 Okay so what we are discussing now is trying to understand the behaviour of a function as a mapping in a neighbourhood of a critical point. So if you recall a critical point is a point where the derivative of the function vanishes okay and of course you have already treated the case when the derivative does not vanish, we have already seen that when the derivative does not vanish then there is a small neighbourhood about the point where the function is even one to one and you can invert the function okay. So we come to the case when the at the case of a point where the function has vanishing derivative and to study the behaviour of the function at a point where the derivative vanishes we needed to understand the idea of branches of a function especially branches of a logarithm and that is what I explained in the last lecture. So let me recall so we recall that we had the complex plane and we had the function f of z is equal to e to the z is the exponential function and this takes values in the complex plane and the point is that we wanted to think of an inverse to this function and the inverse function would be well the logarithm log z okay and of course how this log is defined first of all we should understand in a set theoretic sense that e power z is not a many I mean it is not a one to one function so it does not have a set theoretic inverse okay it is a many to one function okay and in fact if you take any value of z which is translate of z by an integral multiple of 2 pi i the exponential function will take the same value because e to the 2 pi i is 1 okay. So therefore it is not a one to one function and in principle one should not think about one should not say that one wants an inverse function okay because it is you talk about an inverse function only if a function is bijective is injective okay but we want a function which is inverse in the functional sense namely a function that can undo the effect of exponentiation so the logarithmic function is suppose to be a function that under that cancels the effect of exponentiation okay but the fact is that when you try to write out a function in this direction okay then you get not one function you get several functions okay and these are called branches of the logarithm and if you recall how we define the branches was like this what we did was we took we defined first of all what is meant by the principle branch of the logarithm okay and so well let me not say let me call this plane as the z plane and well then this becomes the omega plane or the w plane where w is f of z and well the inverse function should therefore be log w okay and so if I follow that convention here this is the principle branch of the logarithm so this is from the complex plane which corresponds to the w plane and what we are going to do is to define the logarithm first of all you cannot define the logarithm of a z of this of the 0 complex number so you remove this okay. So what you do is you look at c star which is c minus the origin you throw out the origin because the exponential function misses out the value 0 it never takes the value 0 there is no logarithm for 0 so you throw that out and then what you do is well how do you define the principle branch of the logarithm well you say the principle branch of the logarithm is given by take the natural logarithm of the non the positive real number mod w is the modulus of w and add to it that is the real part and the imaginary part is the principle argument of w and what is the principle argument of w it is this angle from minus pi less than or equal to argument of w strictly less than pi okay and if you define it like this log w makes sense as a function and takes values in complex numbers and if you and this is z plane and well the fact is that this is an inverse functional inverse to e power z because you take z and then you take e power z and you plug e power z for w and you will get back z okay so in that sense well it is an inverse function alright now well the problem is that we want an inverse function which is you know we want an inverse function which is not only just a function we want it to be continuous we want it to be analytic because we are interested only in analytic functions now the problem is that this log w it is defined on the whole punctured plane okay that is the plane minus the origin but it is not continuous because the continuity the problem with continuity is at this negative real axis and that is because the continuity is with the imaginary part of the function which is the argument function is not continuous because the argument function on the negative real axis and just at points below the negative real axis the argument is close to minus pi and just above the negative real axis it is plus pi and this is a jump in the argument of 2 pi nearly and this jump tells you that the argument is not continuous across the negative axis. So what you will have to do is well you cut this out okay there are two ways of doing it one is you cut out this and by cutting this out I am declaring that the portion of the negative real axis above the negative real axis is far away from the portion that consists of the negative real axis and the points below it and when I say they are far away then I do not have to check continuity across the negative real axis okay. So in that sense the cutting helps okay so if you cut the plane so as to separate the portion of the negative real axis above it from the portion of the from the negative real axis and the portion below it then it is called a slit plane okay and on that plane log is of course continuous and continuity and continuity on points above the negative real axis is means continuity in a small disc about such a point and for continuity the same will hold for continuity of at a point below the negative real axis but for a point on the negative real axis it will be continuity on the boundary okay. So the neighbourhood will only be a disc only the neighbourhood will consist of an open disc only the lower half of the open disc centred at that point on the negative real axis okay and the upper part of