[Music] [Music] okay so the first topic we'll start with is complex analysis A functions of a single complex variable and I'm going to assume that all of you already have a little bit of preliminary information on this or knowledge about it so it'll be more in the nature of recapitulation so recall that we Define a complex variable Z as x + i y and x and y are elements of the real number line okay now this uh complex plane the XY plane is called the complex plane and we going to talk about functions which are analytic functions in a specific sense of this combination x + i y Okay the reason this is emphasized is because X and Y are linearly independent of each other and of course if you have an arbitrary function of X and Y there's no reason it should be a function only of a particular combination of X and Y in this case it's combination is x + i y remember that this uh thing Z Star x - i y is linearly independent of Zed so the whole thing about analytic functions is simply that you have a function of X and Y which is a function of the combination x + i y and does not involve the combination x- iy so that's roughly speaking what an analytic function is we'll make this idea much more precise okay but before we do that I'd like to introduce the idea of stereographic projection because on the XY plane unlike the case of a real variable X where can tend to either plus infinity or minus infinity on this side and likewise for y in the complex plane there are actually an infinite number of directions in which you can tend to Infinity along any Ray any direction whatsoever now that's a little uncomfortable because we seem to have many many points at Infinity so to speak so the standard trick is to try and put this idea of Infinity on a footing which is more or less the same as that of any finite point and that's done by compactifying this space in other words Imagine This Plane is a huge pancake and then you take that pancake and lift it up and SE its ends together so it becomes the surface of a sphere and this will then be our model for the complex plane the extended complex plane to make this precise one does what's called a stereographic projection stereo graphic projection which is you take a unit sphere that's the equator and the plane of this equator is going to be our complex plane Z plane so on this sphere you have these axes this comes out of the board this is what I'd call the y direction in the XY plane so this is the complex plane the complex red plane this is a unit sphere it's got unit radius and on this sphere on this plane the coordinates are X and Y X coming out here and Y going in that direction and on this sphere the unit sphere I'd like to have coordinates the three coordinates but they satisfy a constraint because the sums of the squares of these three coordinates will become Unity because it's a unit sphere so let's call those coordinates Z 1 Z2 and Z3 so on the sphere and this sphere I call S on S the coordinates are Z1 Z Z 2 and Z3 where Z1 is in the X Direction Z2 is in the y direction and I3 is in what would have been the Zed Direction but I'm going to reserve Zed for the combination x + i y in the third Direction up here now what are these uh things actually in terms of angular coordinates on this sphere well on that sphere unit sphere I can define a polar angle Theta which is the Co latitude and a longitude five an azimal angle five then of course z one as you are all aware is sin Theta cos 5 this is sin Theta sin 5 and this is cos Theta those are just spherical polar coordinates on this plane on this sphere okay now the idea of the stereographic projection is that you take this point the North Pole this and this one is the South Pole here at the other end that's the origin you take this North Pole and draw a line to any point on this plane like this for instance this point here the line joining the point of projection to this point intersects this sphere s at one point and you associate this point with that point in the complex plane okay and now you can see that this is going to be a mapping from the complex plane to the surface of this sphere because for every point in this complex plane there exist one point on the sphere it's immediately clear here that all points lying within the unit circle in the complex plane are going to be mapped or are mapped from points in the southern hemisphere of this reman spere and all points outside the unit circle like this for instance are mapped or are mapped from points in the northern hemisphere of the sphere okay the equator of this sphere is the unit circle in the complex plane the circle on which mod Zed is equal to 1 okay the South Pole is mapped into the origin here and the origin in the complex plane is mapped into the South Pole on the remon sphere okay now what we need are equations which connect the coordinates Z1 Z2 Z3 to the coordinates X and Y on the plane remember there are only two independent coordinates because i1 squ + I2 squ + I3 square is equal to 1 so There's a constraint between them and just as you have X and Y are independent similarly two out of these three are independent coordinates now the coordinate the actual mapping from one to the other is fairly easy to see look at it for an example in the intersection in this z32 plane look at look at it for a minute then it looks like this and what you're doing is to map in this fashion that was y so this is the y coordinate but in the same direction on the reman sphere you're calling it Z2 and you're calling this coordinate here Z3 and this is of course one on this side that's the origin and now by similarity of triangles it's immediately clear that this divided by the whole length on that side is equal to this divided by the whole length out here so it immediately follows that y or rather Z 2 