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Automatisierte Medienanalyse

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Erkannte Entitäten
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so I would like to start and there is something to start with is a correction of the last class namely i if you remember what we uh what we're doing is we're proved uh Lobel decomposition on Laurent series expansion of functions which are defined on an annulus why do we do this while systems that we want to discuss singularities later the and I consider the in the missing inner disk the location of singularities so what we had last time and is a theorem on and a decomposition on the law on the long decomposition given by A will morphic functions so given a function f on an annulus a there was to radius little radio and and capital R. there are 2 functions namely um so if F is so orthographic to holomorphic functions 1 is gene which is holomorphic on and the outer of the disk is given by the outer radius of a is H which is the homomorphic functional be 1 over R so that if I plug in 1 over Z then they will be defined on the complement of the ball of radius the token so this is the decomposition were proved uniqueness basically by an argument uh using your those theorem and then I had to prove the existence and the existence works with um uh writing out the
function for some point in the cell in the annulus there's some intermediate point z we have
some point in the annulus that I should say and we write out f of said
is a boundary integral given by the previous lemma over the boundary was slightly smaller or larger radius of that it's contained in the given annulus and then we what we said is the 1st thing that will be the holomorphic function given by uh by this boundary integral the segment the 2nd term will be at this age of 1 over um a function and it's given by this boundary interval and well 1 thing I did in the last time it is so easy what is it what they it's the edge of the road is 0 well look at this expression here what you integrate in this and the numerator is w W times and F of Z now what is this for W equal 0 well it's 0 so you integrate up 0 so age of so is 0 something I did in the last time so if you use this
expression and define our function on a smaller in universe so the to the blackboard had a so we have all large and and uh the K with 2 radius R and I will carry a and now we have all formulas for some intermediate values of some no color um say that Arroyo and camera enroll location and our sanctions of functions are defined on this slightly smaller annulus while home or do we extend them to read the entire annulus well what we say is uh let's consider tools such as new life see if there's never color you 1 slightly larger than the other kinds of abroad only partially yeah let's look at a somewhat bigger annulus and let's look at the same functions g and h both retrospective blue and you move and with respect to the yellow annulus well and in the coincidence that rule and numerous they will coincide due to the uniqueness theorem you to part 1 so that means that if I started with a function with 2 functions g and h did defining the what I want on a smaller subset and is a unique way to go to a larger subset of larger annulus and since I can actually write out the entire annulus from hour to hour is the union of say countably many such an you lie there and I look at a unique extensions uh by part 1 of my formula so in fact these formula uh I won't work not only for small annulus but they defined function by well I wrote the annexation insect for the entire annulus so that's the correction and of last we the and let me know come switch from the Lorentz what what and
come to the conclusions and you have the OK so given these functions g and age of birds uh theorem which is still we a low MacLite legible the OK in which all OK given given this decomposition here will we would it'd already last time was right out the G and the h function um as power serious and carries out and a holomorphic functions so analytics so I can write them as part of a serious a Z to the end now for uh . B and W to be in In equals OK yeah i stop With 1 since age of the residual the OK and now writing out this uh is minus a sorry a minus in and this is 1 over z when we see that we can merge so this is valid for said less than an hour and this is standard for Dublin you understand uh 1 of our um less than or greater than 1 of our OK and so we can merge these formulas to become um this uh laws serious namely we have that a f of see it as a it is age of 1 of his in the 2 years said which by the combination of the most serious is serious running over all integers a and a minus 10 set OK so this is the the so called low all serious long series since which we already had last time the and the sense of convergence here you see here is the problem more 2 limits the sense of convergence is but we say such a serious convergence is both as if both the serious uh split up and they will converge and effect is part of the as part of this year and all part of the next year and will see that uh is uniform uh local locally this uniform convergence OK so the question is uh sorry this is this goes away was that the question it on there is my make this have but that this is the only 1 ambiguity we had right the choice of edge of 0 so it's up to a constant so perhaps the better formulation actually is to invoke this right of way up here and there is age with a age of 0 equals 0 and then nothing is ambiguous yeah so I this is this is here that that the 1 here yeah if I have and automorphic function which vanishes at 0 and I had