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Complex numbers and holomorphic functions

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Automatisierte Medienanalyse

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OK some the let me start let me start with the class so the 1st thing I want to give a hand to you is a very short introduction to complex analysis so what is it about so actually read English English name complex analysis of tells that um In a fine way uh it's about uh functions from C to C so complex analysis have the it's sort of um the theory of functions sanctions from say F from you'll contained in to the complex numbers C it's not general functions like the German on dearly but it's specifically about functions defined in complex terrain with which a complex where you and so why do we um why do we study that well um it's it's interesting for its only there right it's also interesting and even if you're only interested in in in the real numbers real functions cell and let me 1st say something about forming complex analysis in general is usually considered a AM the really beautiful theory so why is a beautiful so it's really 1 of the nicest pieces of of mathematics essence of uh so many good properties hold so let me give you an example for the various good properties so you know what it means for a function f to be complex differential the the for instance the difference quotient has a difference Gaussian delivered in taking in you of respect to any limit to appoint in you and saw the surprising fact is that this is the same as that f is arbitrary many derivatives complex derivatives and there's infinitely many complex derivatives something which is certainly not correct for real functions you valued functions and also s uh has a power series representation local leave which is usually denoted as F is analytic I really and his local um power series of Taylor series representation serious representations and as we know this is not true for functions from r to r here where x have nice pathological examples like the 2 he to live what is that minus 1 over X squared here where the power of the Tennessee areas the where the Taylor series taking and 0 is the sanction 0 all coefficients are 0 on derivatives at 0 uh 0 so the Taylor series will be 0 solvent uh the coincidence set of a Taylor series with function is just the point we develop what this much nicer so for the complex case uh for each point you find a neighborhood where uh the function is actually their coincides with its Taylor series yeah so very good and at another property is valid has a locally defined as local uh primitives stem from 2 or so each point has a neighborhood that has a primitive here so have to say what is it primitive means f prime is s OK so these properties are all uh to all and uh not fulfilled in the case of uh functions from a 2 hour but they have they are true for a fund for complex differentible functions very surprising and when it takes a great deal of work actually least half of a class to arrive at the sphere and but it will be a few and offered us so that's 1 thing it's particularly beautiful and also it's useful the the and why is it useful while the various things where it gives the um appropriate that explanation the so what's it from explanation the 2 explanation mainly for various phenomena you know we have actually do we know some of this
uh already where uh my mean
why did we introduce for instance the exponential function and analysis 1 uh defined on the on the complex plane well since we want to uh take cosine sine and cautioned singed a this uh functions derived from that so so the relationship between x and the trigonometric functions is explained you know that it's nothing new I won't tell you this once again the here you to the i z is equal to cosine z plus i times signs you should know that and but their various things which you don't know by now and which also can be explained namely and 1 thing is what is sort of addressed here but where does that where as a function f have power series representation where the coincide so this is uh this is uh where quite complex analysis gives a satisfactory answer so where where does where is saying where's the tennis here's where is the power series power series defined serious this defined where does it converge 1 can there's some strange examples in the real which are hard to explain if you have only the real part of the picture but no I won't explain why not anything there are things like famous uh things like the fundamental theorem of algebra contacts this was capitalizing fundamental theorem sides run the the which tells you the polynomials uh and polynomial of degree n pairs and roots and zeros the this an the cover was uh multiplicity such that uh so linear factors actually such that as this function vanishes and so that would be bit that has many many proofs in mathematics and but uh the improvement in complex analysis is quite simple and nice and so we will cover events and also we will learn something about um the variation of improper integrals so I calculation coarticulation also in proper integrals which on the 1st side uh so what i mean the example we will certainly calculated depending on time perhaps some more is a function like this yet well defined as a real function no zeros in the denominator if you integrate 20 uh well if you look