the disc will not be considered as part of the neighbourhood because it has been cut off okay or it has been separated alright. So the point is that if I take log w and slit this plane okay mind you when I slit this plane I am not throwing out the negative real axis I am keeping the negative real axis okay only I am separating the negative points on the negative real axis with all the points above it okay with that kind of slit plane okay log becomes a continuous function alright but then I am not happy with that we would like to have log as an analytic function but you see but to have a function which is analytic you know the definition of analyticity is that it is defined at a if a function is to be analytic at a point first of all the function should be defined in a whole neighbourhood of the point okay. So what happens is that if I take a point above the negative real axis or below the negative real axis certainly there is a small disc surrounding that point where the log function will be analytic okay because that you can verify by any number of means for example you can check the easiest way is you know that these two the real and imaginary parts you know for example it satisfies the Cauchy Riemann equations if you want you can verify that and verify that the first partial derivatives are continuous and so on and so forth. So it is very easy to check that this defines an analytic function of w if you throw out the points on the negative real axis okay and but for points on the negative real axis is a problem because they are boundary points okay for the slit plane the points on the negative real axis is a boundary okay and at a boundary point you never define analyticity because analyticity is always defined at an interior point because a function is analytic at a point the definition already requires a function should be defined in a open neighbourhood containing the point. So you never define analyticity at a boundary point okay but then so you know if you want to think of log w as not just a continuous function but as an analytic function then I will have to take I will have to delete the negative real axis I will have to delete the negative real axis and I will have to and then log becomes you know it becomes an analytic function alright. So I will have to throw out the origin along with the negative real axis then it becomes an analytic function and the point is that well this is not the only functional inverse to the exponential function there are many others and the other branches of the logarithm are given by well other branches other branches are given by taking actually the various branches of the argument function which differ by a fixed integer multiple 2pi okay. So the other branches are given by the principle branch plus 2npi i so I just add 2npi i and get the other branches okay. So all these put together will give you the various branches of the logarithm alright and well now there are two issues that one has to worry about one issue is that I have so many branches and I am not able to see them as it will be nice if I can see them as one function okay. The other issue is that I have problems with defining logarithm on the boundary I mean the negative real axis that I have thrown out okay. So I mean well you know the negative real axis had to be thrown out because I chose my argument to be from minus pi to plus pi and that creates a problem at minus pi alright. But instead of there is nothing special about the angle pi okay I could have chosen any ray okay and I could have called that angle as theta and I can replace this side by theta and that side by theta plus 2pi alright and then I would get then I will get different branches of the logarithm which for which the slit will be along the ray alright. So there is nothing special about the points on the negative real axis okay. So the moral of the story is that I would like to really if necessary I would like to have even a branch an analytic branch of the logarithm on the negative real axis with which I can work okay and how is all this possible all this is possible by looking at the Riemann surface for log z. So I told you that what we do is that well you know we paste together as I explained in the last lecture we paste together you know so many copies of the plane carefully along the well along the slit negative axis. So what I do is that I have I have several copies of the slit of the slit plane and mind you I am not throwing out the negative real axis okay. But what I am doing is that well I take I identify the upper part and the lower part and you know I stick it in such a way I take 2 such copies of course you know this is I am just throwing a boundary but there is no boundary this is just to tell you that this is a piece of the plane which looks like a well disc right. But it is a whole plane that I want you to imagine and I have cut it across the mind you have not deleted the negative real axis. I just cut along the negative real axis in such a way that I have separated the negative real axis and the portion below it from the portion above the negative real axis and the reason is because I want to join the portion above the negative real axis to the negative real axis in the portion below it in another piece okay. So this thing which I have denoted with plus signs on one piece well I will you know attach them to this piece okay and well the portion on the negative real axis and below it will be attached similarly to the portion above the negative real axis in the piece above okay and these are the various pieces which correspond to the different branches of the logarithm okay and therefore the real the resulting picture will look like this resulting picture will be that of a rayman surface it will be so it is going to be well I have one piece like this and well I have attached yeah it is a little difficult to draw it yeah so it is something like this. So if I keep joining it along the negative real axis I will get something like this okay I will get a diagram that looks like this and in this diagram what has happened is that I have taken so many copies