um compute the pose serious than the zeroth term finishes right so that's why I and I have it here and uh so I must use this I mean must use this uniqueness since I claim that this expression holds uh no matter which radius roe I choose and OK so 1 question which arises is what's the formula I mean well how can I express in this is a reasonable question since if you remember what we did for of our serious which a special case of this room slide for power serious and we have this nice expression for the the for the a ends for the coefficients here's this this slide to remind you so that was a theorem that given cautious theorem that if you are given a holomorphic function its analytic it so you can write it out as a power series in effect and the a ends are given by boundary integral boundary over the rate of the boundary of a disk of functions defined on OK in case it's a closed there OK and let me also remind you of proof since this is 1 of the 1 of so well to me it always used to be a miracle what this proof holds that but it's so simple so let's let's just let me just remind you of that here how does it work well use the cushy integral formula which is at the ground of everything here and then used to express the uh 1 over Zadar minus Z term so here's kosher integral formula it contains 1 over zeta mu minus Z in the denominator well then use a expansion an extension of this denominator by geometric serious and plug it in you know so you get a serious here the just by a metrics areas and you plug it in to replace will 1 over zeta mean minus their and if you do that then you get into both of those so you get this year and by reasoning on B M 2 metrics serious you can show that this converges uniformly so uh so we can actually at least on on these complex faults which contained in the demand and so we can um actually um we can actually slipped the order of summation and integration and get to this formula we want OK so that was in short form the proof of a quotient of the air analyticity so all we do is crushing integral formula and geometric series and every uh OK so using this fact we can actually laid out in a similar formula and the case for all the coefficients of wall events namely the following and this for on OK so we have a holomorphic function but from the say and and units to see how almost think at then we can expanded into law into a serious such as the man uh so that 2 formulas here that's number of them matter if of so that is the sum of and said to be an n equals minus infinity to infinity and this holds for all to but the the is that in the end users and uh so actually I should decide on OK let's let's do this and that it some point PC and not it's 0 and then I need is said minus the year in that will be the general case and it's easy to transform OK but what we're curious about is what a B a N well the formula is all I could have should also say this was where this serious converges locally uniformly what steps will 1st to converges absolutely and locally uniformly absolutely means measures and and even uh modulus and locally uniformly means take a convex set within the annulus all their convex that you have uniform convergence yeah Her well OK then ends of moreover on moreover a river following formula and and n is given by 1 over 2 pi i it's virtually the same formula as in because the Firenze of their TB and uh grows say all the of as of the
zeta over it minus speak to the end of this 1 the same time and this works for any role you like uh which is in between and there's no interval of uh and so all over the whole yeah for the the each rule in all of my OK it's not that hard to see that this formula 60 independent of the particular where your role you played in why is that so well that kind this think of think of 2 different values so will 1 and wrote to still contained in the larger units and KY of these integrals the same walls the function is holomorphic Emanuela's so if I use courses into formula with the common trick now as I go forth and back between these 2 circles and have a closed curve and integrate over the domain of franchisal Moffat so that means a written in another war so the integral is 0 so that means uh I can actually yeah I can actually write the integral of this circle is equal to the integral OK so this is no surprise that this would for each well who did that and all the more yeah and on on the right hand well OK so uh the proof this straightforward so with 2 things to show 1 is the formula 1 then the convergence and 2nd uh the expansion for the end of the 1st formula is actually proven by the statements on the left blackboard so I can say 1 is just OK by what it did before or since I have I use power series expansions of the functions of oral morphine and by a theorem which is in still here and there but this film so here I have and I have absolute and local uniform convergence OK so all of those separate a power series expansions so that means that in fact if I put these together it's in the books equation then still have uh absolute and locally in uniform convergence and we have a serious flaw I missed out word convergence and the convergance absolutely behave absolutely in the current word managers absolutely and uniformly note the so that follows at OK yeah so convergence convergence follows from the uh power serious theorem in which is this have a number well here and 26 religions virgins follows from a Theorem 26 I have here a just considering the 2 parts negative indices positive indices separately OK so there's nothing to do initially for 1st half uh it's only formula to we need to prove OK let's do that mn OK and 1 1 way to prove that will be used