 Okay, good morning Welcome to ICTP welcome to the city of Trieste. I'm Fabio Vlaci. My name is Fabio Vlaci. I Was born in Trieste. I studied here and currently I have a position at the University of Florence But I'm always happy to be at ICTP and to give a Contribution to this this diploma courses and this is the third year. I'm I'm giving this course in complex analysis Some information the schedule and the calendar depends on many Factors including in the fact that I have also to teach in Florence Go back and forth and I cannot be here for the whole term However, I will be mainly here in Trieste. My room is the room number 118 119 the same entrance. I write it here. Okay room number Can you read it? This is another experiment. Not only they are taking me on the They take a track on what I'm saying, but I'm also tracking what I'm writing This is an experiment also to me and also for me, but I think it is a good idea to have The black birds say set of pictures black birds and I can And I can give you the copies at the end of the week You know that we'll have 20 20 blocks of lessons together in this course and in other courses you will be supposed to Well to attend the lessons if that's obvious if you wish and also to solve some of the exercise I will give you Weekly approximately Okay so a good way of communication is to use the of course the Email I will have your email address at ICTP and I give you mine. Okay. My room is this I Shared it the office with ever since it just to be the room is in front of the sector of this the room of Alessandra Bergamo one of the two secretaries of the math group and I leave you my email address this is the address From the University of Florence and this is at ICTP. Okay, so the secretary will provide me Your email address at ICTP If you wish in the meanwhile, you can contact me using this. Okay, so It is not easy for me to remember your names and even to pronounce them, but I will normally answer quite Frequently to questions my email. Okay, of course you will also the You can request some appointment in case you need some some extra Information whatever your simply information what I will be happy to to to meet you in my room and but the only Fact is that I have to know it in advance For instance on Friday have to run up immediately after the lesson and next week will have only lesson on Friday Not on Monday because of some overlapping of classes. Okay, they moved it in the calendar somehow they changed it They moved it in the room They moved to say that the classes from rooms from one room to the other and so there are some overlapping. Okay Extra information Nothing I would say As I said, I'm very happy to be To be part of the This diploma program. I think that you already met a director consider this Program one of the best activities At ICT PICTP is a very well known International scientific and I think it for you. It's a good opportunity to be here. You can meet a lot of visitors. You can join the Conferences and of course you have also to to work a little bit. Well back to business So the aim of the course is to give you basic notions and the theory of complex value functions of one complex variable I Assume that most of you already know something about complex valued functions of one complex variable if All of you are very familiar with all the classical stuff Well, we can also switch to something more complicated in several complex variables So it depends on your background. So normally the class is not very Omogeneous in the background. So which is obvious for obvious reasons So I will start quite slowly and if you find it quite boring Please tell me and Okay, I have other accelerate or change the subject. Okay a little bit Well for sure. Okay, what I'm assuming as a background general background I'm assuming that you know basic calculus and real calculus and The some basic facts in topology and some Elements of algebra Okay, so I think this is granted in in case I can give you some some References for the course I have not I'm not following just one book one textbook I have taken notes from different books and this is the idea I will provide you of List of books textbooks for as a reference for the course as soon as I I know what I have to teach in the sense okay, normally We We are asked to we are asked to to provide you basic notions and complex analysis and not in advanced several Variable complex analysis, which is quite different, but if you don't know any notion and and One complex value one complex variable theory. It's very difficult to understand several complex. So so my idea is to start with one complex variable, okay and And first of all Okay, so this is the welcome page number zero To remember to put the note here says first lesson page one All right, so About names, okay, why Complex numbers are considered complex and the real numbers are considered real the two sets are not Either we are all complex in some sense using the standard terminology because well This terminology went in use For historical reasons and different times, but the idea is that we are accustomed to the realm of the numbers even though as you perfectly know the real numbers