for primitive that will be hard but if you use complex analysis in the way I by was specified world at the very end of the class and then you know that is then you can figure that this is so low I don't know that by heart pi of route to for instance so this is surprising since you don't see any complex content in their in their pocket and I will that OK so let me say 1 more thing now would be no um what this class which questions are addressed and let me say it what I had good books and for the references well here's the 1 1 book I find if I were a student I would use this book here OK this is um limit perhaps do this on the overhead to prove in case the works OK well it works but it's or easily readable hurricane and now videotaping will have a problem you know and so you will find it somewhere but it's the it's the effects of switch on of select and so from the 1st
list Mr. system like this that's no chance listed in middle OK sorry the OK so um what we have there we have so this is a book which a if fight act and bosom from in German which has is that has a good pace it's fairly explicit and and clearly written so I would recommend it however it our class and it certain and a very important point or class will go on it would be on a different track and reason different from here however altogether this is a good book uh to have so then there other books of like like our forces the classic and so forth which you can find here also Conway's book is I believe this is a good source if you 1 really something which corresponds more to the lecture then I recommend the 2 items listed at the back of this and the 2nd part of this list uh the book by Hildebrandt just pretty much and what I will do here and or vise versa at and also cunning spare address book uh analysis tool uh basically does the same thing is not exactly the same but it's and it's uh more or less it does what I do at some point it it doesn't even what's abstractly these books these analyzes courses here actually do not cover a much more than the class what books like this uh all I will only cover I don't know the 1st time under 50 or well hundred 70 or 80 pages of this book is set to 0 like class so I don't do that much OK this much about references for the OK so let me know starts um was the 1st chapter of this class so this is about all romorphic functions so um since I did most of my analyzes costs uh in very complex setting and I don't need to start at the very beginning but I will remind you on the basic uh properties um we covered so say analyzes 1 we're abbreviated like this analysis 1 covered up covered following and topics them so so 1st complex complex members c and in particular completeness conquering so while the complex numbers completed what does it mean you have OK thank you have that is the definition and nothing with the supremum so then we cover sequences and serious sequences and serious in C who carry and declare convergence when carry so everything I did there um applies typical applied to the complex case and then we had um theory of continuous functions functions so all the continuity tests uh apply we introduced applied to um to the complex case here like the limit definitional upsilon delta or actually the inverse of open sets autumn um that all applies to the case of complex functions and then we had to um and also when it considered serious of functions hearing and um so particular question when does a series of continuous functions converge to continuous function functions well well what's answer when is limit of of a
series of continuous functions again continuous holy you should write the exam once again th yeah it's uniform continuity is when you need him um OK and then we this particular functions exp sign calls uh defined by power series we the inverse function of the exponential the complex logarithm which I denote with a capital L with all its problem x to so globally invertible I will
not meet these things right away and I'm a polar coordinates the polar coordinates and what else um and then of finally finally we had integration so we have had integrals also say f of TD TE for f complex valued so when S goes saved from a to seek yet defined as the remanent for real-valued functions that the Riemann integral and for the complex case we simply added up there in the real part and i times complex parts as women integrals OK so and in case you ever did not follow uh uh in my class my classes and there was this 1 into so many will will view was not in and the and I was 1 class last year OK OK it's quite a number OK which also explains why we're by rumors so soulful um at sorry and if you want to revise these topics I guess this source is good from page 1 to 40 or something yeah if you have uh if you were in a class where everything was real man I guess you need to go uh back and you can but you could follow the book by fight up was more of course you also would come to look at course notes of my cluster analysis 1 into so here I will only um gives yeah I will go only gives a short summary of those properties actually uh here it's to what we need so the 1st so we give really discussion adapted to our needs so 1st let me 1st say something about C and the geometry of complex multiplication so a text reminder on the com X plus I y notation and is good and it's appropriate here and so what do we have well the complex plane is the set