the power series uh results uh directly um my approach here is I will actually that derived through using 1 so um I do the following here I consider here I consider a function zeta maps tool the integrand f of center uh over Satan minus B to the N plus 1 uh which uh all OK which are given and the expansion 1 for f of zeta here let's plug in the expansion 1 foot if of state and then I get the sum of say k in the integers of a k some and and state of mind as the 2 overcame minus and plus 1 and by something has this is the same as taking this here the and the raising that discovery labeling OK so we I have this expression here and and to this I want to consider for the state in in the boundary so that this is of the singularity so zeta indeed the role yes I will consider this function only on the boundary A. D. row of a circle you can so then this the series of variety this series converges converges uniformly uniformly column on DVD role synsets come compactly contained in any I in and the convergence is locally is uniformly on compact uh sets subsets spoken so that means that I can do the following therefore can once again I want to interchange the order of integration and uh estimation so let's look at that the integral of the to leave the role also say does a minus sign to the n plus 1 is a term which is what we need to integrate them now this I OK here I can plug in uh the sum which gives uh interval of this some of it can plants tend to have someone uh times say minus B to the k and and I'm missing entries in terms of k and since it converges uniformly I can actually have slipped here so this some of us and the integers I can actually interchange the order of the 2 and write this in here and effect this doesn't depend on the integration variable so I can actually uh relocate the integration sign here OK and so this is uh so of some of this uh identities OK and now let's look at this integral here well what do we do well here is a function uh which is uh so this is basically like the integral is that of the k over over the boundary of the circle about written so this means the only the only value of k as such but this does not vanish is uh cable's minus 1 here so this is 2 pi i for k equals minus 1 and 0 otherwise that and by the computation we did long ago right for K positive this is automorphic anyway so it's it's clear right away for k negative do the computation and figure that out OK so that means I can actually forget about the estimation sign and plaque in the particular value k equals minus well minus 1 and I arrives at in a n here and so this is a an integral of the zeta the a Jesus
inter wouldn't want to do also yeah OK I 2 pyruvates like this I wrote this out here you at times to look this is the very claim a n equals 1 over 2 pi of this of this you know what so this is very direct proof you were OK so far about the law of the all serious and we will actually resume um using it to the after the next section in the next section is something next subsections is something I am sort of feel bad about not having told you any of earlier OK it's Fourier series all and this it's not necessarily an application of a law serious about and it can be it can be done uh and this is OK so this is a this is something of a high high value in them physics and also if you want to solve linear uh linear pities and there many applications actually itself that's 1 thing I should tachinid so suppose I suppose we have a periodic function save on the real so perhaps it has a complex ideas but again it's it's would be what it can so meaning that s of uh T is equal to f of t plus 2 cars the for all enough OK so we have and a periodic signal with period with period 2 pi or something like that of come and all this there should actually OK welcome to this much I will show you pictures later can be ordered function not so what we want to do is write this out this function is a sum of the oscillations of sines and cosines to uh talking in real language or e to e to the i in x and y and t in complex language so um 1 2 so the goal is the goal is to extend the the expand this function in no s In oscillations and it's alright in a complex way and these are the 2 of the I n t which is is you know cosine and he plus I assign an anti OK it so surprisingly um from all serious and does this almost with almost no effort for us and however it does it in a various this a very special case so what is a special case well if f a complex plane when hear say of reals where if say all function defined over reals and periodic was period 2 pi and I want this function to the analytic and this is way too strong a to assume it's not necessary at all but if it is analytic then we can uh there's an analytic continuation to some neighborhood of the really exists and effect since the order here it's I can draw the same thing over and over again you and so that means I have a you have a strip of uniform if that is analytic I have a strip of uniform which on which I have an analytic function which is what which is still periodic it's not that hard to show you have the analytic continuation of a periodic function is still periodic so In fact I will just assume that my function is given on some neighborhood of real AIX's is 2 pi periodic periodic and analytical romorphic whatever you prefer yeah so I will consider various special cases and I do this In order to use the law and the loss in this so the proposition goes as follows so I assume that s is a holomorphic function from a string say as equals and here I write it out the man I write it out slightly more general so actually slightly more general than in the 1st version of the course notes so