 And one of the corollary was that if an analytic function has a non isolated 0 then it vanishes everywhere or in other words if there is a extension of an analytic function in a domain beyond it is definition then that extension is unique. So, let us take some examples of this because this is very interesting property of analytic function and the log z function is something we will be very interested in. So, and that also had a peculiarity in for complex plane definition and let us try to see how that works out. So, earlier what I said was that look at the function log z is a 1 to many function and to get a sensible function out of it 1 to many is not something very nice. So, we split this into finitely many we are log of let us give it a specific notation. So, that is a kth starting from 0 the logarithm which maps r e to the i theta log r plus which is that is a unique value and i theta plus 2 pi i k where k is an integer and theta lies between 0 and 2 pi actually this k can be negative also it can take a negative you can be any integer. So, pick any one of this so specifically let us say for k equals 0 pick the function that function is defined over this strip log 0 z is defined over this strip and this line is not included the bottom line is included and this strip is mapped to the by this function to the entire complex plane except 0 is that correct why do why you know why except 0 it will map r equals 1 to 0. So, log 0 is not defined so this maps to the whole complex plane what I need to do here is that when I take the strip I need to take out 0 also from here 0 also cannot be part of this because log 0 is not defined now this strip is not a domain because the domain has to be an open set and. So, we cannot even talk about this function being analytic over because we need to define analyticity at a point we need to look at its neighborhood and that at this edges does not quite exist. So, a clean way of handling it all of this and the fact that 0 is also removed is to just look at this that we say log set this log k is defined inside this precisely inside this strip not taking the two edges in at all then it is a domain and it is well defined inside the domain and it is not too difficult to see and I think I gave it as a assignment problem to see that log z is analytic over this domain this is all wrong this is the range of log that you are very right. So, this maps except the point 0 here to this strip, but that now let us see what else do I need to do fine. So, let us read the argument. So, this is the domain this is the range in this I am the question is that around this line when we approach this line from top or below the continuity is does not exist so again so continuity is not there analyticity is in there to make sure that log z is defined in this fashion is analytic whatever copy we are considering what we do is we take out this line entirely from the domain. So, take the positive real line out from here all the way because this line is what is causing the problem and then it becomes analytic and including the point 0 because if you cannot include the point 0 otherwise it would not be an open set. So, take out point 0 and all positive real numbers then this sort of complex plane with a cut is mapped to this strip now the strip is the becomes an open set there is no upper and lower boundary and then log maps in a nice fashion to this and this I can define for any k the only thing difference is the domain remains the same the range changes to a different strip. Now, another point to notice is that this choice of line to cut out which is determined by the definition I give to this each copy of log that choice is arbitrary we the fact that we had to choose this line is because of this definition that we say log this kth copy of log is equal to log r plus i theta plus 2 a pi i k where theta lies between 0 and 2 pi if I change this range to something else say this then the only what changes firstly this in the range this strip changes it comes to it becomes this strip and in the domain now we take out point 0 and this the negative real axis again this is also arbitrary if I change this to may be minus alpha less than theta less than 2 pi plus minus alpha plus pi pi whatever right then this will again shift and here also the cut line will shift to an appropriate angle alpha angle line starting from origin. So, this choice is completely up to us so, since up to us we will choose the most convenient and simplest possible choice here and that turns out to be this one although the first instinct says to choose this one which is what I did last time, but this turns out to be a even better choice. The reason is that if we cut this out positive real axis then essentially we are saying that log z is not I am not giving the I am not defining log z on positive real axis which is somewhat counter intuitive because certainly log is defined on the entire positive real axis log is not defined on the negative real axis in the usual sense, but certainly positive real axis is defined. So, we should try to include that definition at least when we generalize this definition to complex there is another reason for it as we will see very soon. So, we will choose this instead of this definition we will have this definition of the theta range and therefore, what we the cut that we make is from here to the negative real axis and the strip that we get here are these with me so far. Now, let us try to do an analytic continuation of log z is defined over this domain the missing part is this