are introduced axiomatically and Complex numbers are considered more complicated with some additional information, but sometimes they also contain good properties of real numbers okay So normally real numbers the set of real numbers indicated by are Real numbers I use C to indicate complex numbers normally also I believe that you Identify the real Numbers with the real line with the line Hmm, and it is natural to consider the complex number to be Okay Very good. So this plane is has also a name. Normally it's called gauze plane. So Typically a complex number is an ordered pair of real numbers Hmm, so I consider plane which means are two so pairs of numbers a b of of real numbers Order pairs and this Of course shows you that the complex number the complex numbers are not Very different from the real numbers. Okay, so many properties will be Many properties of the complex numbers will be Will be obtained by using this fact Okay, the point is that the real number The real numbers have also an algebraic structure. We can sum real numbers and we can Multiply real numbers. Hmm. This is in the axiomatic definition of the real What about Operations with the ordered pairs Well, if you want ordered pairs means vectors in R2, so how do you sum vectors in R2? Well, you know, and you simply sum Component wise good, so if I have a b I Define this is the definition of the sum in say That the sum of two pairs is the sum component wise Where the sum here is? The sum in R Okay, so it is quite easy to verify and I leave it on the exercise That the complex number with this sum here is An a billion group. I'm sorry passing to multiplication. Well, multiplication is not as natural as it may Appear because when you have two real numbers, you have Composition given by with some properties, of course, huh? Because the real numbers form a field, so they also group with the multiplication now the idea is that if you have a pair Real numbers or the pair you have a vector right in the plane and there is no Natural multiplication You can multiply a vector by a scholar in fact R2 is a vector space But to define this product you have to some in some sense put a rule and And with the file this to be a c minus vd and Ad plus So I invite you to verify that This multiplication this is definition, okay as associative commutative and for any pair Different form the pair zero zero which represent a neutral element for the sum there exists a Nimbus and inverse means that you take a prime be prime such that Where one zero is the identity? this is probably something you perfectly know and What are the news good news? Well, if you prove all these properties, then you have another field Just good some strange news is that you start from two pairs and you have a multiplication of some special numbers complex numbers which provides you a square Which is minus one when you identify a zero with a in R Like we did here. You can see it quite easily. Okay, but what are the motivations for this? Multiplication This is something artificial in some sense, huh? I want to give you some motivation first of all I want to remind you Want to show you that if you adopt That multiplication You obtain that the the pair zero one has a square which is minus one which is as Minus one in R and this is something completely unusual for the real multiplication any square is non-negative number On the other hand, you also know that if you consider algebraic equations So polynomial equal to zero is fine. You are looking for roots of polynomials. So then you have nudge break equation Well, if you if you consider if you are if you are considering the possibility of finding roots and The real numbers not always this is possible as you know the quadratic Equation one plus x squared equal to zero is no real solution, but Since We have discovered that using pairs. So using complex numbers. We can find a square which is Somehow minus one. We have some hope that we can find a solution also for this quadratic equation first of all Let us start from a more generic setting quadratic equation Well, we are assuming that a is not zero Hmm, then I think that you all know that the two solutions Obtain in the following way and everything is meaningful When b squared minus four times a times c is Greater or equal to zero in the real numbers. Otherwise, yeah, we are writing something which is not real What I want to show you now is that We can transform into a change of Parameters and variables Okay, so my idea is the following we My idea is the following then first I consider a x square plus b x for squared Well, first of all, I can well Yes, I can either I can divide by a everything because a is different from zero Okay, and I make it simpler. So I have x squared plus B over a x plus c over a Equal to zero since a is different from zero Okay, I Can do this then I Add a subtract b squared over for a squared Which completes somehow the square we have here So I have this and this can be written as plus B a But to a sorry squared Okay, and here I have minus Okay So please note that this is exactly what it is here. It's called discriminant, right now I call this Number here D and this number here