divided by y That's this over that this side is equal to on this side what should I write well this portion that's 1 - I3 divided by one that say right so this is to this as this is to this entire length Okay so that's the equation that immediately tells us right away that y = to Z2 / 1 - Z3 and if you done this in this plane here the intersection of this line with this line the plane formed by these two lines then You' have immediately got x = z 1 or 1 - Z3 on the complex plan okay you could go back and write Z1 Z2 Etc in terms of these follows and then you can ask what does this mapping actually look like it's easy to see that for instance x equal to Z1 so for Z1 I write sin Theta cos 5 and for I3 it's 1 - cos Theta and then go to half angles and then it immediately becomes clear that this is equal to C Theta / 2 cos 5 y = c th/ 2 sin 5 so it means that Z is equal to e to the I5 c th/ 2 Z star is e to Theus i c Theta / 2 where Theta and 5 are the polar and azimal angles on this sphere by the way the sphere s this thing here is called the reman sphere you could ask what are the reverse mappings well you you need to exploit the fact that Z1 2qu + Z2 squ + Z3 2 is equal to 1 and then it's not hard to see that the reverse mappings to these two oh by the way I should complete this by writing uh this implies that Zed equal to Z1 + I 2/ 1 - I3 Z star is Z1 minus I I2 1- I3 okay and what are the reverse mappings it's not hard to see directly from here that Z1 is 2x / X2 + y^ 2 + 1 Z 2 is 2 y / X2 + Y 2 + 1 and Z3 is equal to X2 + y^ 2 + 1 / - 1 1 + one okay which you could of course also write as 2x that's Z + z Star ided by mod z s + 1 this is equal to z- Z Star over I * mod Z 2 + 1 and this guy is mod z^ 2 - 1/ mod z^ 2 + 1 okay so there's a one toone mapping between the complex plane and the remon sphere the question is what's the point n going to get mapped into under this map okay it's immediately geometrically clear immediately that as I get closer and closer here the point in the complex plane is going further and further away okay so it's evident that no matter which direction you approach NN you're going to be hitting Infinity along some Ray in the complex plane but all those points are getting mapped on to n okay so I can call n the point at Infinity like in fact I can call it Infinity I denote it by infinity and what this does is to bring Infinity to a status which is similar to that of any finite point in the complex plane okay so very often I'm going to denote the complex plane by Zed this is all this is all this is the set of all values of Zed such that the modulus of Zed is finite and if I include the point at Infinity I'm going to call it the extended complex plane and very often I'm going to denote C Tia equal to the extended which includes the point at Infinity okay but the reman sphere provides me with a model for the extended complex plane okay is this clear that it's geometrically obvious intuitively that this n now represents the point at Infinity so that's why very often in complex analysis I'll Loosely say that there's just one point at Infinity this Infinity okay I might tend to it in different directions I don't care but what I mean by that is this point here this point n okay so this compactification of this complex plane enables you to extend it to include the point at Infinity puts it on on the same footing as anything else and then of course we can do calculus on it very uh rigorously without worrying about this Infinity what this Infinity is one could ask on this uh point on this sphere do I have a notion of distance on this sphere well there are many ways of defining distance on it one of them would be to define a great circle distance now what's a great circle distance between any two points on the surface of a sphere you pretend that the one of the points is the North Pole and then you look for the distance along the longitude to the second point and that's the geodesic distance the shortest distance lying on the sphere you could also Define distance in this case as if I have two points Z1 and Z2 in the complex plane in the plane here then Z1 minus Z2 modulus is what I would call the distance between these two points on the complex plane okay on the other hand I could ask what's the distance corresponding distance on this sphere well it would be the cordal distance between these two points so I draw a cord through this hollow spere from one point to the other and calculate what that distance is when the two points on the complex plane are zed1 and Z2 and then a little bit of algebra shows you that this distance cordal distance between Z1 and Z2 let's call it D of Z1 Z2 this is equal to twice turns out to be twice modulus Z2 divided by square root of mod Z 1^ 2 + 1 Z2 2 + 1 turns out to be this quantity little bit of algebra you can substitute these expressions and then you end up with this expression expression for the distance notice the presence of these two denominators here okay now what does that do that makes the distance between any points on the sphere finite including point the point at Infinity okay because you can see now that this satisfies all the properties you need of a distance function for instance D of Z1 Z2 equal to Z if and only if Z1 equal to Z2 there's no other way this distance can vanish okay and then you know that this distance D of Z1 Z2 in general is non- negative that's obvious from here okay and the distance from any point to any other point satisfies the triangle inequality Z 1 Z 3 is less than equal to D of Z1 Z2 plus d of Z2 Z3 and this is also well D of Z2 Z1 is equal to this it's symmetric Under The Interchange of these two points so the this distance as defined here satisfies all the requirements that you make of what you call a distance function so it's a