 So, today let us and we are looking at that log z as a function as an example right and then we saw that it is not as nicely behaved as the log over reals. So, we will come back to log z again because there is a that infinitely many possible log functions can actually be stitched together into a single log and exactly how to do it we will see in a short while. To do that we will have to look at the concept of analytic continuation which is what I will slowly do. Now, there is one more theorem that I think is important which I should prove it is very easy to prove actually. So, we saw that for analytic functions we have infinite differentiability right. So, if a analytic function f is analytic then f prime all is differentials f prime f double prime and so on remain analytic. We also saw that it is for an analytic function is integral is also analytic that was part of Morera's theorem that if you define f z as then this function f is both continuous and differentiable and therefore analytic whenever f is analytic. But there is one more way you can define integral of a function from an integral from an analytic function which is suppose you have a bivariate function which for one on one variable is an analytic for every value of the other variable and then you integrate over the second variable. So, for example, suppose we have so f is a bivariate function the first is a complex variable second is a real variable and it is continuous on some domain and more importantly for every value of t in the range a to b f z comma t is also analytic on the domain d. The reason I am choosing real variable here instead of continuous is that most of the time will only be looking at functions of this kind and when integrating over the real variable. So, this is enough and the proof here is really simple then if you let capital F z to be integral of z t or dt then capital F is also analytic. So, it is a integral of this small f over the real second parameter which is analytic. So, this is the theorem the proof as I said is fairly straight forward. So, what I think to observe is that capital F is continuous and that follows by the continuity of this function. So, if you look at f z and f z plus delta z that is as delta z goes to 0 that the integrand here converges to f z t and therefore, the integral will converge to f z. So, continuity is straight forward and now what we need to do to prove that capital F is analytic well if you go by definition if we show that this is differentiable then we are done. But there are times when we can use alternative characterizations of analytic function for example, the one we showed last time which is that a function will be analytic if it is continuous and on any rectangle its integral is 0. For the in this case that characterization is really handy. So, since so let R be any and we say x is parallel because that makes it even simpler then since f small f is analytic for any value of t what we get is this is 0 fix any t in this range this is analytic and therefore, this integral is 0 that is why Cauchy and this happens for every t therefore, this is also 0 no what I am saying this should be t z. So, this is also 0 because integrand itself is 0 here now what is the left hand side of this equation equal to if we can swap the two integrals we are done. But can we swap the two integrals that is bit tricky over reals we know if there are two integrals over reals in finite range we can swap them that is a classical state theorem. But here one integral is over reals the other is over complex and that too over a like rectangle. But here we can simply use the fact that this is an x 6 parallel rectangle this integral I can split into 4 integrals and these 4 will be real integrals on x when y is fixed on y when x is fixed x y. So, I can write this complex integral over a contour as 4 finite real integrals and then I can do the swap and then add the 4 complex a 4 real integrals again into a one single complex integral and therefore, we get that this delta r a to b this is this and this is equal to the integral of capital F which is 0 now invoke Morera and capital F is analytic. So, essentially we what we should intuitively take out of this is that analytic functions are very nice functions they have very well behaved have all differentials all integrals of certain kinds and. So, we can play around with them with reasonable amount of freedom at the same time there is I will want to strike a note of caution that do not take them for granted the analytic functions can throw up rather unexpected behavior as we shall see also. So, what are the examples analytic functions you have seen well you have seen that all polynomials are analytic all sine cosine exponential logarithm also at least in a strip is analytic it may not be analytic everywhere and there is a job here. So, that is a discontinuity so it is not analytic there. So, I said last time that I will show how to make it analytic and that is that I will do later on once I discuss power series then I can do it because I will need power series to discuss analytic continuation and then I will bring that back in. So, if I forget remind me now one some of these functions polynomials are pretty straight forward functions, but if you look at exponential function for example, it is a slightly more complex function and we know at least over real that exponential function has this infinite power series representation using the Taylor expansion similarly log sine cosine they all have this infinite Taylor expansion as a power