I assume an arbitrary SLU strip not 1 which is centered at the relay CSAs so here is the and so this set means that the imaginary part is where a is in between I of the points a and AB so this is my said this on considering and the interesting case for the Fourier series is the case that a is negative and B is positive then I have a strip containing real race but let me formulated is a like this since there is no difference actually so have a function defined on the strip taken complex values and I wanted to be an analytical model more and periodic and the other meaning meaning that influence z plus to pi equals that of and so all said in this that OK this is my assumption so if I haven't extension then then as it stands in Fourier series the experiments as a so called for we can have serious the French mathematician for the the 1st half of 19th century uh what is the point is that f of their equals the sum of a global In each of the I N z for n equals minus infinity verbs to infinity and uh this holds for all of the fall said and then ask you can and moreover these coefficients a and B are given an answer I know but uh as a following by the following integral and interest and take that a little easier to just to make writing it out a little easier so let me just assume I'm in the set in this setting that they have real x is contained so and in that case I can write out a n equals 1 over 2 pi the integral of uh from 0 to 2 pi of my function f of t which is now defined for real t and I multiplied with the oscillation the 2 reminders i and t the t OK and I consider unfortunate and run out of space there uh so I consider how they considered as integral while this here tells me it's an average right it's 1 of the length of the interval so this tells me average is the easy part and this I consider a weight so I can take so this I consider a away so it means that I integrate up but I Average the weighted functions and weight I use is almost the same here except for over minus signs of minus sign makes a gives a minus sign here if I pull it out and all its sign function calls even so nothing to do there OK so this is this is the claim so I can write out the quantum physics I would say I can write out my function F as the superposition of waves right you my
superposition means just the sum and wave since these are oscillating functions OK the so an proof to so I have it's almost OK even tell you the idea when they have to carry it out so what do we do that we transform ever transform all of f onto enantiomers or more meaning transform the domain of various distress so transforms the x this will be correct on many others and and a standard and if by long serious there war serious act on this and use this is this is the entire idea so why does this work well by period this peridicity if the uh in the map this structure in a way that it's uh covers the annulus multiple times then I will still arrive at of well-defined find function that's the idea so let's do this now consider In the following semantic Web which meant is good for for mapping strips on to analyze the quality exponential but no M observed the this is a horizontal strip between 2 imaginary values so we must and look at it the exponential we in Keegstra sector of my late so if I met if I consider this map what is it do well let's do graphically here if I have here from my a 2 I've Beaminster's S and method under under that maps to each we isolated what happens well I get to the annulus the when I get to an annulus but are due to the eyes here the order the inner the smaller imaginary value actually maps to II a which is the to remind us a and the to remind us a is larger than e to the minus B so this is really the larger value is equals e to the minus any the and a smaller value would be the smaller value would be r equals e to the minus B right just plugging in new e to the I I a is e to the minus and the and e to the minus a is b is larger than a given the 2 minus a is larger than will minus B OK and actually since since any z in here and is that here's a here's the to the i z modulus equal to be truly on imaginary part III to VI imaginary part the right it's a real part of z + imaginary part of Z which is in fact and missing on the were you have the book and and so what is this well and from what it was we only have to look at the real part which is e to the minus im said smoking so that means that in fact fact and so e to the minus according to always struck man is exactly in between these 2 numbers so that means that this strip maps onto onto the annulus that aa on so then no can it maps on MIPS that the strip the tool while sexually onto meaning subjectively to they are that the length of our I define this here walking and now what can I do with uh with my function F will now this is done in a way that that actually my as defined here but being periodic with respect to 2 pi addition well such that its use gives rise to well-defined function here since an increasing the way you by 2 pi here means just turning once around so I get a well-defined function on the annulus by setting so claim now the playing there exists say to distinguish it led to a word is capital R. cattle from a part to see a function defined on the annulus wave the world such that at the image point is in my given f it's an at the at the original point so f infrastructure f in the annulus we can lefty finally annulus OK so what you have to check for this well 1st it's this is instance I don't define f of an argument i have to show this is well defined and uh actually I claim that S is horrible stick the well it's either gonna each who watches all she tag it who cancel was that 