This sorry this variable new variable here call it u. I have u squared equal to D When I'm considering something which is not So some some quadratic equation which is not solvable in the real I'm assuming that b square minus 4ac is negative, right? So this is this number here is D here minus it comes D on the right-hand side is negative Okay, so put Last line Put minus D is positive and I consider as new variable x u over Square root of minus D and this is meaningful because minus D is positive and I have okay and so I'm reduced to this equation So this is somehow the quadratic equation which cannot have real roots But then I consider This is not a standard notation, but please tell me It is familiar to you then I consider the ring of polynomials with real coefficients and of course x square plus 1 is 1 of this polynomial now I consider the ideal Generated by x square plus 1 This is the set of all polynomials of this form x square plus 1 times a x Where a x is a polynomial, all right? What do we know about This ideal Well, we know that this polynomial x square plus 1 cannot be split into polynomials of degree 1 It is irreducible. So the ideal is prime good Very good. So this is the background in algebra. I'm requesting since it is prime You can consider the quotient right Okay, so let me just write here. This is irreducible as a polynomial in R and hence a sorry I is Prime Therefore I can consider this quotient here which turns out to be a field This is a field now How can we represent the elements of this quotient? Well, you take a polynomial You divide it by x square plus 1 and the rest is what you have as a representative But as a rest you have a polynomial of degree at most 1 so the Representative of this is the class Sorry representative in a class is a polynomial of degree at most 1 Okay, so I write it here just once formally and B and a are real numbers Because B x plus a is a polynomial Rx Okay, good So hope everything is somehow known or Affordable Now what I'm doing is well, I know that this is a field but What if I sum two elements and This quotient I sum two elements like this and I remember that this is By definition The class represented by the sum of the print the representatives, right? So it is quite easy to Obtain this nothing new let us ask to Multiplication and this case the things are a bit more interesting Because when you multiply the two representatives then you obtain Representatives I have degree one at most one. Okay, then in some cases you have a in a product As a result a polynomial of degree two, so you have to take the quotient, right? All right, so I Forgot to put this here But what I have is Bd x squared Right plus BC bc x plus ad x Plus a And this has to be divided by x squared plus one right so I Do like this I add and subtract Bd Okay, so that I have here Bd x squared plus Bd minus Bd Plus and then I sum up some the coefficient of x I Have ad Plus bc and x plus a And this is Representative and this quotient so that I have Bd x squared plus one Which council because it is huh? and What is left is ad plus bc x plus a c minus bd it is Do you have a question? Yes good? Sorry, I didn't understand the question I can repeat the question when I understand it What okay? That that's correct. That's correct, but formally that's what you are doing. You're yeah, that's correct, so Your colleague is saying that Since you have this quotient over here Hmm Anytime you find an x squared you can replace x squared by minus one right X is just I but I is not is not in the business now. I'm yeah, yes, sure This is formally the the the construction of a quotient of polynomial so of real polynomials real Polynomial with real coefficients, but I agree with you what I have done is nothing But well to make it appear here. I added Bd I added Bd here and Remove Bd here So that Correct instead of x squared, but I think that formally this is what you do want to do a good idea Thank you for for pointing out x turns out to be something similar to I But I has not appeared here Even even in the in the in the previous notation, I never use I This was on purpose just to okay to make the introduction more Anyway, it is correct. So this If this is correct and it should be Because I'm using I'm using something that well, I will do a lot of mistakes. So please Stop me when you don't understand what I'm writing and feel free to make questions but This gives you a motivation while multiplication is The one we have chosen for the pairs ordered pairs of real numbers So if I consider the representative a Plus bx To be put in correspondence with the pair a b then I discover that what I've done is the following then I had C plus dx D That the multiplication is what I have here. So I See minus Bd ad This is the motivation. So just to to use the notation six the pair of the ordered pairs of real numbers with the Some and multiplication introduced so far and be in fact considered as this quotient here good news and this New field larger field. We can solve all quadratic equations Because the formula we have Can be solved because we have a square Of a number which is negative But bad news Since we are working with essentially