of real numbers in what real vectors in our tool with vector space edition no and with a certain multiplication which out you know what is its it's if I . A B E with C D is said then so this right now I think of these as vectors in our tool then it's uh I it's a times C minus B times t and come makes terms AD and plants C. the right this is this is by definition a complex multiplication and we know that our 2 with these 2 operations is this field the scope of them few of the OK and no um the standard notation X plus I the why for complex numbers z uses the following thing so because we have the basis and not to extend a base is 1 0 0 1 that we want to denote by 1 and I some particular i is equal to 0 1 since we set for the 2nd basis vector that's all by this multiplication law as we all know i squared is should then be minus 1 related M and then the riots z equals militia I forgot thing which is perhaps and this kind of
payments for some the
OK sorry please remind you where to forget something um so the notation z equals x why for such a such a number of true written as OK now and use the bases X times say 1 0 plus y times 0 1 um single 1 I want to write as usual in the form x plus I y so what is it used well if you think about it no the local the it is the and the the OK so what's behind this notation z equals x 2 + I wanted to remind you it's the fact that and the embedding of the fields are tool every that would do the right down there sorry I should what did I do there this should be erased from the video hoops would see uh no correct limited to correct the are 2 0 yeah OK and I should write see there this is what is the field and this should be seeking that that's that's what I was missing else correct or k so if I consider if I consider them the map from our into c so if we have our Standard and times and map it onto C is about half so with plus vector and times as above right if they consider this matter that X goes to X the rope then OK 1st of all it's compatible so this might then say this is a field homomorphism so what does that mean it means that some the addition and multiplication operations are compatible so if I 1st ed all mounted lie here and then embed or if I 1st embedded and then add or multiply here I get to the same result yeah this is meaning that I don't want to define this really uh in a mathematical definition but that is that is really the meaning of field homomorphisms writer goes it respects the addition and multiplication yes so that means that for instance for instance if you take a and B in R and multiply them yeah like you get a times B is a real number and on the other hand you could 1st go take a and B as vectors in r 2 or in CEE so this goes to C and then it corresponds to visit uh . 8 0 In a 0 and b 0 over and now here what is a multiplication well if the same components and the D. there uh vanish then all what you get uh is the 1st entry 8 times well there is a time see sorry here's a times B E cell here we get a times B 0 on the other hand if you take this number and map it by this embedding that this embedding to see you get the same result in itself
so this here the achieve here commutes and same thing for addition but it's even what this institute component CN but uh this is something which is easily forgotten when you there used to be you know when you think about why you use limitations such as z equals x plus I what this is the real background OK so and the of mind you 1 iterations for complex numbers so what we have we have the conjugate complex columns you gates complex number well let me just draw a picture is this is the man and the conjugate C. by well is that this was x plus I why this is X minus Y as you know and this this relates to real and imaginary part by addition so if I ate z and the bar get tool real parts of C Net services C plus and similarly so if I subtract I get to imaginary parts so subtraction means the minus the bar home care fissile pyou remember the the so any of those who have not intended for class was 1 into it this is a new for anybody of you OK nobody dares to say to find good OK so you will know that and so on so we have a modulus the Taka uh see below the modulus rude of expect plus y squared and the modulus is compatible 1 and the other formulas z times uh C. by our mood also OK and and the modulus is compatible with multiplication so we have spent W times seem odd as w modern times the more home carry and 1 more thing the most complicated item here is the argument since uh there's an issue of the well-definedness um what is it so it's say consider a number of nonzero complex number then and you write it in the form say z equals the true very are mean let's take a real number times the to the I find this far larger than 0 and high in our then sigh in our called the argument it's the angle of rotation in this section the polar this is the polar decomposition is you know right so here we have an angle phi defined up to multiples of 2 pi and users the IR here which is equal to the the the OK this is the polar representation and uh phi here is called an argument so far it contains and argument and if you want to nail it down to specific number when you make must make a choice like always for angles and say I don't know what phi is uh in my mind is pi to pi say and then if you make this choice and then we call this the argument of see here for course the the argument of C and it's a discontinuous function all these problems OK so this is my terminology and now let me um well now there's a theorem for who theorem to state and I guess the best thing is there if I do this after the break so uh is anybody who doesn't want to break hello OK good so it's the donors depends on how you ask and so on so let's make a 10 minute break and let's see if it we can leave the door open as much as we can the
