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# Complex numbers and holomorphic functions        Embed Code
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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
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
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
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>
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 