line. So, let us try to extend the definition of log z over this line to do that let us pick up a point very close to this and it will be useful to pick a point which is a very close to minus 1 on the real axis, but not quite on it with at the same time lying on this circle of radius 1 somewhere here. So, let us say let z naught and so that is a point we pick z naught is absolute values 1 and this argument which is the angle it makes is minus theta minus pi plus epsilon and now let us take a tiny circle around z naught and the circle will have radius. So, this distance is about epsilon the distance of z naught from the real axis approximately epsilon because we assume epsilon is very small and let us take a circle of radius to epsilon with center at z naught and expand log z around z naught as a power series. So, what is the power series of log around z the first term is log 0 of z naught itself plus what about this higher degree terms the coefficients will be the derivatives of we know that this is analytic around z naught the coefficients will be derivative of this function at z naught. So, what is the first derivative of log 0 z 1 by that why you know log x is 1 by x over reals, but why do we know that log z is 1 by z over complex numbers what that is right. So, that is why we what we can use here just the last times that gives you very simply that log z must be 1 by z sorry derivative of log z must be 1 by z because if you derivative of log z is an analytic function because log z is analytic and if you look at that analytic function log z prime minus 1 by z this function is 0 over the entire positive real axis. So, therefore, it must be 0 over this entire domain and that is one more reason why we want to include the positive real axis inside our domain because then we can just lift this definition of log derivative of log x to derivative of log z good. So, derivative of log this is 1 by z which is evaluated at z naught is z naught minus z naught plus then the second derivative is the same it just continue the same same same trick and. So, we get minus 2 by z square and divided by there is a this is the first derivative minus 1 by z square and that is divided by 2 factorial. So, that is becomes minus 1 over 2 z naught square plus and so on that is a positive expansion fine and now let us use the fact that first let us plug in for this what is this log 0 of z naught use this definition that is equal to log r which is 0 that is by choice of z naught. So, what I get is i and theta is minus pi by a plus epsilon plus 1 over z naught which is e to the i minus pi plus epsilon. Now, let us say for you are only going to look at those z's which are at distance delta from z naught. So, what do we get? So, such a z will have this form delta being the absolute value and then phi will be vary over the as the circle moves at between 0 and 2 pi. Now, as delta tends to 0 we can ignore this higher degree terms because you will see that in this here delta multiplier here delta square multiplier all other thing these are they have absolute value 1. So, this absolute value of this term is delta square or delta square by 2 actually then delta cube by 3 and so on. So, they have become smaller and smaller. So, in the limit we can throw them all away and just stay with this of course, if delta really close become close to 0 we can throw this away also, but that I do not want to do. So, this is approximately now let us follow the trajectory of this point as we move around that circle. So, when let us say delta is 0 then we are at somewhere little above minus pi we are somewhere here when delta is 0 minus pi plus epsilon. Of course, delta is I am not going to say delta is 0 delta is going to be little bigger than epsilon actually twice epsilon also in that case where are we. So, this is pi plus pi minus epsilon epsilon here we can almost ignore in this phase case situation and then. So, pi plus pi. So, if I have if I am here let us say this is the and if I choose pi to be 0. So, at this point what is the corresponding point here pi 0. So, you get e to the i pi which is minus 1. So, you subtract delta from there. So, minus pi plus epsilon minus delta where does that that is somewhere down here with me so far. So, we are actually this is this power series is taking me out of the definitional region of oh that is imaginary sorry that is a real. So, that is right. So, now this does not take you down it takes you here still inside. So, now we start moving in the circle as we move in this circle what happens to that point that also moves in a circle of radius delta. The center of that circle is minus pi plus epsilon and as we traverse around this circle this traverses around this circle of radius delta and completes the circle once we complete here that also completes. Now, given that delta is little bigger than epsilon which we can always choose we do exceed this strip and go below it and then come back again. So, that is nice, but the question is this power series that we just defined is this convergent in that disc that because unless it is convergent in this disc and none of this will make sense, but is it convergent. Obviously, it is convergent because epsilon is very small. So, it is value as you go here this is the power series expansion. If you look at even the absolute value as you go down here it starts becoming negligible. So, it clearly convergent series you know what point did you not this one that was fine why is it. So, this is the power series we are looking at when z is in that circle of radius