of the plane which is just so many copies of the punctured plane I have not thrown out the negative real axis but I have done the curtain paste along the negative real axis in such a way that I get a continuous surface alright and if I do this continuously what I get is rayman surface okay in the sense that it is a surface namely it is locally homeomorphic to the plane okay because it is consisting of various sheets as you may think of it each sheet being a copy of the punctured plane okay so it is locally homeomorphic to the punctured plane alright and it is certainly housed off and it is second countable and therefore it is immediate that what you get here is a by definition rayman surface rayman surface is supposed to be well topologically at least it is supposed to be something that is locally homeomorphic to the plane and which is housed off and second countable and of course on top of that you also need a collection of compatible charts to be able to do complex analysis on that to be able to decide when a function defined at a point neighbourhood of a point of the surface can be called an analytic function but then you get the charts because these homeomorphisms will allow you to identify the certain sheet with the punctured plane so there is a projection like this so here is a projection this is a projection map so we say that this is a covering of the plane of the punctured plane so you know I can remove the origin here and what happens is that this is a covering which we say is a countable covering because there are as many copies as there are integers and because each copy corresponds to a choice of n which is the domain of the branch of this logarithm alright and the moral of the story is that once you take this so this rayman surface which is the rayman surface for log z then the fact is that from this rayman surface if you take the function log I can define a function log w on this surface which is a single valued function okay mind you this surface extends in infinitely in both directions I have just drawn of three sheets okay but it extends infinitely in both directions and on each sheet I have the corresponding branch of the logarithm alright so but the point is because I have pasted it in a nice way okay the argument function the argument function on one piece if I take only for example if I take the principal argument the principal argument function is on this on the piece on which it is defined it is discontinuous at points on the it becomes well if I did not separate the if I did not cut the plane just for points on the negative real axis and below the negative real axis the argument function takes values minus pi and below and just above it takes the value plus pi but then what I do is I join this portion of the negative real axis and the points below it with another function for which the argument above is again values close to minus pi and lesser than minus pi and by doing this what I have done is I have extended the argument function continuously so the moral of the story is that once I do this bluing process the various argument functions okay from each of these which correspond to the various imaginary parts of the various logarithm functions the branches of the logarithm functions they all fit into one continuous function. So what happens is that you have one single valued function log function okay and that single valued function lives on this ringman surface on each sheet it corresponds to a particular branch of the logarithm on all put together it is a single logarithm and it is an inverse to the exponential function okay. So in fact I should also write argument of say argument of probability right. So the moral of the story is that the moral of the story is the following the moral of the story is that if you have a function which is an analytic function okay and then you can think of an inverse function you will not get one inverse function but in general you will get several inverse functions they will be called branches of the inverse functions and these branches will live on a ringman surface and the ringman surface will be got by doing cutting and pasting along what are called the branch loci okay or the branch points of the function. So for that is the reason we say that the negative real axis we choose as the branch points for the branch cuts for the so called logarithmic function. And of course the branch point is usually called the origin okay because the branch point is a point from which the branch cut originates. So the branch point is origin the branch cut is the negative real axis and you know by doing this cutting and pasting of various copies along the branch cut you are able to get hold of a ringman surface on which all the branches agree as a single function they all show up as a single function. You are able to stitch them all together to get a continuous function not only continuous function log w will actually be now an analytic function on this ringman surface mind you this is a ringman surface. This is a ringman surface because from every piece or every sheet you have the projection which identifies it with the complex plane okay and if you calculate the transition functions that the transition function will only be a translation by multiple of 2 pi i and you know translations are of course holomorphic. So the moral of the story is that these homeomorphisms that identify each sheet with the punctured complex plane they give charts and the compatibility of charts is automatic because if you calculate the transition functions they will simply be translations by you know multiple of some multiple of 2 pi i but you know every translation is holomorphic and the condition for charts to be compatible is that you know well the condition for charts to be compatible is that the transition functions are holomorphic okay. So the moral of the story is that you get a ringman surface so on this surface you can ask whether a function defined on the surface is holomorphic and the answer is that if you take log w on the surface it is holomorphic okay and it lives throughout the surface and it