 Mathematically, I mean that as you take, see f is defined on delta d which is an open set and there is a boundary. So, as you take the limit of the value to approach the boundary from any direction, the value of the, as you follow any path. So, essentially it is a continuity that you have, you take any path approaching the boundary and look at the value of f along this path and look at the value of f on the boundary. So, this limit converges to that value, it does not matter which path you follow. So, that essentially the value of f on the boundary does not behave strangely, f is continuous in d well that is hard to say because that would mean that if you are approaching from outside also then it should be continuous, that you cannot say, f is continuous on d bar as long as you approach it from inside then it is fine. So, that is why one cannot say there is continuous on d bar, but one has to say it this way. So, that is the theorem and proof is a pretty straight forward almost directly by Cauchy's integral formula and we do induction on. So, for m equal to 0, this is just the Cauchy integral formula. So, that is the starting point. So, let us assume this is holds for some value of up to m minus 1 and now we want to prove it for m. So, now what do we know? We know that m th derivative of f is defined as and of course, this limit should exist and all things should be around only then this is. So, let us just because now we know that m minus 1 is derivative not only exists is given by that formula. So, let us use this. So, what is f m minus 1 z plus delta z that is equal to m minus 1 factorial over 2 pi i integral over delta d f w by w minus z minus delta z and similarly for f of m minus 1 at z. So, finally, subtract the 2 and now let us use binomial theorem to expand these 2 or well not we want to expand these 2 or we will expand this out as splitting it as w minus z and delta z and expanding this out we get j equal to 0 to m m choose j w minus z to the j. Let me write it as m minus j and minus 1 to the j and delta z and if you look at this the first term for j equals 0 is w minus z to the m that cancel with it. What is the second term? m choose 1 which is m times w minus z to the m minus 1 times minus delta z and second and higher terms will have delta z square or higher powers of delta z. So, I will write it as m minus 1 factorial over 2 pi i plus. So, this is the first term that survives and the rest I will just take as delta z square times something some expression in w and z and that is a polynomial expression. So, if we just split these 2 integrals out what we get is take the sum as 2 different integrals or sum of 2 integrals we get the first term would be m factorial over 2 pi i delta d f w over 2 minus z minus delta z to the m times w minus z delta z d w that is what happens to the first term. And the second term is in fact delta z is can come out the integral is over w and for the second term let us just notice that it is whatever it is it is some delta z square times something and now knowing that this is the quantity we are looking at the difference divide by delta z let us divide the right hand side by delta z and take the limit. And as you take the limit delta z going to 0 this vanishes and what happens to this again delta z is independent on integral. So, at delta z goes to 0 you get the m z is this goes to w minus z and so it is pretty straight forward. And it all follows from this Cauchy's integral formula directly essentially which in turn follows immediately almost immediately from the Cauchy's theorem about the contour integral. So, analytic functions have this really nice property of being infinitely differentiable. So, there are no discontinuities or any kind anywhere as long as the function is analytic on a domain. Of course, it is quite possible that the function is not analytic at some point and where there will be a discontinuity even at the basic level. So, we will look at that aspect as well that the function being analytic at most of the places, but there will be points where it will not be. In fact, that is the kind of function we will mostly be interested in because zeta function is one of that kind. But before we try to understand that we will let us you know clean up the basics of analytic functions. So, fine. So, analytic functions are infinitely differentiable they have this property that in any domain where the function is analytic its value on side a point of the domain is determined by the integral along the boundary of the domain and they satisfy this Cauchy's integral Cauchy's theorem that integration along the boundary of the domain of an analytic function is 0. Now, what one can may ask what about the converse an analytic function has this property of Cauchy's theorem what if a function is 0 when integrated along the boundary of a domain then is it analytic that would be the converse of this theorem that converse of Cauchy's theorem and actually a very strong form of converse actually holds which is a Morera's theorem. This is a very useful theorem when we try to prove some further functions analytic. So, we start with a continuous function on a domain because that is a minimal we will need anyway and if this function the integral of this function along the boundary of an excess parallel rectangle. So, r is an excess parallel rectangle. So, it is the sides are either parallel to x axis or y axis. So, if for every such excess parallel rectangle inside the domain t this integral is 0 then f is analytic on t. It is a strong converse because if f is analytic on d then for any closed curve inside the domain d this integral is 0 whereas, here we only require it being 0 on along the excess parallel rectangles is that clear why this is a strong converse we do not even need it the assumption to be that f is 0 along every closed loop. So, the proof is quite simple but this is a neat trick. So, what we do is suppose this is domain this is a domain d pick up a point any point in the domain call it z and let us call it static point z 0 and pick up any another point here let us call it z and pick up yet another point which is very close to z call it z plus delta z. Delta z will as the expression says will eventually go to send it to 0. Now, let us define a way of integrating between two points say between z 0 and z. So, we will how do we integrate between this I am going to since f is given I will integrate f from z 0 to z but whenever we try to do it we have to define a path of traversal from z 0 to z and the path will choose is very simple this is probably