delta and the expansion is like this every successive term the absolute value is delta to the k over k for the k plus first term delta is very small very close to 0. So, the absolute value of this power series converges. Moreover, this power series is also uniformly convergent because if you look at the cut out first few terms look at the remaining term absolute value of that because there is a big delta to the power k sitting in multiply as a multiply to all of them and then everything inside converges to some finite value. This whole thing is less than a finite value times delta to the power some power and since delta is less than 1 as that power increases that goes down to 0. So, the power series is actually absolutely convergent not absolutely is even more is uniformly convergent for any delta less than 1. We choosing delta which is very close to 0 little more than epsilon epsilon is also something which is very close to 0 delta is also let us say 2 times epsilon yes this is precisely at this goes around a circle of radius 2 epsilon at z naught. So, the and since epsilon is very close to 0 delta is also very close to 0 certainly less than 1. So, everything converges very nicely this is somewhat funny thing that is happening now here because it is the range is very nicely defined I mean you have this you start at this point look at a power series at it reasonable. In fact, you can expand the disc size to quite a bit actually you look at a power series defined on this disc given by the usual definition of power series that is uniformly convergent power series defined very well and log maps the function the log value according to the power series is defined on a goes on a nice circle here as well or a disc here. The problem is that it does not quite match with the definition that we had earlier fixed of this mapping of log 0 z to this strip because this strip is being violated and we actually jumping into the strip below. So, what kind of values is it taking? So, this is a strip below this is minus 3 pi. So, it is certainly taking a values in the strip below and so that is one issue the second thing is this circle is not only going hitting the line which we have cut out, but also part of the domain below. Now, we have already have a definition of log 0 z in the domain below we did not have one on the strip with line which you cut out what we do have a this definition. Now, what is this definition now what is this value at this domain below actually I goofed up I should have taken z naught a little below this because that is when minus pi plus epsilon is correct otherwise it should have been plus pi plus epsilon see that this angle is plus pi and this angle goes to minus pi. So, that is but that does not change the argument. So, it is this circle is you know sort of cutting across this line which you have removed and going into the both sides of the domain on one side it matches with the definition of log z, but on the other side the values provided by this do not quite match with the definition of log z because if you are going on the positive side according to log 0 z what we should get is the imaginary part should be plus pi and then a little bit more and then there is some little bit here also not much because delta is small in a instead what we are getting is a minus pi and then something. So, this power series does not even agree with this function on the domain it only agrees with the function on part of the domain but that is even stranger because we just proved last time that if two analytic functions agree on a contiguous domain then they must agree everywhere. So, what funny thing is happening here first thing is this is not quite a domain for the power series although it looks like it is a circle which is a domain but remember that this line is cut out the in the original domain the definition of the original domain as this line is cut out. So, what this circle really corresponds to is two half circles where the middle line has been taken out. So, it is not a domain because it is not connected and since it is not connected it is possible that that property is violated that property does not violated only when the domain is connected. So, it actually gives you a very nice example where when a domain is not connected that theorem fails to hold that is one conclusion from this but we would like to get even something more out of it the exactly what is happening or after all this is artificial that we are taking out this line for the sake of definition we did it but this is kind of artificial and then we have to have we have this power series is giving the all the nice values on this line which we have cut out as well and if you now think and look at it carefully that what is happening on this side it agrees with log 0 z. Let us ask the question what does it agree with on this side does it agree with some other version of log on that side and the answer is very clearly yes on this side. So, let me just write it here on this is just a formal statement of this fact that this disk below the positive real the real axis agrees with the and on that region power series agrees with log 0 z and now on the other section log log what log minus 1 minus 1 means that this trip and that is what it actually comes down to this and if you see that this is going to get mapped to this thing now this is a very interesting observation because what this is saying is as we traverse around this circle so we start with this domain. Let us imagine that you are you have this infinitely log functions