is holomorphic okay and that is the that is what happens here and that is what happens for a general function also sufficiently good function you will see that function will when you try to write out the inverse function you will get several branches these branches will be well defined and holomorphic if you make branch cuts along the so called branch points and if you do branch cuts properly and do cutting pasting process then you will get a ringman surface on which all the branches will define a single valued function okay. So this is the general philosophy alright and the point that you will have to understand is that well one important thing is the non-vanishing nature of the derivative of the function that you sort to invert. So in this case I am trying to invert e power z its derivative is e power z that never vanishes okay so the non-vanishing of the derivative is something very very important alright and so what happens in more in more generality is the following. So well the more general situation is like this so here is a theorem which we often would use the theorem is that so let me state it let me state it like this let f of z be an analytic function that is never 0 on a simply connected domain D okay then there exists an analytic branch of log f defined on D okay. So see here I am what I have done here is I am trying to get the logarithm of a complex number alright that means I am trying to invert e power z right. But more generally suppose I have an analytic function that never vanishes alright suppose I have an analytic function that never vanishes on a domain and assume that the domain is simply connected that means the domain has a property that it has no holes geometrically it means that given any loop in that domain that is a simple closed curve in the domain you can continuously shrink it to a point without going outside the domain okay which essentially means that you draw any simple closed curve simple closed curve of course means simple means that there are no self-intersections okay. So a closed curve is a continuous image of an interval with the initial point equal to the final point okay and you take any such simple closed curve at the fact that it can be shrunk continuously to a point is the same as saying that the region that it encloses is also inside the domain there is no part of that region which goes outside the domain which means that puts a hole in the domain so there are no holes that is what simply connected means and the point is that if you have a function which is nowhere 0 not that which means it is not 0 at any point of a domain is simply connected then you can find an analytic branch of logarithm of the logarithm of that function on the domain okay. So the proof of this is well pretty easy probably you have already seen this in a first course in complex analysis but I can recollect it for you so you see so here is my well here is my simply connected domain I have drawn a bounded domain but it need not be but the domain is the this region which has no holes okay and well and you know and what is given is that there is a function w equal to f of z which is never 0 on this domain alright. So what I do is I do the following thing what do you mean by saying that I have a logarithm of f of z and analytic branch of the logarithm it means it is an analytic function which is a logarithm of f of z namely that function is what if you exponentiate that function you will get f of z that is what it means alright. So you have to find a function h of z which is analytic in this domain okay and such that h of z is log f of z but what does that mean it means that e power h z should be f z okay. So if you exponentiate that function that you are trying to find it should give you f that is what it means. So well how do you construct such a function it is very very easy what you do is you fix a point z0 in the domain alright take any other point z and it is very simple take any curve gamma from z to z0 and you define an integral fix an z0 in D let for gamma for z in D let gamma be a curve from z0 to z of course by curve we mean a continuous image of an interval alright define h of z to be well integral from z0 to z along gamma so I will put this in brackets and of course I am going you know what I am going to write I am going to write D log f z okay which you know where of course you know what D log f z means where D log f z is supposed to stand for f dash of z f dash of z by f of z d z okay. So I define this function alright this mind you f of z does not vanish okay and f is analytic alright therefore f dash is also analytic and f does not vanish so f dash by f is also analytic alright therefore the integrand is an analytic function alright therefore this integral is well defined. So first of all you must understand that first of all for an integral to be well defined I mean the integrand should be continuous function alright and the integrand here is f dash by f okay the continuity will not be a problem for f dash because f dash is a derivative of an analytic function you know derivative of an analytic function is also analytic. So it is also continuous but the problem is there is an f in the denominator and that should not vanish okay but then we have assumed that f is never 0 on the domain therefore there is no problem about f being the denominator therefore you have a quotient of continuous function therefore it is continuous. So in particular it is also continuous on the path gamma therefore this integral is well defined there is no problem about this integral okay. Now what I want to say is that this integral is independent of the path okay see if I choose why is that true because if the integral is dependent on the path then this h of z will not be properly defined because to define h of z I have chosen a path okay which is a path that connects z0 to z of course inside d I am not going outside d alright but this h of z should not depend on the path if it depends on the path then it is not a well defined function but I claim it does not depend on the path why because you know if I choose some other path