with vectors We're not on the line. We're in the plane and therefore vectors cannot be ordered like Points on a line. There is an orientation on on the on the real line You can compare say one number is greater Smaller or equal to the other but this is in the case of the real numbers for complex numbers This is not completely meaningful So historical remarks, thank you for so it was Okay, this was known and People use some strange symbols like the unfortunate symbol Square root of minus one, which is of course Not good so As far as I remember in the first time the letter I appeared was in 1777 It was Euler who said well because this number cannot be real He said well it is imaginary and he used the letter I and they wrote in a in a letter and Adopted this symbol Then numbers became Known as imaginary numbers because they are not real but also real numbers are not real in some sense They are imaginary as well So, I think it was Gauss who decided to call them complex numbers because if we If you use this notation, this is the number zero one Okay This is the pair zero one which with this multiple with the multiplication Introduced so far is the one the number with whose square is minus one and I definitely prefer this to this This to this this is something Okay misleading Okay, so if we accept that this is I as you probably all know then the pair ordered pair a b which is the general element of Set C is also a times one zero plus B times zero one when you regard the vectors as vectors in as a vectors of Vector space of the real vector space a and b are scholars in this sense. So then we have this number here Which can be regarded as one In are and this number here. So the two generators Okay, the ultra normal basis and this number here is I So that any complex number a b has also this representation a Plus bi which is very common and normally the letter for the For the complex number easy why is he Well, because we already use X and Y. No, well Probably because Z is the First letter in the world salad in German, which means numbers Okay, so these numbers, but we cannot use Z capital Z or blackboard Z To indicate this set because it was already used for the relative numbers. Okay Okay, good. Now We can represent Then any complex number in what we call the Gauss plane the Gauss plane Consider a to be the first coordinate and b to be the second coordinate, right? So this is the number Z a plus I b which means this Okay, good. Oh Normally the Okay, I'm observing remark. It's obvious if B is zero the complex number is real Okay, so if a is zero And the complex number is no As we say, it's no real part because a is called the real part and B is called the imaginary part So purely imaginary, okay? Which means? respectively that the point to Z is on the real axis Second case it is on the imaginary axis on the vertical axis good all right, so some Facts which I leave you as exercises page six point Wow, so What I want to okay consider this map and This one notation is quite natural say t and s are Real numbers and on the right hand side you have a complex number Okay, so Call it them the question is are this functions Injective one to one whatever Do they preserve the structure because are is a group? field So are they homomorphism I'm not indicating on purpose if They are homomorphic if they are group homomorphism If you know whatever, okay, so please I invite you to think of this too. So if we identify the real part and the imaginary part with the Projections on the axis And vice versa if starting from the real number Go to a complex number. What are the algebraic properties preserved and what are not? Okay? At the other exercise is the exercise one say now the exercise is the following Okay for Algebraic general facts, we know that C is a field, okay So any element different from zero Okay, s s an inverse okay reciprocal Okay, so I invite you to consider the generic reciprocal to write down What is the real part of the reciprocal of a b and the reciprocal of? Sorry an imaginary part of this reciprocal as well, and I also invite you exercise three as to find real and imaginary part of One over z squared when the z of course is different from zero You take any element in the plane Normally use accent for the real part and why for the imaginary part since we are using the identification with the Gauss plane of Complex number take a complex number different from zero zero means the order pair zero zero, so It is not possible for Z to have both real and imaginary part Zero, but they can Okay, and Something more to I as well Okay So I invite you to make some calculations, and this is a way to better understand How to work with the real numbers I would sorry with complex numbers Okay before continuing let me also tell you that Historically after the complex numbers there were some attempt to extend to other sets the multiplication in order to have an extension of complex numbers as It was done for the real numbers real numbers complex numbers and something else But it was very hard. In fact, it was almost it was impossible to find a Multiplication then it was proved it was impossible to find it in in R3 So that is well, I