at and I think that the OK I would like to continue please get see this located so I'm impressed by the physicist since training told me how to keep the door open and I had to guess that this is an improvement and the severed also from OK this is good and out maybe also is 1 of us of a piece of information um theory about the exam and I was tool for analysis um we had been cells what ionization and there were various complaints uh which had to be handed in in writing and uh in a week from now next Tuesday I will bring along uh all these exams and the complaint so you can check what we did for them uh and I will also tell you what the over all the outcome of the result of the exam that was OK so if you want to see if what happened to you complain you can see it in a week from that OK so let me come back to complex numbers so the the OK so medication of complex multiplication a the is an R-linear map on not to 1 are true for instance an example if you take the maps to i z yeah what is been well the 1st basis vector 1 0 goes to the 2nd 1 I and II goes to the minus the first one so the matrix is as matrix the 0 minus 1 1 0 which is exactly 92 Q rotation net modification with uh i is 90 rotation as we all know and 90 rotation has this matrix and is a non linear Melbourne our tool will in general we may ask uh what distinguishes so will question would be what distinguishes distinguished uh complex multiplication from an arbitrary uh are linear map or not to from the from more among all minimal among the all linear maps from I 2 2 i 2 and this is my 1st proposition I list various equivalent properties of complex multiplications uh all let me start with 4 and are map our linear methods and say 5 from hot to to uh to and it has a matrix what's matrix kept away um so matrix as an analysis tool is always with respect to the standard basis uh you this is I guess the last time the the basis of C or a true soulmate treats this kind of sigh considered completely real mean right now the following is equivalent the following it is equivalent the a so various properties no style for instance fine OK particular property 5 could be it's complex linear is the linear that is I can factor out complex numbers not only real numbers so that means uh I fi of say w z equals W times files of said for all w said in C. you this is I assume this for real numbers W but here I climate for complex numbers if particular case of this is that I can stick . 1 complex number I himself 5 of i z equals uh i times the phys ed yeah I always had without mentioning I always use the identification of ah and to see yeah so this is perhaps the number not true but i times phi I mean by identification with I 2 is before OK so this is a special case when WE was I and this is sufficient for you can almost see right away by ah linearity you get uh you get back to the to the W so for all of this is for all in C. so this like guess is no surprise and then a equivelent to it it is that 5 is represented by complex multiplication it is complex multiplication that is so there this uh complex numbers say a plus 5 B of standard convention that these are real numbers um and see ways well phi of z equals C times the form of In equals C times and and I can also uh writers out if you want this is a uh so if have assigned to the effect of the solid is a x uh real parts minus complex parts my plus i times mixed on the Y plus b x OK and um she the next property is the when I look at my matrix a and I say anything that's right in the form of the I think this is better than what put on the web In a B C D E rendered on out like this if a matrix looks like this when I say it must satisfy that b is minus C and a equals the and what the Pompei whoops state equals T and but because of the equals minus B and this will be quite important later so boxes these will be coating remember equations in interested in a week from now so so in fact that means it is awful form a minus b b a this is actually the very much application matrix by uh C. equals A. plus I. B. you can just check take any of this form times X Y and you get exactly this after this 1st component the 2nd component yeah so this is also very obvious but it would be important and now they become to the geometry as announced that geometrically c corresponds to a dilation rotation well rotation dilation I don't notation dilation which is it to be a sprinkling OK so what this rotation that means Mary is the dilation factor lambda and a rotation angles the such methods in equals lambda times rotation matrix that In case so it's not a general linear map but it's a particular 1 namely uh as namely pretation relation and the last property of 1 to list in this that's uh fee is angle preserving however I write it out in the form of following form which is media typical way to phrase it he is the so called weakly this can be skipped weekly conformal an orientation preserving and by definition this is uh the