the simplest possible move around in move parallel to x axis from z 0 until the point we reach just below exactly below z and then move vertically. So, that is the path if I have to integrate from z 0 to z plus delta z I will do this go all the way up here and then go up. So, whenever I am going to integrate from z 0 to z 1 where 2 point z 0 and z 1 are given it will be this kind of a path to be followed and let us define f of z to be this integral from z 0 to z of f w. So, this integral is well defined since f is analytic f is continuous it is a finite integral. So, it is well defined now similarly f z plus delta z is the integral z 0 to z plus delta z f w and now the path followed is this one then what is f z plus delta z minus f z is this integral or the difference of these two integrals where this is following this path and this is following this path and now we use the assumption let us complete this the second one is negative which means that in the first one we are going from here do not have a colored show we are going from here to here in the second one since it is negative. So, the sign is the direction of traversal is reversed. So, there we are coming down from here to here and then here. So, I can split this integral that is easy to split this integrals first one as going from here to here then here to here and then going up there the second one is coming down from here to here and then going from here. So, these two parts cancel each other out. So, what is left is coming down from here to here then going from here to here and then going up now we use the assumption that along the any closed any along any excess parallel rectangle the integral of f is 0. So, look at this rectangle the integral of f on the boundary of this rectangle is 0 in the integral there we already have covering three sides of this rectangle fine. So, let us add the fourth side also add and subtract fourth side the fourth side would be addition would be going in this direction and if you add that then this whole thing is 0 that part and you subtract it which means integrate from here to here that survives and then this from this point to this point also survives. So, that is all the and that is all there is survives. So, which means that this expression on the right hand side is equal to z to z plus delta z f delta 2 and remember this important thing here is that whenever I write integral from point 1 to point 2 it is in that mode of traversal horizontal then vertical which is indeed being followed here from z to z plus delta z I first go horizontal and then go vertical. So, all these things cancel out and ensure that we get this integral divide by delta z what is this equal to. So, if I just take this f z and integrate from z to z plus delta z of d w that will only give me delta z I will take that later. So, I am just writing it in this fashion and then I just integrate this part if you just look at this part f z d w integrated from z to z plus delta z that will give you delta z f z delta z and that will divide by this and then you get z plus then you is continually write then take this on the left hand side and take the absolute value. Now, this by continuity this value is bounded by a small amount right. So, let us say for small enough delta that this is bounded by epsilon and where epsilon will go to 0 as delta z approaches 0. So, this is epsilon and then this integral now this is in absolute sense z to delta z will have at most contribute most two times absolute value delta z what is that this right this is this is less than epsilon I am just saying about the rest of the this will come out and then the just the integral z to z plus delta z of d w because that is now inside the modulus. So, I cannot say that it is exactly equal to delta there, but is bounded by two times delta and therefore, this is less than equal to 2 epsilon and now take the limit. So, the derivative of capital F is small f that was the whole idea of showing this. So, capital F is a function which is differentiable on d it is straight forward to see that it is also continuous on d by its definition actually since small f is continuous on d it just translates this property to the integral which is capital F. So, capital F is both continuous and differentiable on the domain d therefore, it is analytic on domain d since capital F is analytic is infinitely differentiable. So, its first derivative which is small f is also in both continuous and differentiable and therefore, small f is analytic. So, now we get this characterization of analytic function in terms of there in contour integrals on a domain z was arbitrary point in on domain d. So, naturally because it is a domain and z naught I can choose and which is a it is an open set d. So, any point z I can always by nature open set there is a disk which is entirely contained in d and within disk I can pick my z naught and z plus delta z because anyway delta z has to be very tiny and then let me prove one more very interesting theorem for analytic function today which will which is actually again quite remarkable it allows us to extend definition of functions on real to complex numbers. Instead let us look at some examples of analytic I think we already saw all the polynomials are analytic by the very nature e to the z is analytic continuous and differentiable right. We can define of course, you have to do it by first principle you cannot just extend the definition of differentiability over reals and plug just assume that it is differentiable here. So, you have to just look at e to z plus delta z minus e to the z and take the limit of delta z going to 0. What is the definition of sin z y z is a complex number right. So, that is that is a way to define sin actually e to the i z minus e power minus i z to y. Similarly, cos z e to the i z plus e to the minus i z divided by there the fact that they are analytic follows essentially from the fact that this is e to the i z is same as e to the z it makes no difference it is only a. In fact, that is also interesting point to note that if you look at the plot of function e to the z it is actually a 4 dimensional plot because z has two parameters x and y and this will also have two parameters being this being a complex number. So, it is a 4 dimensional plot, but without worrying about the values of this if you just look at the z plane the x y plane which defines the complex number z and look at the corresponding e to the i z this plot of this is just 90 degree rotated from this right because if you z is like i x plus i y this is i z is