have R. I have R2, which is the complex one with some efforts I can extend everything now three, right? Well, this is not possible if you want to have an algebra so division office if you want to have reciprocal right and left This is not possible But it was by accident that Hamilton discovered this is possible For in R4, right For the quaternions he introduced the quaternions. So the quaternions. This is possible However for the quaternions the quaternions were discovered in the late 80s. So quite recently When Hammeter was looking for a solution R3, of course, this is the normal situation mathematician is trying to prove something and he proved something else, but Anyway, the the quaternions are now quite known and studied even though They are recent the theory is Developed at a certain stage using and applied in in physics very much in physics because Quaternions are not have a multiplication, which is not commutative It's not antiquated. It's not commutative and So when trying to generalize and generalize there is another set of the quaternions from R R2 R4 R8 So seven imaginary units are added to a real part and This set is known as a set of octonions or Cali numbers and This set is really odd because multiplicationally Introduced in the Cali in the Cali numbers is neither commutative nor associative So everything is very much complicated, but I'm sorry, but luckily we will deal only with complex numbers Okay, and with some analysis on complex numbers I am an interested for other reasons and analysis on over the quaternions and Well, you can imagine it's difficult to to define anything without the commutativity property think of Well Derivation for instance In fact, there were several there have been several times to have a notion of derivation and The class of functions discovered to be derivating Differentiable say in the quaternionic sense is either very small or very large or very bad It's up to you. So only recently there were some good Developments of the theory and well, maybe in one of the next lecture I will I will I will show you some some facts on the slides and I will send you the Because I like it. Okay You're not supposed to do anything on in a non commutative field which is odd Not the social TV. It's it even is even unaffordable for for many mathematicians. Okay, but What do you have to know is that there are no extra algebras? after quaternions or octonions Okay, so you don't have the possibility of Divide on the right on the right on the left different inverse reciprocals And So the complex numbers are somehow the good numbers After the real numbers and they are the only ones Okay, okay, sorry for this Parenthesis Now as a matter of fact it is natural to consider given a complex number Z Complex number, which is symmetric with respect to the real axis same real part opposite imaginary part and this Complex number is The noted by Z bar and called conjugate. Okay, so if you want to use the notation AB The conjugate is the pair a minus b This is something new because for the real numbers the conjugate coincides. So a to know actually two complex numbers Sorry a complex number which coincide with the with the with its conjugate is in fact a real number Okay, which means that the imaginary part is zero And why conjugate? Well because if you conjugate twice So if you consider the conjugate of the conjugate, well geometrically this means that you are taking this symmetrical and then Algebraically you can also see this a b a minus b a minus minus b that is a b and as I said if Z and Z bar I use Z bar Terminology is Z bar Unfortunately bar means many things in mathematics, but well in complex and I typically Z bar means the conjugate if this Occurs then Z is a real number. Okay. It's good Now something very easy to prove is the following take the complex number Z and sum it with its conjugate So the imaginary part cancels and you have twice the real part So is X if Z is X plus I Y and X is also the note in this way a real part of Z real part of Good Similarly if I subtract to Z See bar Okay, what I I have here is well X plus I Y minus X plus I Y right minus minus So I have to I Y that is to say that the imaginary part Of Z Y is Obtain in this way Okay, this is simply a relation So you can express the real and the imaginary parts of a complex number in terms of some subtractions of The number and its conjugate times a constant Okay What is also very important to to observe is that if you multiply Z and Z bar You can either adopt this notation here and then make a calculation following the rules given for the multiplication or remember that if Z is a plus I B Z bar is a minus I B and multiplication of real and imaginary part is simply the multiplication of In the reals and remember that I squared is equal to minus one so This is the new ingredient multiplication becomes like this so Z times Z bar is a squared minus I a B plus I a B I'm using the commutative properties you see how Often in naturalities and the thing of me working in the only quaternions and struggling with the Okay, and then I have B B B squared minus I squared that is minus minus one So this plus B squared Pardon me. Sorry. I