following property what is the weekly conformal mean well it's angle preserving in the following sentence that the of w he of Z is k I do not want to divide over a norm of the w fuse that but I want to write it on the right hand side and since this is traced in real language sigh make about half of norm that persists after identification This is the complex modulus and now write out the same thing for w itself and I so angular would divide over z and w modulus but a write them out here which is all in the order w the services for all of WC and see if this is this refers to re weekly conformal and the orientation preserving as you probably know is to the fact that the determinant of is non-negative and did also whatever you like a this non-negative the maybe 0 0 so note that if a complex number 0 we multiply with all the matrix A 0 or whatever uh then uh of course with determinant is also 0 OK so let me a story the little should learn this so before I say something in her about proved let me just note that the angle all off 2 vectors so should they should make sure right if if uh so this is an of this WC on nonzero then the angle of w z so just think about a real vector space the angle is defined as a scalar product over lengths that was known as right and so if I am in this case I will be allowed to take these um these factors uh as denominators these here as well and then this property here then the van star uh means she's angle preserving means um the star means that the time is spent of deserving but the way it's written down is also valid in cases that the denominators are 0 right if I multiplied like here this is all can be also written out if C is 0 uh so that's why I prefer this form which is little is yet so this means so really uh meaning of weakly conformal is really it's angle preserving and is not only a preserving but it's also intention during at the same time which is not implied In an acceptable of an angle preserving but not orientation-preserving map would be uh C. maps to the bottom reflection in the x 8 since here is angle preserving but not orientation preserving just fiction and to reflection big success OK so it's a little tedious and to prove all of these items so I won't do that but me see all that we quickly so you solve this clause goes to 10 past um so let me not to to match of this and so actually the properties a 1 2 3 and 4 our straightforward straightforward in the sense that the worst the most you have to do is choose the basis and evaluate separately on the basis so 1 2 the these are called not hard to prove and if you have problems uh let me know and I'll tell you how it goes and actually also that 5 the property red it's about rotation dilation and that that implies the other is is also clear sense is just compare the matrices for instance you have these matrices has the property that on the diagonal of the 2 elements are there be a dilation rotation has the property that of the dying of the 2 elements are equal and we entered I know they uh the negatives so this is clear so what is actually the more difficult it is to go uh to state here what the and lambda are so for instance if you go from I don't know 3 to 5 because let me just do it the quickly what you need to do is uh the angle phi is just the argument can be chosen as an argument of the of the complex numbers C so I said the equals 2 arguments e and uh and uh lambda to want to see and then you're done yeah just check that this sort of light and OK for see nonzero and if C is 0 then uh just to choose lambda equals 0 and and 5 whatever you like OK so perhaps I don't write it out it's this comes from the rear polar representation loops and sorry upon component representation uh often also see factor is lambda times the lambda lambda times the 2 VI oxy and so this is just what you need to prove and into the to be c is exactly this matrix after identification so this should be clear now a little more difficult a little more difficult is the proof of this of 5 the of 6 year of the angle preserving property and actually I let me just decide for now but that I won't do that so 5 to 6 requires some attention since as a geometric property way it's done just let me say how it's done is make you consider to vectors z and w which are linearly independent and together with the origin they um span a triangle and no colored chalk this band a triangle I like this and then you look at the since the map is angle for observing I've only give you the idea check notes to see the proof written down in detail the fact that it's angle preserving means that the if I plot so I should this should go pets in some other direction and the fact that it's angle preserving means that say feels w and v of c and will have the same angle here and that means but you can also look at differences see this this vector here for instance the miners w and as so you can take any other pair like W and C minus w or beast tool and also is the angle preserving property and then you should check that all the angles are equal and then you see that you have similar triangles the and if you have similar triangles that means that in fact uh you have the the it is is by way of the other direction I'm sorry uh I'm sorry I'm really sorry the sorry I assume this angle preserving property I'm sorry and I derive uh but its rotation dilation right if it's anger preserving triangles are so