say gamma prime alright then if I you know that if I go along gamma and if I come back along the reverse of gamma prime alright then I get a closed loop and on that closed loop if I integrate this if I look at the integral of the d log fz that is f dash of z by fz I will get 0 my Cauchy's theorem because f dash by f continues to be analytic there because you see f dash is already analytic f is also analytic the quotient is analytic wherever the denominator does not vanish but the denominator never vanishes the denominator is f it never vanishes therefore the moral of the story is because of Cauchy's theorem this integral does not depend on gamma it does not depend on what path you choose to connect z0 to z so long as you make sure that the path lies inside your domain okay so by Cauchy's theorem theorem you know integral over gamma from z0 to z d log fz is the same as integral over gamma prime from z0 to z d log fz well actually you know I should be a little careful because I have used z here I should not use z in the integral you know maybe I should use zeta alright it is better to use zeta and then I will also change this to zeta so that it is easier because z is supposed to be the fixed value the fixed point where I am trying to define the function h alright. So let me change this to zeta let me make a difference between the variable of integration and the limit of the integral so well here it is this is better notation okay and by Cauchy's theorem this is equal to this because the difference is 0 by Cauchy's theorem okay. So this tells you so h of z is well defined so it tells you that h of z is well defined now you see now look at h of z now again let us apply another theorem namely Morera's theorem to conclude that h is actually analytic you see if you look at the function so you know Morera's theorem is the kind of converse to Cauchy's theorem okay what does Cauchy's theorem say it says you take a function as analytic if you integrate it on a closed curve simple closed curve and assume that a function is analytic inside the curve and also on the curve then the integral is 0. Now the converse to Cauchy's theorem will be if you have a function for which whenever you take a closed curve the integral is 0 then is that function analytic that is the expected converse but for the expected converse which is Morera's theorem you need to put the extra condition that the function you are starting with is already continuous. So you see in this case you see I am integrating the function that I am integrating is f dash by f okay and f dash by f is continuous throughout the domain f dash by f is continuous throughout the domain right and its integral over any closed curve is 0 because it is analytic therefore it satisfies the conditions of Morera's theorem and therefore this integral is an analytic function okay. So let me write that in the proof of Morera's theorem we see that h of z h of z is differentiable with derivative f dash of z by f of z okay and so the Morera's story is so in particular the proof of the theorem tells you that h is analytic okay. So thus h dash of z is f dash of z by f of z and h is a branch h is an analytic branch branch of log f in D okay. So it is easy to define I mean the looking at the logarithmic derivative it is easy to define a branch of the log okay and branch of the log which is analytic and a simply connected domain. All you need is a simply connected domain alright and okay I need this now what we will do is that we will try to study the function z going to z power m at the origin okay and what we claim is that you know this is how any function will behave at a critical point up to you know translations and up to some simple transformations okay. Any function at a critical point will behave like the mapping z going to z power m at the origin okay that is what I wanted to explain. So the function f of z is equal to z power m and I will assume that m is greater than 1 okay. So look at the function of course if m is equal to 1 it is the identity function there is not much to say but you want to look at this function and if you look at it then the derivative f dash of z is m z to the m minus 1 m is at least 2 and this is 0 only at z equal to 0 namely the origin so it has only one critical point. So the origin is a critical point is the only critical point with critical value 0 the origin is the only critical point and the critical value is 0 okay and of course I would like to study the function this function at the in a neighbourhood of the critical point namely in neighbourhood of the origin and the reason I am doing this is because I have this theorem about the availability of a logarithm for a non vanishing function on a simply connected domain I can use this to study the behaviour of any analytic function at a point which is a critical point okay. So this is the foundation so let us understand this case alright so well the first thing that you that you can see is well I have so I have this so I can draw a diagram similar to the one here okay that where I have constructed a Riemann surface for log z okay namely a surface which covers the punctured plane on which log is all the branches of the logarithm live together as one function okay. So I am going to do the same thing here so what I am going to do is I am going to just take the function ow is equal to f of z is equal to z power m okay and this is a function that goes from the complex plane to the complex plane and this is the z plane and this is the w plane and well of course the critical point is the origin alright. So I will single that out by putting a circle a small dot circular dot here so the origin goes to the origin and you know that the inverse function if I think of the inverse function it has to be z is equal to w to the power of 1 by m this is what the inverse function should be it should be the mth root of the variable that should be the inverse function alright because this way the function raises the variable to the power m the function in the other direction should take the mth root but the point is that you do not have one root you