each of each define over the entire complex plane with negative real axis thrown out and stack them on top of each other. There is one which corresponds to log 0 there are infinitely many planes below it infinitely many planes above it. So, this is the one we are looking at and we start at this point and start moving on this circle and as we cross this thinking of this keeping this in mind this mapping as we cross this when we are looking at the mapping given by log z then the point we end up with is not on the same plane instead is on the plane below right again keeping in mind that when we say that traverse on the plane we are doing it in the context of this map given by log 0 z and then we expand it like a power series and then we traverse around this and as we move around keep looking at the log value of this as we cross this line number one the definition of log value log changes it becomes log of minus 1 z and we have actually moved down the plane to the points in the plane below. Similarly, if you start with a point here and start circle and moving along the circle or any curve leading cutting across this then what will happen is as we cross this line this definition will change to log 1 z and we will move on the plane above. So, a nice way of visualizing this and that really sets up all these log functions in one context is to view this map log as not taking a complex plane to a complex plane. But instead taking a very strange surface which I am going to define shortly to complex plane and this strange surface is formed by defined by taking infinite copies of complex plane cutting each copy. So, you take infinite copies of complex plane stack them on top of each other cut out the negative real excess from all of them. So, make a cut so that it sort of visually this these two sides can move apart then take this side twist it and take this side twist it up and join the two edges. So, this side let me try to draw it my skills in drawing are not very good. So, let us see if I can get it right this is of course has not come out well at all but I hope you get the drift that you take cut out this side and then try to fold it down cut out this side fold it up and join the two ends the two edges all the way from 0 to infinity which means that this actually this point and this point should really touch each other. So, that is why it is a very strange kind of surface that you have this infinitely many planes they all collapse get collapse at 0.0 they are really this all sort of held together at 0.0 then they go out like this where they are adjoining strips this is joined to this and this one is folded up and the other one is folded below and that is joined similarly that is folded down this folded up and join. Now, if you traverse start any point and traverse on this strange surface you will follow the rules given by this on this surface log z is one function there are no infinitely many function just one function which is defined everywhere except the 0.0 which is nice and it is completely analytic over this entire surface and it maps this entire surface to the entire complex plane. So, the range is very nice just the complex plane the whole complex plane the domain is this strange surface this surface is called a Riemann surface. So, Riemann was the one who realize that you can visualize such the one too many functions as being one to one on a different surface which is and the name sort of came from Riemann and there are all strange kind of Riemann surfaces depending on which function you are trying to analyze. Oh yes there is a complete geometry on Riemann this Riemannian geometry is defined over Riemann well one of the things it does is to define geometry on Riemann surface uniformly convergent oh you see you know how you converge sorry yes yes but there are still infinitely many terms yes but they convert because it is sort of a geometric series see once you take let us look at some term at delta to the m and this whatever power this is this is all absolute value one it does not matter what this is sorry and there is something divide by when there is a m below there and there are signs plus minus also not factorial the log expansion of log log is x to the m by m right so plus so just let us just change make to make the things worse change all the signs to plus one change the all phases to one so the series looks like delta to the m plus one by m plus one and let us throw all of these out also they are also reducing the series. So, that is the pain geometric series which sums up to delta to the m over 1 minus delta no it is not at all because you see that the strangeness of this that these points they all are collapsed into a single point they are not distinct points the zeros of this because this the when you cut of course you can as you go down you can twist it more but at close to this you cannot really know no place to twist so they all need to come together to get joined there and then they sort of branch out and so if you traverse you make it you start on the surface and travels in its make a circle travel in a circle you actually go up in a spiral go down on a spiral well traverse circle around zero circle elsewhere may not be that bad something for your homework not an assignment just to keep you busy consider the function square root z that is also one to many function is one to many over reals also so it is certainly one to many over complex numbers now construct the Riemannian surface for square root z so that over that surface is one to one analytic everywhere