i x minus y. So, x goes to minus y y goes to x. So, that is the just the coordinate shift that happens between these two. So, which clearly shows that all continuity differentiability these are all maintained is just a simple rotation. Now, sin z cos z therefore, all naturally defined and differentiable and therefore, analytic this is going to be a very interesting function for us log z. How do we define log z? Well, in the real world it is the inverse of exponential function. So, can we define it as log z is w where equal to e to the w that will be a natural definition for log z, but there is a problem with this definition. It is a one to many map log z if you define this why because e to the z is not a one to one map it is many to one map. If you look at the x y plane and see what for any z on this plane what is e to the z let me just mark out 2 pi 4 pi. So, in this strip between 0 and 2 pi e to the z is 1 to 1. However, in this strip which is between 2 pi and 4 pi e to the z values are identical to this strip. Similarly, in this strip the values are identical because the e to the z is e to the x plus i y and as y takes value of y from goes from 0 to 2 pi is a cycle back. So, e to the z function itself behaves please differently I mean the exponential function on real in real world you just assume it just keeps on growing that does not happen here. In fact, exponential function in real world can never take negative value it can take negative values if value of y is between pi and 2 pi its value would be negative. In fact, if you think about it for a minute e to the z maps this strip to the entire complex plane except 0 I think it will never hit 0, but every other value is mapped to. And so, if this is mapped to then again this strip again maps repeats the same set of values with maps to again. So, it is e to the z is a map which is many to 1 in fact maps infinitely many points to the same complex number. Now, when you divide the inverse of this which is log z the question is there are infinitely many w's which are for which this holds. So, which w do we assign this we can say that there are infinitely many log functions one for each particular strip right we can say that first log is when z is mapped to w where the y coordinate of w is between 0 and 2 pi. The second log is when it is mapped to y coordinate is between 2 pi and 4 pi third log and there of course, we have to do something about the negative side also here which is not a very nice view to have infinitely many log functions that make sense. The other alternative is to say forget about the other strips we just say that focus on one range of values and this seems like a the main strip between 0 and 2 pi seems like means naturally we are in more inclined to think about this strip as the right strip for us. There is mathematically there is really no difference between these strips but for us somehow between value between 0 and 2 pi seems more natural than a value between 2 pi and 4 pi. So, we say that log z is always going to refer to this strip then it is one to one there is no confusion about the definition of log z, but then this view is also somewhat arbitrary of course and there is which is fine I mean we can be arbitrary if we are defining a function we can define it the anyway which we want. But there is another problem that this function is no longer very nicely behaved what do I mean by that even the continuity of this function is not present everywhere because suppose you are mapping this e to the z maps this coordinate system to this coordinate system and log z being the inverse reverses that. So, let us pick up a circle of radius 1 in this coordinate system and let us start traversing from let us start from this point and traverse this circle counter clockwise and let us see what happens to the log value there when we adopt that log z is this corresponds to this strip. So, what is this point this point is 1 0 what is the corresponding log value to this y would be 0 here and x would be 0. So, it is this point 0 0 actually we cannot include both ends of the strip also. So, we just have to say take this line and throw out this line because otherwise 0 and 2 pi will also have a conflict fine. So, we start at this point and let us start traversing it counter clockwise as I said. Now, as it traverse the modulus of the radius remains 1 which would mean that here the x coordinate will be always 0. And as we increase go this. So, the angle goes up which means that we start traversing up y axis. So, you rotate go keep on going keep on going keep on going come all the way up to here. So, you reached somewhere here and now in this plane I can just loop back, but what happens here you from here you have to jump back here. So, a continuous motion on this plane leads to a non continuous motion on this plane which is a not a very nice property. So, we will see how to fix this. In fact, there is a very nice way of fixing it and the way it works is that as you move there the you keep moving up the y axis and as you traverse cross this x axis here you will move from this function to the function which maps this plane to this one. In a very natural way if there is no we do not have to introduce any artificiality while moving there that is called analytic continuation and we will see exactly how that happens. So, I mean this. So, it would bring in this infinitely many different possibilities of log set which I earlier first called define, but instead of making them any arbitrary choices it would tie them with certain ways of traversing the plane here and that will bring in this all these copies of or different types of log set functions and passion together. So, but by the way why is log set and analytic we have not figured that out. We have not shown that inverse of an analytic function is analytic. No, no inverse is one over the function is analytic, but this is inverse of it. That is that is probably the simplest way of doing it that you have to show that this satisfies Cauchy Riemann, but to do that also we have to z we have to substitute a z equals x plus i y and then you will separate it out into complex and non complex parts other imaginary and real parts. You can write it in polar coordinate that is probably a better idea. Yes, use Cauchy Riemann polar coordinates and then you write in polar coordinates and then check it out. So, that will ensure that this is analytic which I will leave to you to verify and as I said this function is going to be really really important to us and the reason we will for that we will see later.