have in fact m of them and these are different branches and the branch point is the origin and the branch cuts are all raised the branches are defined on sectors alright. So let me draw the diagram for let us take m equal to 3 okay if I take m equal to 3 then you know pretty well that you know if I take this ray which is well it is 2 pi by 3 is this angle alright then as z for this whole domain namely this whole sector with angle 2 pi by 3 is going to be mapped on to the whole plane alright that is because if I write it in polar coordinates z power m if I write if I write z as r e to the i theta then you know z power m will become r to the m e to the i m theta and if m is equal to 3 I am going to get e to the i 3 theta. So as theta varies from 0 to 2 pi by 3 3 theta will vary from 0 to 2 pi so this whole so sector okay is going to be mapped to the whole plane okay and similarly you know if I take the other sector namely if I take this which is going to be 4 pi by 3 okay then this sector beginning from here to here that is an angle of 2 pi by 3 and that is again going to be mapped on to the whole w plane okay if you remove the origin then the origin there is also removed alright and well similarly this sector which again is an angle of 2 pi by 3 alright that is also going to be mapped on to the w plane alright and of course so you know this whole copy of the z plane is being mapped thrice on to the w plane when m is 3 and you know when it is m it is going to be mapped m times and it means that you know if you take the image of a for example if I take the unit circle here I take the unit circle here and I travel along the unit circle and you know I look at its image there what I will get is well I will get the unit circle but it will be traversed 3 times if m is 3 and it will go around m times for a general m okay and this also tells you something for example you know if I have so you know if I take this point 1 the unit circle will of course go to the unit circle only thing is it is going to go here 3 times as I go there 1 around 1s you get actually if you look at the point 1 it goes to 1 but then if you try to take the inverse function then you get the cube roots of unit right because you are taking the when I take m equal to 3 I am trying to base I am trying to look at values of 1 to the 1 by 3 which are the cube roots of unit and so what I will get is I will get these 3 values I will get one value if you call it as omega the other yeah so the problem is this omega conflicts with the w there so probably I will use so I will use a funny omega which is the omega with a dash on top of it so this is an omega with a dash on top of it which is a complex cube root of unity and of course you know the other one is omega squared okay and all the 3 points 1 omega and omega squared all the 3 will be mapped to the point 1 okay. So you know what is going to happen you know that there are going to be 3 branches for this function alright there are going to be 3 branches for this function and the 3 branches again are going to how are you going to write them down so that they are analytic they are continuous so we make use of this theorem okay we make use of this theorem so let me go back to so let me rub this part of diagram and go to some heuristics when you define powers okay so you define z1 to the power of z2 how does one define this in complex analysis or more generally analysis you define it as e to the log z2 log z1 I mean this is how you define it okay so because if you know rules z2 log z1 should become log of z1 to the power of z2 okay and e and log has to cancel this how you define for example if a and b are real numbers okay you would define a to the b like this the only problem is that when you define this log this log has several values and therefore if you take z1 to the power of z2 you get several values okay and all those values come because the various values of this log. So in the same way you know if I want to follow that example and I want to define I want to define the function w to the power of 1 by m okay then how would I do it I would define in the same way w to the power of 1 by m to be e to the 1 by m log w this is how I would define it okay and now the question is how do I define log properly you know that there are so many branches of the logarithm and so and you know all these branches of the logarithm they all differ from the principle branch by adding multiples of 2 pie i okay. So what you do is that you define one branch using the principle logarithm okay so what you do is so this tells you that you define w to the 1 by m okay and when I say this maybe I will put here I will put a lower round bracket 1 to say this is the first branch this is one branch I am defining and I am defining it as e to the 1 by m log w okay I can define it like this where log w is the principle branch of the logarithm where log w is defined as ln mod w plus i times principle argument of w the argument of w of course varying from I make a well branch cut from minus pie e to this okay and you know that if I throw out the negative real axis you know that principle logarithm is you know an analytic function and therefore I get an analytic function which will be one branch of w to the power of 1 by m okay. So on this on the cut plane on the on the plane on the cut plane c minus negative real axis which is well minus infinity to 0 okay I throw out the negative real axis w to the power of 1 by m subscript 1 in brackets is a branch of w to the 1 by m okay it is a branch of the mth root function alright and now so once you have this one branch then how do you get the other branches the other branches are given by well I need some notation so I think I will get let me look at it for a minute w to the 1 by m so you know you are going to get m branches alright in the case of w is equal to z cube you get 3 branches okay one branch and under each of those branches the image of one will be under one branch it will be one and the other it will be w which is a complex omega which is a complex cube root of unity and under another branch yet another third