respectable distance function satisfies the triangle inequality and so on now what does it mean to say the distance to the point at Infinity well D of Z Infinity equal to so set Z1 equal to zed and Z2 equal to Infinity then of course the limit when Z2 goes to Infinity it's all this is all that contributes and that cancels against the Z2 here and you immediately get 2 / squ < TK of mod Z 2 + 1 okay so what's the distance between the origin and the point at Infinity the cordal distance it's two and that's in fact the codal distance between the South Pole and the North Pole on the sphere it's just the diameter of the sphere so this thing here is extremely useful the idea of this cordal distance and one can make a lot of progress using this okay we're not going to do much more with this but simp to point out here that there exists such a distance notion of distance and it's got an interesting structure we need that dist for anything we do but I'm I talk about it in a problem set or something like that yeah it's useful this okay now let's get to to analytic functions and let's see what we mean by an analytic function so very roughly speaking an analytic function in some region so we have a function f of x and y let's call it and I said it should be a function of Zed is an analytic function I'm going to use this term analytic and later on I will qualify it in certain ways it's analytic in some region in this complex plane if if it satisfies a couple of relations if this F of Zed is uh U of X comma y + I * V of X Comm Y where u and v are the real and imaginary parts of this function then this is analytic analytic if the koshian conditions are satisfied okay I'm not going to prove the koshan conditions here it's fairly straightforward to do it but I'd like you to tell me recall what these conditions are and then we'll try to interpret them analytic if the Koshi remon conditions are satisfied what are these conditions yeah the conditions are Delta U over Delta x equal to Delta v/ Delta y Delta U over Delta y minus Delta V Delta X so right away you see that an analytic function must have partial derivatives first order partial derivatives the first order with respect to both X and Y both the real and imaginary Parts okay so that's a prerequisite you need those partial derivatives and they should be continuous and that's more or less all that you need for a function to be analytic but what it really means these conditions what it really means is that this thing here as I said is not a function of the other linearly independent combination it's a function of X and Y but it's a function of the combination x + i y with no reference to xus i y so we could say that an analytic function is something for which Delta f over Delta Z Star equal to zero see that okay no dependence on uh Z star at all so it's by the way um we know that uh you have to tell me what is x in terms of Z and Z Star it's this Y is zus Z star/ 2 I okay just as Z is x + i y and z star is x - i y i invert those these are the the other inverse relations now what would this imply now I can take the Z Star write it as x - i y and then use the chain rule of differentiation so this would imply that Delta f/ Delta x minus I Delta f/ Delta y equal Z and now you put FAL to U + IV and equate real and imaginary parts and you get the koshian conditions okay so quick way of asking what's an analytic function is to see if this function is nice and smooth has first order derivatives and so on and then check whether it has any dependence on Z star or not and if it doesn't and you can express it purely as a function of Zed then you say it's analytic you don't have to check these conditions out each time is this an analytic function I won't even call it f of Z I'll just call it f is that an analytic function no because you see by the way this immediately tells you that you can't have a function it's analytic in a certain region if in that entire region it has no imaginary part at all if it's purely real or if it's purely imaginary with the real part being identically zero or the imaginary part being identically zero in a whole region you cannot be analytic and that's indeed true because X is Z Plus Z star /2 and it involves Z star of course right so it cannot be an analytic function how about y not an analytic function how about you know sometimes you write this as r e to the I Theta by the way this Theta is not the same as that so let's get rid of this you write it in polar form in this fashion where R is equal to square root of x^2 + y^ 2 and this Theta is tan inverse y/x write it in this form so tell me is R an analytic function how do you write R in terms of Z and Z Star well r s is x² + y^ 2 right so it's z z star so this is R is equal to it's mod Z and that's not an analytic function it's got this dependence here what about Theta is Theta an analytic function tan inverse y/x now I write Y and X in terms of this how do I write Theta in terms of Z in terms of Z and Z star well Z star is r e the minus I Theta so if I divide Z by Z Star the r cancels out and I get e to the 2 I Theta right so this is equal to 1 / 2 I log z/ Z Star does that involve Z Star so can it be an analytic function no not an analytic function X is not Y is not R is not Theta is not and so on so analytic functions have to have very specific structures not every function is analytic is this an analytic function well let me give you let's let's go ahead let me give you a few more example as we examples as we go along but one consequence of what this these equations tell us one consequence is immediately that D2 u/ dx2 plus d s u/ Delta Y 2 equal to Z and that's also true for V okay so in the region in which some function f of Zed with real part U and imaginary part V in the region in which this function is analytic the real