1 indeed f is defined by the f defined but affected what if they have 2 numbers such that E to the 2 numbers z and w such that each whereas because each we I W then this means the head by the uh since we know that he is 2 pi i periodic this means that Z minus w is uh into is much integer multiple of 2 pi and if that's the case then uh by pre-orders city of F so there it does matter assigns a WOZ here I get the same value by the uh theaters cities saw by period this city punitive that came up I get that f of said equals that stuff and so that means I get the same right and sensible left inside as well-defined OK and 2nd F 4 what taken it work so when 100 see that's well we plug in the logarithm here and then you see that after sitting a hazard to the logarithm of W you see that and section f of w and if you have the logarithm yeah and and logarithm is a useful mosaic and the composition of almost a functional form of the factor is the order a solemn offic sorry and so that everything is fine except for the room isn't isn't defined right have globally so a way around is well we don't need anything global since we talk about a month this cities so actually local argument is fine so I don't have to say which others might take I only say that I can uh given this function ii it has a local inverse why does it have a local inwards love the exponential has full rank right it always has rank 2 so I can inverted locally by the and by the what is called inverse mapping theorem right so long to say this uh um well OK that maps to each to the i who said it is locally invertible invertible since uh since the derivative is always non zero so say by a function which i colleges kappa capital L looks like a logarithm that is not a logarithm and then I ride then I can write that s also uh w so is equal to s also all apply placarding easy to the L quotes W and this is just by my definition uh where is it here and it's F of S L composed of composed of their own on w and actually is locally inverted when I missed out the most important words here with a holomorphic functions right it's uh by the inverse function theorem the inverse of the holomorphic
function is uh again if it exists at all lost at this Slovenian but l on the wall 3 so on who OK and so that saves the item and this is holomorphic pedestal morphic postings lotic 5 so 5 when the goods so I I have do so instead in place of my function f mapping mapping to the uh to the complex numbers I now have my friend Capellanus doing the same thing such that this more commutes and now I can use cues that on um preexpansion slash so formats so what they get a w it is equal to the sum from minus infinity to plus infinity a N all w to the n who can all that which holds for all that we do again the annulus OK and so what I have so and now we write what my w is that I get the desired expression for the left name these that so all your money and I get to that effort said the quilts well would have to do f capital f of you to the z according to my definition and if the plot this then I have some of at the end of the 2 the I n Z and this is the very running over the integers and this is the very expression for the blind which we can see right now which I claimed hopefully yes right and now this works for all it is in S to the er who so it's simply taking the last serious here and transporting it overbear or vise versa uh which gives you the Fourier series and now i have to verify the formula for BA and and when we do this well it's time for a break but let me just do this please and so what the formula I want to use is here so let's plug plaguer then what do we have well all we need to do is take my w of the to the and i t there so so using tools all I get the following so or little keener OK so I get to the that F of the VIC ANN sorry it's more than 1 computer OK a n is this formula with I applied to capital left which is what it long serious fossil again this formula just for cattle if we into bill of uh OK and now I can In under the additional hypothesis that a is negative and is positive this animals will contain the radius 1 so I can sit at a radius of 1 year and it's about 0 so this is complete and it's f W over just using W here w minus 0 so uh all w to be and this 1 GW uh and this is now plugging in w equals e to the i each reaI center I get 1 of the 2 pi i integral over from which it can now right it is in this goes from 0 to 2 pi thank since has F over from the from the to be t over to the I n plus 1 times c prime sky to be on the derivative of each realities I and times E to the i IT D T and now this nicely Council uh is 1 factor here cancels the 1 and be on the occurrences lists on here so if everything is so I'm left with the 2 we I n t in the denominator which is an E to the minus i and and is hopefully exactly what I claimed eaten 1 over 2 pi F. F. society which is half of the to the t which is a little if of the times the to the i to the minus side T. t which verifies the formula for the coefficients and saw after the break i will discusses a little life 1 question before is Sebastian students here who is so no OK to this OK so let's have 2 minutes breakfast yes so I would like to continue solve and if they see my program forest of today is I will explain you few facts of a decomposition hopefully extension and by way of course you is familiar with the 4 year and willfully fully a serious of periodic functions OK what what did you do it would tell me signals yeah yeah so you did this in physics so lose computer science OK and the other somewhere in the background same think yeah OK yeah and mediate merge OK so it's probably a good idea is um I mean this is the