series. So, at least some of this power series therefore, are analytic the question I want to address is which power series are analytic because power series are a generalization of certainly polynomials and they also consume or contain all the functions we have seen so far all the analytic functions we have seen so far because all of this analytic functions can be written as power series. So, let us our discussion on power series and see which of the power series are analytic. So, definition is very straight forward a power series sigma k greater than 0 a sub k and here I am adopting notation from the real analysis. So, this would be a Taylor series expansion around the point z naught. So, I am just lifting that notation for this and as we will see this will be the right notation for us as well. So, z naught is some particular complex number these a k's are also complex numbers and such that infinitely many a k's are non-zero because if only finitely many a k's are non-zero then we have a polynomial then we do not have. So, that is a general definition of a power series and it is if we just using binomial expansion expand this out. So, we can always write any such power series as sigma a prime k z to the k. So, these are equivalent definitions this makes it little easier to understand that we are looking at power series around the certain point which is z naught. Now, the moment you have an infinite sum there are all kinds of issues which arise we have to address the issue that when is this convergent for any given point z whether the power series is convergent at z or not is becomes an issue which is not certainly not an issue with polynomials they are always convergent. So, and that will be a very important notion for us when we look at power series in terms of analytic functions as well. So, let us put in some basic facts about convergence of power series before we dive into its analysis. So, let us just talk about convergence therefore, there are several things we can talk about convergence, but I will just talk about the notions which are of real interest to us here. So, I will only define two types of convergence one is a over s sequence of numbers. So, sigma k a k is just a sum of complex number we say that this sum is absolutely convergent if the sum of the absolute values of this complex numbers adds up to some sum does not diverge. Now, for sums of numbers absolute convergence is kind of a sanity check because if a sum is not absolutely convergent then funny things happen with that sum. For example, if you look at k to be plus 1 if k is even and minus 1 is k with odd and this sum is not absolutely convergent or if you add it up you can start with a 0 a 1 which cancel each other a 2 a 3 cancel each other. So, you can conclude that the sum is actually 0, but if you take a 0 out and cancel a 1 with a 2 then a 2 with a 3 and a a 3 with a 4 a 5 with a 6 then the sum would be plus 1. So, you can depending on how you bracket the sum you will get different values. So, that is really not a very nice summations. So, we do not want this well of course, you never want this to happen, but it does happen sometime, but we will not want to get into there. So, we will only try to look at absolute convergence of a series yes. Does strange way we are occur for all the infinite sums which are not absolutely not necessarily there are I guess if you look at this sum then absolutely in absolute term this diverges, but if you add it up I think it actually converges to a sensible value irrespective of how you bracket, but I do not take my word for it convince yourself that is a case. So, there are cases I believe when you do not really necessarily need absolute convergence, but that is like when things are not absolutely convergent then things can be very messy. So, it is pretty safe to look at absolute convergence then the nice things happen, but that is only sum of numbers what about sum of well these are sum of powers of z actually. So, these are like not just numbers, but these are a whole actually this represents infinitely many such sums one for each value of z and we want to say something together about this whole set of sums. We can for example, talk about absolute convergence of such a series as well finitely. So, we say that this sum of this series is absolutely convergent and here we have necessarily have to talk about a particular region in which the value of z lies and that is typically the natural way of defining the region and we will come back to this later on to justify this choice is to say that z minus z 0 the absolute value is smaller than some number and this is smaller. So, if it is absolutely convergent in this region if whenever you take any such z that the corresponding sum of numbers absolutely converges that is a natural generalization of absolute convergence there to the power series. Unfortunately, absolutely convergent power series is not strong enough for us because absolutely convergent power series can at times I have behavior which is not very nice. Yes, for every any z in this region show you an example of an absolutely convergent power series which is not very nice. Let me give you the other definition then we will try to see the difference between the two. So, we call this power series to be uniformly convergent in a region if the following property holds that for every m greater than equal to 0 if you look at the difference between the power series and the truncation of power series up to the first m plus 1 terms. So, this difference is bounded by in