead of it being a slit plane you make a Raymond surface and on that Raymond surface you will get a function which on each piece that is a designated subset of that Raymond surface you will get all these branches on every piece. So how does one do it one does it in a very nice way see you have so let me explain so you have so let me draw like this this is a slit plane and I will just draw it like this and here I have log z log w okay and what you must understand is well if you then what I have next is well I have log w plus 2 pi i this is another branch this is also defined on the slit plane okay this is I have added 2 pi to it alright and so on it continues in both directions alright so what I should draw about will be something like this it will be log w minus 2 pi i that is another branch of the logarithm that again is going to come that is going to be again an analytic function on the slit plane. So I have so many copies of the slit plane and I have for each copy I have an inverse function which is given by a branch of the logarithm all the functions they all differ only by integer multiples of 2 pi i okay. Now what I do is I do the following thing see I do a pasting construction and the pasting construction is like this you see if I look at log w and look at the imaginary part it is the argument you see the argument is negative here it is negative at these points that is just that is on the negative real axis and below it is negative and above it is positive and in the negative on the negative side the values are close to here the values are close to argument close to minus pi okay and above the argument is close to plus pi alright. Now look at this one look at this branch if you look at this branch the argument is going to now change from so I have added 2 pi so it is going to change from plus pi to plus 3 pi okay. So what will happen is here the argument is close to plus pi and of course it is lesser than plus pi okay it is close to plus pi and not lesser than in fact greater than or equal to plus pi that is what it is because I have added 2 pi to this okay and if you look at it above if you look at points here the argument is close to 3 pi and of course less than or equal to 3 pi strictly less than 3 pi because it changes from minus 3 pi less than or equal to the argument of this branch strictly less than I mean plus pi less than or equal to argument of this branch strictly less than 3 pi okay. So below it is going to be close to plus pi and greater than or equal to plus pi and above it is going to be close to 3 pi but lesser than 3 pi alright and now you look at these two slit regions look at these two slit regions okay look at these two slit regions is anything bothering you. If the negative axis is removed is it not I mean the argument below that diagram we strictly greater than plus pi instead of greater than equals now you are right I mean if you think of it as removed then I should write strictly greater than pi if I think of it as removed then I will write strictly greater than or equal to strictly greater than or equal to plus pi okay but if I do not think of it as removed then I write it like this okay but I want to do the following thing what I want to do is I do not want to remove it because I want to paste it. So what I am going to do is I am going to do the following thing look at see think of this piece you think of this piece like this okay so this is the piece of the complex plane here the values of the argument are greater than or equal to pi okay and above they are well going close to 3 pi but look will concentrate below here the values are greater than or equal to pi and that is these are the values which are the values here the values here are argument is close to pi and strictly less than pi okay. So what you do is you see you paste this piece of the negative axis to that is the lower piece the lower this slit plane you take the lower portion of the negative axis along with the negative axis and simply join it to the upper portion of the portion above the negative real axis on this piece okay. So you know basically if I put them together I hope I will be able to I wonder if I will be able to draw a need diagram so you know so just for the purpose of identification let me put let me put something here so I am identifying I am pasting the lower thing here with the upper thing there so I will put plus here and I will put a minus here okay and so you know so I am going to so let me write this here so it is from here to here I am doing a paste okay why I am doing this pasting is because now if I paste these two slits together then as I move across the negative real axis then the when I am on this piece the arguments when I am when I am on that piece the argument the argument values are close to pi and lesser than pi and they are approaching pi but then after I pasted it they will continue on this piece they will achieve the value pi and they will continue okay so by pasting the upper portion of the imaginary axis with the lower portion of the imaginary axis and the imaginary axis together what I have done is I have gotten a surface on which the argument seems to be continuous even on the negative real axis you see what is happening the argument has become now a continuous function including the points on the negative real axis just because I have cut the negative real axis the portion above the negative real axis from here away and I have joined the portion consisting of the negative real axis and the region below it to the portion of the negative real axis above okay. So and you know in fact in the same way what will happen is that you paste the if you look at the values of the argument here the values of the argument here are close to minus pi and of course and well greater than or equal to minus pi that is what happens here alright if you look at this okay I have taken away minus 2 pi so I will get if I take away minus 2 pi from this I will get minus 3 pi to minus pi so what will happen is here the at these points below the negative real axis and on the negative real axis the argument will have values close to minus 3 pi and of course greater than or equal to minus 3 pi this is what I will get okay and what will I have above I will have values close to pi minus 