complex version of the function f goes from the strip to see and uh interesting cases that really makes is container Mr. so uh the real version the real version and can be derived from it uh so far exceeds that of F of the assuming that as low as safe the floor plan and from the real to the real 2 pi periodic can um instead f of t can be written s some I use greek letters in order to distinguish them from the a but usually this is also an a a naught and assistance will be in a and so on I can make you write it out as call science and art and science are and how in and is run snow old from in units 1 to infinity and perhaps you place in and you take in there actually the anal Watson's cosine of 0 is 0 this works fine and a friend who knew have fared OK this is the claim and in a minute I show you why it's true so the zeroth coefficient is the average of a function and all the others are again weighted averages so 1 but OK not quite averages HT 1 of a kind reverend 2 pi F. of T and now it's weighted with cosine and and all of us all data in waste on 1 over and with sigh and tease him OK and these formula but are a direct consequence of those in makeshift written up to you but I'm too lazy to do this on the board why this is true it's basically a so the like a that I linear Trajan 2 variables so uh this is again in the same plane so let's look on the uh on on the device yeah if I write out the form of a complex the formula for a in the 2 pi times 1 over 2 pi times the weighted global test and I plug in what the 2 i and T is written that it was real functions called mainly because I'm and t-minus i times sine entities or then I realize that my given coefficients which erode also board a friend and
beta and come up here so we s real real valued they come up is the alpha and and minus phi beta in and shake it's a half since here observes that vary in the complex formula I get 1 over 2 pi but here has uh 1 of applies so that explains effect a half uh in front of us and when eye beta and for a minus end for real formula have the same I do the same thing I plug in what b to over plus I interiors there we only differences but plus there in front of the I signed and so I get the alpha and plus I beta in and then I write out the uh whatever instances the box formula here a force that is equal to some new to the island so can't and plug in I write this out separately for the positive and negative terms and doing this and plugging and plugging in other words each we i and he is and what a man is according to that formula arrive at this line here yeah this is the a formula this is low formula and the cosine in science come from a to B I N T E to the minus identity and now if you recombine you quickly see that you arrive at the desired expression namely the all the real real parts that are given by alpha and times cosine and all by minus 5 squared data in times signed and which is this term and similarly so here at the same time actually so up a tool perhaps go away and I get this as real part imaginary parts damage due to the fact that his minus and has a plus the red and they're all the same thing and same thing there so uh by this super computation you see but that yeah I don't know it's slower I don't have what if you have any inside why this is a good computation tell me but I'm not convinced up to this moment so In any case you get the real version which is with what's all use there is that f takes values in the reals right in order to do the decomposition into real and imaginary part so if you have this you have the now you have 3 real version of it and in fact this so we have so you can read it as follows you can say if if f is given I defined the alphas and betas and this then I get all the following decomposition of S into multiple frequencies here well OK and in fact this is valid in much more generality than for f analytically and I should point is out but I will not prove it here so this holds this means that if f is given enough at the Alpha Betas of this then S this this or say stuff it's this a good way to and then start downloads to yeah for and for if much more general than an analytic namely want hadn't clever way that against of saying there's namely air for instance for continuous functions or continuous with uh would finally many sections so how our how others stressing a holds more much more than generally much more generality in much more engine the T here for instance at for if it continues but also which I will show you in a minute that's what 4 and continuous continuous with finitely many exceptions except finally many points or and was some experience you see that these will not really be the good assumptions actually in a way and this decomposition always works well will then ever these integrals here defined now when these integrals defined well you talk about this the next term in integration class uh but basically there in fact that we say is not a source F. Strayer indigo ethenyl to school integrable and what is precisely means uh and as we will see then we will see in the force to the things he explained it will functions are more the important and not the probability theory and quantum mechanics everywhere grade was a year so in a way whenever these are defined and results so they're in and so this holds in but in much more generality than analytic functions and I have some examples here it's what well it's good to see the real test is 1 of here in the Earth and Moon huge good what good is an example of