absolute value this difference is bounded by epsilon m whenever z lies in this region. So, epsilon m is independent of the actual value that z takes here and further as m tends to infinity this epsilon goes to 0. So, in other words what this is saying is that this truncation of power series becomes a better and better approximation of the full power series in this range. Now, I claim that this is a stronger notion than that do you believe it any show that uniformly convergent series is also absolutely convergent that value of epsilon. Now, how do you work there let me give this as a home exercise instead of spending too much time on this. So, this is a general statement what I am looking for here is any implication if you can show that this implies that that is one part of it and that this is stronger. So, that means one example which is absolutely convergent, but not uniformly convergent or if this does not imply this then one example which is absolutely convergent, but not uniformly convergent. So, just make sure that all bases are covered good. So, this is the key notion of convergence for us. Now, here is a very useful theorem about power series before that let us define. So, let R be the largest positive real number such that this sequence absolute value of 8 k times R to the k. So, just think of this as sequence of numbers and see where this sequence of numbers going and if you actually if you choose a very small r there is there is more likelihood that this sequence will actually converge to 0. The smaller r you choose the more likely you will have that this sequence converges to 0. So, let R be the largest positive real such that this happens sometimes it is possible that there is no such largest thing we can always get a slightly larger one. So, I should not say this is largest positive real. So, let us say this R be the limit of you may have to take a limit then R is this number is well defined it can be 0. If the a k's are grow so rapidly that it is quite for example, if a k is k to the k for example, then no r can send this to 0 this sequence will actually keep no diverging. So, then when then capital R is well defined it is 0 small r being 0 will certainly make this happen. If these grow very slowly then you can get away with the may be a very large r also. For example, again if a k is on the other side is 1 over k to the k then you can get away with every r. So, then the radius of convergence is infinite yes you want sequence each individual number should keep on shrinking and go towards 0 will see this theorem is the motivation that this number defines the region of convergence for the power series. So, let f set p a power series then for not for f is uniformly convergent in this region where r is. So, yes it looks a little odd here by I am just looking at sequence of numbers and seeing where they converge that sequence converges, but the connection will be fairly straight forward the proof is reasonably easy here. So, let us show that this is the radius of convergence. So, I should have added one more step we will do that may be in the next theorem. So, I need to show that for any small r less than capital R if you look at this region the series converges uniformly. So, pick a number s between small r and capital R fine. Now, since s is less than capital R by definition we know that this sequence this sequence converges to 0 right that is by definition that is the definition s is less than capital R. So, this sequence by definition r is the limit of all positive reals r. So, that this sequence converges to 0 that is the definition this is I am saying r is its definition r is radius of convergence of power series is defined to be the largest number or the limit of the lim soup of all the numbers for which this quantity converges to 0. So, this is the definition or if it is larger no it would not see if it converges to any positive number instead of 0 then when you add up you get infinite right. So, you have to take a big sum. So, you need it to go to 0 not the sum not the sum it is just a sequence individual numbers just looking at the limit of the individual numbers. Now, so this sequence converges to 0 we know that fine. Then therefore, if you look at this quantity of course, is same as less than equal to this quantity. Now, this is less than equal to now by definitions in this converges to 0 this quantity is clearly as an upper bound for all k I can write in absolute upper bound on this quantity. So, this is less than equal to some c times k greater than equal to m and what is this this is just geometric series this is a fixed number non 0. So, this is some c prime times. So, this is your epsilon m which is independent of the actual value that z takes inside that region. So, that is property 1 and property 2 was that as m goes to infinity epsilon m should go to 0 which also happens here because r by s is a number less than 1. So, as m goes to infinity this goes to 0. So, that is proof of the theorem. So, this converges to 0. So, this clearly means that there is an upper bound this quantity for this series is upper bounded by a some number it does not diverge it is converging to 0 or converging to some number which means that there is an upper bound to this series. So, let c be that upper bound in absolute. So, I can write each one of this as less than equal to c. Therefore, this holds and then we are done all right. This should be greater than m you are right. So, this is m plus 1. So, this converges in reasonably straight forward fashion uniformly in that