while triangles uh mapped to similar triangles and if this holds then and then in fact we have a uh rotation dilation based a little more to say to that which makes it a bit complicated about the rank of the map and so forth so perhaps I will not go into that and so only say see nodes OK and actually we will not need to prove freely but I think it's nice it's this ties to know to have his interpretation and to know and maybe also say red uh consequences is that if 1 of his properties holds 1 2 6 in 1 through 6 quotes then the rank profits mapped she is I've 0 or 2 so the real for information here it's not the 1 year that it could be 0 1 2 it's never 1 there is a question please I OK it it's angle preserving but it's not orientation-preserving Pepcid cement of poor handwriting hey is left problem here and it's Yeltsin's if you and the orientation this this is 1 and this is by this gets mapped to war and I have uh I box which is minus I looked and so this this is right and this is left so as not orientation preserving or you just put down the matrix which is the 0 1 1 0 0 and you see that the determinant is negative negative 1 the OK yeah so angle preserving is a little more general than what we have is angle preserving and orientation-preserving which makes it little complicated OK so far so far for linear algebra the I the so why are we interested in all this stuff well the it depends on if you're aware of what the main message of analysis tool of so when I talk to teachers degree student I realize that this was much clearer to me uh printer you sigh didn't make make it really the so let me say this quite clearly so what do we what was main inside of analysis to it was that the difference differenciation it is the approximation with linear maps and most of the stuff we did there was a sort of a corollary to this effect here for instance the chain rule yeah you multiply matrices offer differentials and with 1 another area but that's the composition of linear maps no surprise all these other things like the inverse mapping theorem uh and the implicit mapping theorem I actually told you 1st uh in in the language of linear algebra and then I told you the nonlinear case and finally what else did we do 0 of the transformation formula for integrals for the determinant of the Jacobian yet again this is as I as a try to talk to tell you this is again a clear in the case of linear maps and so it follows for the monitor their case and what is the linear map and question it is the differential right so that is the that was the main the main message of analysis to just to make it explicit once again so why are we interested these linear maps will views of a differential of complex differential let's now this is how the differentials look like yeah in a minute and we'll talk about the differential right now so and this is why linear algebra is just the basis of an analysis the yeah it's it's more than that but for the purposes of our analysis here it's just a part in a particular case that the function agrees with its linearization OK so as to make this hopefully entirely clear let me uh remind you also see notions of 2 French ability in the following um there fewer this the following theorem it says where if you you is a set in the complex domain in a minute that will be open but now it's arbitrary is a function to the complex numbers and I pick a point in you and then I say um then In following I could lend the following notions of 2 French ability of our because of the land the In and it's also much of the in a minute um stopped whatever free notions of 2 French ability we used 1st of all the difference quotient has a limit solve the limit the limit C equals f of z plots arrangements have submitted over over the age and so I should like I forgot the limit limited H 2 0 exists 2nd we have a linear approximation property that as follows there was a which can be written as as f + as of the half of Z plus C times h plus the remainder term depending for each set for each said depending on age where this is low of a wage which means but we have to limit limited our estimate of h over n h equals 0 4 page 2 0 so of course here I need to use specialized to um admissible values uh for H right so this is engaged which in the search that uh and on stage is a new In all these cases there was a for these years to be defined enciphered property there was useful when we um memory um came to the but chain rule that's the the coveted only um the uh sense of differentiation which is uh there exists already exists the functions finally uh fire from you to see continues continues it the point z such that um well this almost this threat this bit here is replaced by find so we have f of the plus a which includes careful that class um see time the of that class in each time stage the and again I meet this convention here and uh these are all equivalent find these free properties are equivalent solemn distances so fundamental and it's a nice reminder from analysis 1 I hope let me just go through it it's so simple I hope that I can convince you of this now if so how do I get from 1 to 2 new well if I want to have this here this representation here all I need to do is write out what Ozzie of H ys namely set plus in which 9 hours of of said um minus C so that's prints write it down are that of stage because as well that was image minus and c F of the minus H