2 pi which is minus pi and lesser than minus pi so above I will have argument values close to minus pi and lesser than minus pi okay and now you see here if I take the top portion of the negative real axis the argument has values lesser than minus pi and tending to minus pi and those values are achieved here in the bottom portion of the negative real axis and the negative real axis the argument values start from minus pi and they increase. So what you do is in the same way you paste this to this you make a paste like this okay so what you do is you take the negative real axis and the lower portion of the negative real axis and paste it to the upper portion of the negative real the portion above the negative real axis in this piece okay and that is you see that is just what you did here okay so if you now look at this procedure what you are doing is for every piece what you are doing is you are taking the negative real axis and the portion below the negative real axis and you are pasting it to the piece to the portion of the negative to the portion above the negative real axis in the slit plane I mean in the sheet in the piece above okay and if you do this infinitely okay you can imagine it so let me draw a diagram it may not be the best diagram you know if I do this I will get at any finite stage I will get something like this I will get something like a and of course you know the origin has been removed so there is a hole okay and you see I get something like a spiral continuously spiral staircase kind of thing alright and so I get a surface and this surface will of course if I do this if I do this of course you know I have drawn a boundary here there is no boundary okay in fact I should put dotted lines because this is just this is a whole plane here there is no boundary but you know I am just drawing it so that you can visualise it it is a little hard to visualise but with a little bit of thought you can okay. So what you get is you get you know copies upon copies of the plane okay being cut and joined on the negative real axis okay with the origin removed and you get a surface. Now this surface is a Riemann surface why because what is our definition of Riemann surface a Riemann surface is something that should be locally homeomorphic to the plane it should have it should be as a topological space it should be you know connected it should be how star should be second countable of course all these things are true the way I have pasted it is going to be connected it is going to be of course host of because you take two points I can separate them by open sets if they lie on the same piece of the cut plane then I can certainly separate them by two open sets if they lie on different pieces okay which are called so these various pieces of the of this of the plane the cut plane they are all called as sheets of the Riemann surface okay so you have infinitely many sheets as many sheets as there are integers okay because you have all the you have as many branches of the logarithm as there are integers and every it is only by choosing an integer n that you get a different branch of the logarithm from the principal branch by adding 2 n by I do it okay so you get this this thing is certainly a Riemann surface okay it is a Riemann surface it is connected host of second countable and certainly I can make sense of there are natural charts because each piece is just the slit plane. So I can define and the way I have defined the way I have cut and paste it the argument becomes a continuous function. So what you must understand is now you see if I can write A R G Z A R G W this is a this is a continuous function argument is a continuous function argument which was originally not a continuous function on a single copy has now become a continuous function that is because I have carefully cut and paste various copies. So this argument which is multiple valued function which is not continuous I have cut and paste the domain so many copies of the domain so that I get a function which puts together all these branches of the argument function. So you know therefore argument function is a continuous function on this and beautiful thing is the log function will also be a continuous function on this and it will be analytic on each piece it will give you a branch of the logarithm corresponding to that piece okay. So the beautiful thing is that you know so the and then you know I have I can write a projection here onto the W plane okay and you know now you know from the Z plane to the W plane I have the function e power z alright I have this of course when I project it the origin will be missed the origin will not be there because origin has been removed alright. So look at this diagram I have the exponential function if I try to define the inverse function I do not get it here but I get it above I get it on a Riemann surface which sits so which projects onto this so this is called the Riemann surface of log z okay this is called the Riemann surface of log z let me let me put smaller I will put smaller because it says it is now a single valued analytic function on this Riemann surface it is analytic because analytic is local and if you want to check analytic on every sheet I know it is already analytic on each sheet so it is an analytic function okay. So it is a single valued analytic function it is inverse for this function so the beautiful thing is here is a function it is a many valued function okay the inverse you do not get from the target you get on a Riemann surface that covers the target in fact this map in the language of covering spaces is called a covering map this is a covering map it is called an infinite sheet at covering of the punctured plane okay and the moral of the story is if you try to look at the inverse to a function you will get so many branches and the if you want to think of the inverse as really a function you have to go to a Riemann surface of the inverse function there it will be a honest analytic function which will be an inverse to this okay so this is the picture so this is one of the nice things about Riemann surfaces they allow you to think of various branches of an inverse function as a single function on a surface okay alright so I will stop here and continue in the next lecture.