part and the imaginary part separately satisfy laasa equation in two Dimensions what do you call functions which satisfy lass's equation harmonic functions so the real and imaginary parts of an analytic function are harmonic functions okay now this set of equations is telling you in some sense that if you give me the real part the imaginary part is more or less determined by in principle solving the koshian conditions okay so what an analytic function is is like this suppose U is harmonic in this region in the complex plane and V is harmonic in this region in the complex plane then U plus IV is guaranteed to be an analytic function in the intersection of these two regions in that intersection region both u and v are harmonic functions and they can be the real and analytic imaginary parts of an analytic function so in that sense if you solve the koshian conditions given you you can find V but you have to specify the region you have to specify the region in which things are analytic and now let me jump a little bit and explain that if a function is analytic in the whole of the finite complex plane it's called an entire function if these conditions are satisfied in the whole of the complex plane for all mod Z less than infinity then F of Z is an entire function lots of examples of entire functions is this an how about this is that an entire function yes or no yes it satisfies the Kos conditions for all Z all finite Z how about this do you think it satisfies well you got to write Zed as X plus I but more simply this function is differentiable and it's clear that it doesn't involve Z Star so it's very much analytic you can in principle write x + i y whole squ and then you differentiate the real part and imaginary Parts see if they check satisfy koshian they certainly do so that's an entire function how about the is that an entire function yes certainly most certainly how about uh F of Zed equal to PN of Z some polinomial of of degree n is that an entire function arbitrary polinomial yes certainly it's an entire function how about uh F of Z = to e to the^ Z do you think that's an entire function well we need to write the definition of e to the Zed in terms of a power series and then ask whether that series is valid for all Zed or not is that true or or false is a series what's the definition of e to the Z by the way for what Z is this valid less than all finite Z mod Z all fin so is e to the Z an entire function yes e to the minus Z yes all it does is put a minus one to the end it doesn't do anything inside how about e to the Z Plus e the minus Z divided 2 yes so what do you call that by the way C that's entire so e to the z e to the Z cos Z what about Sin hyperbolic z c z is that an entire function yes yes indeed it's only the odd part of this series and the other thing is the even part so the function itself is an entire function it's odd part it's even part they're all entire functions right how about Sin Z or cos Z do you think that's an entire function can you write it as exponentials it's e the i z plus eus I Z / 2 and sin Z Zed is an arbitrary complex number by the way S and S hyperbolic cos and cos hyperbolic they're all the same function I mean essentially they're analytic continuations of each other so regard all these functions are defined by power series and as the argument of the power series being a complex variable Z that's the correct way to define look at all these functions all these are entire functions everything is an entire function how about tan Z well that's that's not is not immediately clear at all it's not clear that it's an entire function certainly is singular at some points it blows up right so that's not entire how about Z to negative Powers they blow up at Infinity right at at zero Z equal to Z so those are not entire functions pols yes but rational functions no because you have is where the function blows up right now there's a theorem called Li's theorem which says that if a function has no Singularity everywhere it's analytic everywhere in the extended complex plane at all points on the remon sphere then it must necessarily be a constant there's no other function no other function at all so if you have a function that's entire and at Infinity it's not singular but continues to obey the Kim conditions then that function must be a constant trivial constant okay so what's the conclusion immediately for all these functions for all these normal entire functions these are not constants they functions of said non-trivial functions but they're entire so the conclusion is they cannot satisfy the koshian conditions at the point at Infinity they must be singular at Infinity they must be singular Infinity so all these functions that's why I was very careful to write mod Z less than infinity because at Infinity there are singularities and we'll come to what kind of singularities they are Etc in these cases they're all what are called essential singularities okay now the same thing goes through for this this is a monomial you have an arbitrary polinomial here but these things blow up at the point at Infinity they have singularities at Infinity those would be called poles of various kinds okay so we will classify these singularities quickly but the fact is that you cannot have an entire function which continues to be analytic at Infinity without being a constant it has to be a constant but these are lots of examples here all these power series Etc would all um be entire functions but they'll blow up at Infinity they have bad behavior at Infinity okay now before we go on to the Behavior Uh power series there's something else I wanted to mention and that is to ask what's the derivative of an analytic function that's one more way of looking at the koshian [Applause] [Music] [Applause] [Music] conditions [Music]