I'm sorry, you sure sure. Yes. I was I was Talking about commutativity and forgetting about the sign. Yeah, right So what is left here? And this is important to remember is That this is a real number and it is a non-negative real number which represents in the Gauss plane What the length of the vector? Okay, the modulus of the vector. So if this Z this instance from the origin is This number here, which is normally also denoted by Modulus of Z squared, right? So the modulus of Z as The square root of a square plus B squared and this is the square root of a non-negative Real number. So it's a standard square root Okay Some very easy Ah Properties of the of the modules which I if you don't feel comfortable or familiar with this I invite you to to verify or to think about it is the following some properties on the Modulus of Z and the modulus of Z bar is the same this Well for several reasons, but Think about this Second point is that the real part of Z Well Minus the real part of Z smaller equal to sorry Just the opposite This is true, but the real part of Z is in between Minus modulus of Z and modulus of Z and similarly The imaginary part of Z Okay When we have a quality here, which means that the real part of Z and the modulus is the same What about Z purely? It's a real. It's a real number Good now the obvious fact is that if you take two num if you take a couple of Complex numbers Z and W then this What is true the product of two complex numbers has a modulus which is more to the two module good Let me also tell you that I forgot to write it down before what is the conjugate of the product the product of the conjugates and What what about the conjugate of a sum good? What about The sum The modules of the sum less than the sum of the module. Good. This is the Triangle in a quote very good So last exercise as well consider Z plus W squared Modules squared plus Z minus W squared and What is this some parts cancel right correct? Exactly exactly so it is Two models is square plus model What is important to remember? Okay, and in case to use but if I repeat some if anybody Fills that this is new Well, it's a good. It's a good training to make the calculations once in your life at least To be to feel confident, okay, and this kind of stuff Good now what is the next step? well, we have That's that any complex number is can be considered as a vector in our tool and we Automatically used adopted the Cartesian coordinates to represent what the point so we apply the the vector on the origin and We represent the vector knowing the position of the ending point, right? So in some sense We are using a misleading terminology because if you consider an ordered pair You're considering a point not a vector vector is column. Okay, but just a matter Transposition of our matrix Okay, if you write a B like this This is clearly a veteran R2 and when I write a B like this Well, this should be a Point but well we identify the point the ending point of the vector Okay, this is the vector of course you cannot use very many symbols over Z because otherwise It can it can be Confused with the notation of conjugation, right? But if you think of the vector and this is the point okay here good well We have already studied I think that well the Cartesian coordinates are not the only ones you can you can use for vectors. You can also use polar coordinates What where are the advantages and disadvantages of this choice? Well in some cases you have advantages and some others you are created disadvantages first of all You cannot represent any vector of the plane using Polar coordinates. This is obvious the origin has no Polar all in modules is zero but not the the angle and the angle is not defined So there is no There is no correspondence with With the Cartesian Cartesian coordinates, huh? however, if we are taking Z and see Different from the origin we can define the Polar coordinates and namely Z Okay, write it this way or a Z plus I and Z We have this angle here Theta, this is modulus of Z is the length of the vector the only vector which has modules zero is The zero vector remember that we define a modulus of Complex number to be the sum of two squares of two squares of real numbers So in order to be zero it has to be Simultaneous is zero a like the real part of the imaginary part, huh? Good. So Well some very basic Stuff and calculus tells us that where there are two functions, huh? To cosine and and sine which helps you a lot to to make the correspondence. So the real part of Z as the modulus of Z times cosine of theta and The imaginary part of Z is Modules of Z times sin of theta Well if Z is not Is not the origin To see we can associate a pair of numbers are non-negative Raw and the theta which is between 0 to 2 pi But as you all know that the difficulty is that there are several Determination of the angles, okay, so we have to make a choice and this is one This is the preferred choice normally preferred choice of the angle and in this case It is called argument. Okay principal argument But you can have very many because these two functions here are periodic and Of period 2 pi Okay so This fee is called argument Sometimes also the