I divide this by age the and this has a limit on the right hand side namely the limits see soldiers means that there is tends to 0 full stop OK so how I get from here to there to this statement about the function well I need to write out what uh fee of suppose H is it's at least as obvious nearest so all the of z + page a replaces replaces this part here divided this part here divided over age so it's uh SC plants odds yield image over H and since this is the limit 0 just means that the function years continuous it h equals 0 for it the point of the function the is continues it said corresponding to achieve the 0 right so and to get from we to 1 what do I need to do well I want to ride out this limit here by using this property the continuity of the hot what do I do well I'll write it out In the following f of the plus H minus F of singing which equals the of z plus h times h and I divide here and I can take is a continuous function I can take the limit it equals to 0 and this exists uh and uh I set this equal to C right the of the that the of C equals C right so is made easier it took a long time and the analysis courses but this is this is so I hope in most a nice reminder so all these 3 properties of the same difference quotient and has eliminates linear approximation and and the Carretero carteri version of the linear approximation and of course as only in the case that we are able to divide like an off few CV can write out the 1st the difference quotient In the limit 1 of currently the OK and no the let me tell you what holomorphic functions are definition of basic definition of this class is home care and was 1st and we know that already there 1 of 1 through 3 hold and then f from Newton's see is differential but context a frangible thanks differential that see and you current thing I gave that's nothing new about what is new here is the same part of the definition uh um well it's the following property a function um and sell to the decided side of this way if s um and the the in in C 1 function it is so is C 1 of you return the which with which I mean that it is it has a continuous derivative considered as a real functions so let's write this out here so II therefore has continues the continues partial derivatives the partial derivatives this system and if you was open and you it's all moved then uh me write out form suppose sorry suppose OK man if f is differentible at all times if you like a little more thick cell in if uh um right there's um than therefore is homomorphic if and only if f is complex complex differential different should look at all see and you OK so it's more or less like task is differential yeah weird and the real analysis course where could've distinction after venture point ever differential uh go without addition meaning that it's a friendship at every point there's only this all of assumption here well um this is an assumption I need and after few weeks I can raise so let me know to this so in fact what I do here contrary to them the Convention in many books it is I assume a little too much and later I show that it's a relevant here and sigh I talk about this extra tension that it has partial derivatives which are continuous yeah which is not necessarily readmitted necessary in order to state this um but I want to I want to have this assumption you will see uh soon while away need said here social let me know about the requirement requirement and f in C 1 if and C 1 and is not standard slow for instance if you look at this book here it's not a it's this is not an assumption being made that it's not standard but we will show there later we show uh um it's them it can be omitted the script yet so we start with a slightly stronger assumptions namely that differential that the partial but the partials office functions are continuous and assumption you've seen a lot in analysis tools and and uh that we want to make here as well and it will be clear next Tuesday way and it's a movement important assumption or it's it's a viable assumption to make OK cell examples examples but um what functions like I said to be an are different I holomorphic the use homomorphic well and then the well this is 1 of the homework or of a problem session questions uh to remind you why this is sold so I won't tell you right now and um f of the equals z bar is not holomorphic it's not even complex differential it is not complex dif fer should ball it any point there didn't see why this is so all of all let's look at the difference quotient the yeah what is this well plug like in this um the z plus h bar minus in bound over H which is uh what this Council sound since the buyer can be separated it's h power over age and was what is H power over age well if you take a real age then the bar does make a difference is it's warm for engaged in are and if you take an imaginary H purely imaginary of the type I y where the bar makes a minus sign so this is minus 1 4 H. in kind solve it means to be a limit of a difference difference quotient doesn't exist yet I can show I can give you 2 different sequences where uh um where a the limit is different now faces nonzero there whoops OK so so limited also as H 2 0 and here it's a complex limb at any age no matter if it's purely real purely imaginary what it should be true for any age should have the same and it does not exist of OK so 2 examples so this is a non will more in particular