branch it will be omega square okay so you get 3 branches. So if it is m you will get m branches and what are how do you write those branches there you can write them as e to the 1 by m log w plus 2 pi 2 okay maybe I should not use i because i is always a complex cube root of unity I mean square root of unity so let me use j or matter I use even n alright so then I will have 2 n pi i okay these are the various branches of the logarithm alright and you will see that essentially you will get only n distinct ones you will get only m distinct ones okay you will get only n distinct ones because well because you know if I start with so if you know if I start with so I will start with w to the 1 by m if I take and here if I put n is an integer okay then these are the various branches of the logarithm and therefore I should get the various branches of the m through function but mind you there are infinitely many branches of the logarithm there are as many branches of the logarithm as there are integers alright but then if you take the m through function there are not so many there are only m branches and the reason is because there is a 1 by m that is appearing outside this n has to be read mod m because it will repeat n mod m will repeat okay so you will get only m branches so let us write it down see for example when I put w0 1 by m is my original notation 1 so this is really bad so I just change it so that my w1 is actually w1 so which means I will have to put okay so I will have to make the following adjustment I just put it as 2 into n minus 1 by i if I do this alright and I put n equal to 1 then w1 is going to be well my w1 it is going to be e to the 1 by m log w alright this is what I get when I put n equal to 1 alright then I put n equal to 2 I am going to get e to the power of 1 by m log w plus 2 pi i alright and this will go on up to w to the power of 1 by m d well 1 to m when I put m what do I get I will get e to the power of 1 by m log w plus 2 into m minus 1 pi i I will get this and mind you all these are distinct they are all distinct because there is e to the 2 pi i by m which is getting multiplied alright this function is this function multiplied by e to the 2 pi m every successive function is the previous function multiplied by e to the 2 pi i by m okay so these are all distinct functions they are all mind you they are all analytic functions they are all branches of the root function alright but then if I put m plus 1 I will get back w1 okay because if I put m plus 1 you know w1 to the well if I calculate w1 to the m m plus 1 what I will get is I will get e to the 1 by m log w plus 2 pi 2 m pi i okay and then you know if I simplify e to the what 1 by m into 2 pi 2 m pi i will give me e to the 2 pi i and that is 1 so this will this will again go back to w to the power of 1 by m the first branch okay so you will get a repetition. So the moral of the story is that if you calculate w to the power of 1 by m n what you will just get is you will get w to the power of you will get the branch that corresponds to n mod m this is what you will get as n repeats I mean as n varies over all integers so essentially you get only these you get only these m branches you get only one you only get m branches and of course the problem with all these branches is that at the wherever you have cut it on the that is on the negative real axis they are all not going to be analytic okay. And again if you want to think all of them and the reason why they are not going to be analytic is because well the even if you try to make the argument continuous by separating the lower the negative real axis and the portion below it from the portion above it if you do that you can actually make the argument functions continuous but even then you know you are not going to make the function analytic on the negative real axis the only way to do it is to construct the Rayman surface for the root m root function okay and the Rayman surface will be what you will get is the Rayman surface will be as many sheets as there are branches okay. So in the case of the when you considered the logarithmic function which is the inverse function to the exponential function then the logarithmic function all the branches of the logarithm could be realized as single function on the Rayman surface for log z which is which is which has as many sheets infinitely many sheets as many sheets as there are integers because these are the as many branches of the log that you have but here you have only m branches of the of the mth root function. So what you will get is you will get an m sheet at Rayman surface I will be content you draw it like this and well and you will get on this Rayman surface the function w to the 1 by m will make sense as a single valued function okay which is analytic and you can see all the m branches of the mth root function living here as a single analytic function okay and of course I should tell you that this is of course not these are not just m disks and again I should tell you that this is a this m sheet at Rayman surface is an m sheet at cover of the punctured plane okay it is what is called a covering space okay in the sense that if you take any point here if you take any point here and you take a sufficiently small neighbourhood then its inverse image will be m points with m neighbourhoods which are homeomorphic to this under the projection map and if you call this as a projection map and for example in this case you can see them as you know you can see them as three you can see them as three points here for a point there corresponding to the images of the three branches okay and this is what show up as three we should show up as three separate neighbourhoods above in the covering space okay which is the Rayman surface for function w to the 1 pi m okay. So now I will end with the statement that this behaviour of the function f of z at the critical point z equal to 0 is will be the basis for the behaviour of all of any analytic function at a critical point the behaviour will be like this up to conformal change okay that is what I will explain in the next lecture.