principal argument Because we have this restriction between 0 to pi Z A row is modulus Is the modulus And remember The modulus of Z is 0 if and only if Z is 0 so that in some sense you can extend polar coordinates saying that when the row is 0 No determination of the argument, but well, it means that we are considering the only the only Vector, okay, which has dispropped this The origin good now What is okay Relations between the two coordinates, we have already one so is Z is a plus b row is Definition the square root of a squared plus b squared Okay Let me let me Say this way starting with polar coordinates. We immediately obtained a real imaginary part Take raw cos theta and you have the imaginary the sorry the the real part raw sin theta and you have the imaginary part So now we are doing the opposite Okay description we start from the real imaginary part a and b. I will want to have raw and theta. We are assuming that Okay, and be Anytime speed this way And b are real numbers But not Okay, if this is the case then theta is also determined and Is well remember that I said that the real part of Z is raw cos theta and Imaginary part of Z is raw sin theta that is this is a and this is b Okay, I'm assuming that both numbers are not zero. So in particular a is not zero and I can take the Ratio b over a b over a is meaningful and Raw and raw cancels. So I have sin theta over cos theta b over a sin theta over cos theta. So theta is Arcton of b over a Well, if p is zero well, it's not big big problem because well Arcton is defined for zero. It's zero, right? So if p is zero Still okay Okay. Yes Okay, you are correct Good question. Okay is pointing out that I I Am taking an inverse function of the tangent function, which is not the define uniquely Multi-valued function, correct. All right. So Since I'm trying to have The argument the principal argument. I'm considering the Arcton to be the function defined on the real axis and With value in between minus pi over 2 and pi over 2 Okay, so the principal Or if you prefer you're correct, you have to add something like mod 2 pi Okay, good question, but this is exactly the problem with polar coordinates In any in any in any setting It's a matter of choosing of choosing sorry out choosing it's a matter of choice of the principle either you take In between 0 and 2 pi or minus pi pi Okay, good. Thank you however, what I'm saying is that be equal to 0 is not a big problem for The inversion of that and the tangent the problem is then when a is zero Okay, a and become both the zero is not as is not considered. Okay, because in this case We are taking the origin and the origin is no argument But when a is zero well depends on the sign of B To to determine an argument if B is positive Pi over 2 if B is negative is 3 pi over 2 Okay, okay Okay, if a is zero be not zero Right be positive means theta is pi over 2 be negative if theta is Okay, now This is not a This is to say This cannot justify all this calculation in the world we don't have I don't see any advantage to take this angle Angle instead of for two to real numbers with all the restriction with all the problems of the argument Well, the very good observation is the fire take two numbers complex numbers using the Polar coordinate so put an index I Sorry an index one for the first So row one theta one is the coordinate of the Z one and Row two theta two The coordinates the polar coordinate. Sorry of the second complex numbers complex number Well, let me make this calculation, which is probably familiar to you Take the multiplication and we have Row one row two and then cos theta one cos the two and minus sin the one sin the two plus I And I put another parentheses cos theta one same to plus sin theta one cos Okay, and Therefore in the product of two complex number using polar coordinates you obtain that the module I are The the module I was to the two complex numbers are multiplied to obtain the modules of every which is something we already knew, okay What about the argument? Here I recognize that this is cos of the sum of the angles good and here is Sinus of the sum of the angles therefore in polar coordinates the product as this coordinates multiplication of the module I and the sum of the argument and help my time, but Okay, I think that this is something you can Verify so take several complex numbers using this representation and verify that if you multiply then J 1 to n Z 1 Z 1 Z n gives you This complex number Okay, by induction you can prove this In particular we have that Z is raw cos theta plus I sin theta Z squared is all squared cos 2 theta plus I sin 2 theta and More in general e to the power n is raw to the power n cos n times theta plus I And time theta and this is known as the more formula Okay, and this is very helpful in the calculation of roots of equations Okay, quadrat cubic equation quadratic equation say take this equation This is an equation in the complex Setting not something you can compare to them. So we are looking for a number whose square is equal to mine Is whose square sorry is equal to I I is a complex number? well adopt Polar coordinate for Z and I use the more formula I Know that Z squared