the non-homomorphic the bar OK and also we have consequences the often notion of holomorphic on amorphous city or of differential bill notions column of the city the inference in T is preserved under law all reasonable operations like under a sum of functions products products um compositions In versus well defined compositions inverses and we know the appropriate formulas for that side and you have to state the quotient will lower the chain will whatever here wins whatever rule is to differentiate him and all a number lost properties which I don't have time to ride out which is fed to them if taking the differential is a linear map that polynomials are of frangible and um well let's leave and also rational defintions of automorphic redefined and so that's the end of today's class and I hope to see you all next you say thank you
Nachbarschaft <Mathematik>
Subtraktion
Punkt
Komplexe Darstellung
Klasse <Mathematik>
Derivation <Algebra>
Physikalische Theorie
Gruppendarstellung
Reelle Zahl
Inverser Limes
Holomorphe Funktion
Funktion <Mathematik>
Leistung <Physik>
Mathematik
Kategorie <Mathematik>
Differenzenquotient
Stellenring
Arithmetisches Mittel
Stammfunktion
Menge
Flächeninhalt
Sortierte Logik
Koeffizient
Potenzreihe
Funktionentheorie
Ebene
TVD-Verfahren
Multiplikationsoperator
Komplexe Darstellung
Klasse <Mathematik>
Gruppenoperation
t-Test
Überlagerung <Mathematik>
Gruppendarstellung
Theorem
Trennschärfe <Statistik>
Inverser Limes
Inhalt <Mathematik>
Wurzel <Mathematik>
Analysis
Funktion <Mathematik>
Sinusfunktion
Bruchrechnung
Fundamentalsatz der Algebra
Erweiterung
Mathematik
E-Funktion
Rechnen
Ereignishorizont
Teilbarkeit
Integral
Arithmetisches Mittel
Polynom
Sortierte Logik
Tourenplanung
Beweistheorie
Potenzreihe
Trigonometrische Funktion
Funktionentheorie
Grenzwertberechnung
Subtraktion
Folge <Mathematik>
Punkt
Komplexe Darstellung
Klasse <Mathematik>
Gerichteter Graph
Physikalische Theorie
Überlagerung <Mathematik>
Weg <Topologie>
Logarithmus
Vorzeichen <Mathematik>
Uniforme Struktur
Offene Abbildung
Inverser Limes
Analytische Fortsetzung
Analysis
Funktion <Mathematik>
Vervollständigung <Mathematik>
Obere Schranke
Exponent
Kategorie <Mathematik>
Güte der Anpassung
Komplexe Funktion
Inverse
Reihe
Physikalisches System
Stetige Abbildung
Umkehrfunktion
Forcing
Menge
Mereologie
Potenzreihe
Ebene
Nichtlinearer Operator
Kategorie <Mathematik>
Komplexe Darstellung
Klasse <Mathematik>
Zahlenbereich
Basis <Mathematik>
Vektorraum
Term
Gesetz <Physik>
Gerichteter Graph
Integral
Multiplikation
Zahlensystem
Menge
Rechter Winkel
Reelle Zahl
Mereologie
Basisvektor
Körper <Physik>
Geometrie
Funktion <Mathematik>
Resultante
Subtraktion
Komplexe Darstellung
Klasse <Mathematik>
Iteration
Zahlenbereich
Drehung
Bilinearform
Baumechanik
Ausdruck <Logik>
Gruppendarstellung
Zahlensystem
Multiplikation
Reelle Zahl
Theorem
Stützpunkt <Mathematik>
Zusammenhängender Graph
Auswahlaxiom
Nichtlinearer Operator
Addition
Parametersystem
Topologische Einbettung
Homomorphismus
Winkel
Vektorraum
Kommutator <Quantentheorie>
Stetige Abbildung
Helmholtz-Zerlegung
Arithmetisches Mittel
Rechter Winkel
Körper <Physik>
Garbentheorie
Innerer Automorphismus
Lie-Gruppe
Aggregatzustand
Standardabweichung
Länge
Gewichtete Summe
Spiegelung <Mathematik>
t-Test
Gesetz <Physik>
Richtung
Lineare Abbildung
Zahlensystem
Gruppendarstellung
Negative Zahl
Kettenregel
Vorzeichen <Mathematik>
Gruppe <Mathematik>
Umkehrung <Mathematik>
Analytische Fortsetzung
Addition
Multifunktion
Kategorie <Mathematik>
Winkel
Partielle Differentiation
Biprodukt
Polynom
Menge
Polarisation
Rechter Winkel
Sortierte Logik
Beweistheorie
Ordnung <Mathematik>
Diagonale <Geometrie>
Orientierung <Mathematik>
Folge <Mathematik>
Logischer Schluss
Subtraktion
Wellenpaket
Komplexe Darstellung
Klasse <Mathematik>
Basis <Mathematik>
Bilinearform
Äquivalenzklasse
Loop
Differential
Rangstatistik
Reelle Zahl
Äquivalenz
Holomorphe Funktion
Abstand
Analysis
Erweiterung
Matrizenring
Determinante
Differenzenquotient
sinc-Funktion
Aussage <Mathematik>
Schlussregel
Offene Menge
Zeitdilatation
Resultante
Matrizenrechnung
Punkt
Physiker
Kartesische Koordinaten
Drehung
Element <Mathematik>
Theorem
Funktion <Mathematik>
Bruchrechnung
Parametersystem
Approximation
Theoretische Physik
Reihe
Ähnlichkeitsgeometrie
Billard <Mathematik>
Teilbarkeit
Dreieck
Linearisierung
Konforme Abbildung
Arithmetisches Mittel
Umkehrfunktion
Geometrie
Standardabweichung
Aggregatzustand
Gewicht <Mathematik>
Quader
Gruppenoperation
Zahlenbereich
Gebäude <Mathematik>
Transformation <Mathematik>
Physikalische Theorie
Ausdruck <Logik>
Multiplikation
Inverser Limes
Lineare Geometrie
Zusammenhängender Graph
Leistung <Physik>
Schätzwert
Quotient
Relativitätstheorie
Vektorraum
Kette <Mathematik>
Stetige Abbildung
Integral
Skalarprodukt
Flächeninhalt
Basisvektor
Mereologie
Normalvektor

Metadaten

Formale Metadaten

Titel Complex numbers and holomorphic functions
Serientitel Complex Analysis
Anzahl der Teile 15
Autor Große-Brauckmann, Karsten
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - Weitergabe unter gleichen Bedingungen 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nicht-kommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben.
DOI 10.5446/34035
Herausgeber Technische Universität Darmstadt
Erscheinungsjahr 2014
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik

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