is raw squared Cos 2 theta plus I sin 2 theta Okay, and then this equation is equivalent to to this So that we have raw squared has to be one Okay, if you want you have to compare the real and imaginary parts to Remember that to any complex number is an ordered pair of real numbers. Okay. It is a one-to-one correspondence So two complex numbers Corn sight if and only if they have the same real and imaginary part So in this case I have to do this I consider that the real part on the right hand side is zero right Okay, and on the right hand side on the left and sorry on the right hand side is zero But on the left hand side is low square cost to theta on the left hand side The imaginary part is this row square assigned To theta and on the right inside is one So if raw squared is equal to one it has to be one Seen has to be one. There is no other choice and Cost to theta has to be zero so the only Possibilities left are theta equal to pi over four Okay, or T toa equal to plus pi So three over four correct therefore This the the two solutions of this quadratic equation in the in the complex number is So theta is either Minus Sorry Sure one plus one plus four is five right Sure. Yes, I add pi to pi over four. Thank you Sure sure and I'm it's important to remember that four plus one is five is no three Pardon me. Okay. I Don't listen your colleague pointing out that I made a mistake and I of course I Recognize it was a mistake p over four solves this both This two conditions, but also By sorry pi over four solves these two equations But those are pi over four plus pi and I forgot to to to add four to one to obtain five over four Three or four is not correct Okay Okay, check it. All right so last exercise as We solve this what if I have this To solve Okay So natural answer is of course one is a solution, which is a complex number. It is also a real number Okay is there any other solution in the complex numbers for this cubic equation with real coefficients a Bit different from the other but invite you to make use of the morph formula and we'll we'll see you'll see so we we can see say now that beside the Real solution one there are two other solutions Okay, and these two solutions are also conjugate Okay, so I think that I stopped here So probably some of you are It's really bored when it is first introduction, but So just to give you an idea what we are going to do is We have a set With some algebra very good algebraic properties We want to make some geometry and the knowledge on this set. This is the main Subject of complex analysis, but Of course, we are also inspired by what we have done on the real numbers and On the real numbers. We have extended the real numbers To the extended real line Adding two symbols plus infinity and minus infinity Okay, because when you are considering limits of Sequences of real numbers. Yeah, in some cases these are convergent and others are not and when they diverge to something Which is very big in modulus then you have to to use a symbol. Okay to describe this We have to do the same, but we have to add something to a plane Okay, so the next next task is to extend the complex numbers to the Riemann sphere and Then to define met a way to measure distances between points Such a way will introduce a metric structure on this extended plane Okay, and From the metric structure, we also obtain the topology which do not will likely be Good topology started from this we can then Study the theory of convergence in C and Sequences of points so anything okay, but this is this is then The project for the next lesson to introduce to extend the complex plane to a complex Riemann sphere and to introduce a met a way to measure distances Okay, are you all familiar with this stuff? Please confess you already know this very much Yes Are you all Okay, have you all studied holomorphic functions and one complex variable? Oh No Well, there is at least one or singularity some this okay in several complex variables No none, but me even in one Complex variable or in several complex in several okay, but in one complex you ever read So Have you studied harmonic functions? Have you studied Riemann surfaces? No, just just a little bit If I say maybe use transformation is it something you maybe use maybe use conformal map, okay But conformal mapping is more general Maybe use maybe use transformation. Maybe use No Outomorphism of the plane of the disc No, well just to know I will it's not it's not you can learn it Moreira theorem if I say more Mobius Okay fractional in them Okay Poles residues So the class is not very homogeneous. I have to confess For some of you part of the course will be somehow trivial, but I'll try to to add some some new Comments on what you already know Otherwise it is difficult for those who don't who haven't ever heard of holomorphic function theory Something in several complex variable Maybe I ask the other the other stuff what is better to do for you Okay, okay, we'll see Okay, we'll meet here on Friday morning same and then Next Friday, right? So not to not in two days, but in ten days Okay, because on Monday then you are full of lessons. All right Okay, I Think that we can stop here