a function OK and these computer examples actually uh I do not consider that a fundamental demand from 0 to 2 pi but from pi minus pi to pi but all these integrals are the same just by shifting functions there is no difference OK so here is an example of a function f which is constant between 0 and minus and 0 tools minus 1 and 2 plus 1 between 0 and pi and then it repeats since its chaotic with minus 1 and so forth so within the uh uh section which is this in the light blue you see a fundamental domain uh length to pi and now what we see is now let's look at the sum MSE where I go on the uh 2 1 2 3 and so forth step up is after step so the 1st the 1st n o well 1 thing you can see right away from the center the 1st thing is this was what is the average of a function was they were near so the 1st step will be the ethanol it should be 0 the alpha 1 OK here we have the F 1 is a and yeah it's a sine function exceed cosine functions due to an all even um consideration will not show up for this example so everything will be in here so science time assigned times 1 key and this is the mode and purple and no so this is the 1st this is the n equals 1 term is goes away at attractions should point of this is beta and data 1 yeah so this is stated in terms of up to n equals 1 and now I go on the 2nd term in serious invoking the uh uh well actually it invokes as you can see where should be added to use the where is it will be here in this year that down there are 2 hrs it should use formulas new formulas uh are who it looks different I checked is in my area and Austria's however lose higher hope His blows for the time saved cocaine so actually this is the formula and you don't get a frequency tool term uh a lot use due due to some symmetry considerations he actually serve to fit to lift the sine functions that's cool yeah I think of what you need to do get in order to get closer to the function well with assigned free x is function is good since it pushes up uh pushes up function where you needed yeah so all its of x 2nd term wall of 3rd term actually and now uh we can still keep on going uh actually only have frequencies CFI 5 5 times the 7 times and I think the last 4 we have is 9 times it's still warm I am OK so this is how much we computed and you see that the approach function pretty much here by the sine functions with different frequencies and the the correct amplitude so intensities near and this is already a function which is not in this category but it's here continuous with uh or sign at the main exception points in the fundamental domain so I forgot to write this down near in India 0 to 2 pi arranged Soto discontinuities here and it's the fact that the Fourier series what it does at the time actually use uh mouse and what it does at the discontinuity is it maps to the half value so here minus 1 plus 1 over 2 which is 0 so in the in the limit it will stay here it's in OK however it will always overshoots in is a nice there's a nice theorem telling and that it will always into overshoot with the same amount no matter how high you go there is it smaller some some monster anyway so this means that such a signal that you are constant to minus 1 0 1 and is this is the sum of minus the masses sign all yeah in fact sine waves so here we have some more functions is sought to know where yeah where our church go back on it's own it's all the blue graph indicates a sort of so linear increasing from 0 2 pi over 2 2 times and again you can go into this zeroth mode is rarely ever rich in this case the ever which is section on 0 it's this is the zeroth term which she wasn't visible down there that's you no yeah i will being and now we go to higher modes to x so this edit now like what's written out there 3 E not for i in the air quality only eating precancers some of them you do since since evolved and again it approximates know pretty well go use their already with uh an a turns and again you see that it averages out be step of buying cheap this community of function and metal and it also you see that it always overshoots is could use all and you can also use this also if you like to you use the complex uh expansion which would look like this in our in the present case yeah it's it's the transformation which you see on this slide but which is carried out here OK and what any function would do this is assigned a example of its land she will engage users OK which is piecewise linear but not had not so symmetric and the zeroth mode and see already and those who know what those savings not see the formula for more nap sheets work Schnall and now let's go to years since then you know nastasi fast bouncy everything that I right and so this is finding said let's go straight line and you see the same thing so as usual manager Alex to come it all and mathematical programming uh want me to do this but for me the Fourier decomposition is a completely obvious you to across states right in across states if you will uh um at a natural sound using by string or if you miss a whatever yeah as saying that it's actually it's actually a composition of uh of uh my but you can think of it as and the composition of sine and cosine waves and fact this is what I want you that's since How would you figure out was what intensities are renew and physics the when you want to experiment that you do a resonance experiment here you have a starting yeah like you take a general interest the pedal or whatever you know and user testing or whatever right and check out the intensities alpha and the beta and and something in the year of our little resonators and in fact what we hear can hear from the way on the alpha and beta bands in a way it's actually a little more complicated so but we hear such intensities and in our brains brains we we actually do this formula here so uh and if you have any experience with music and Acoustics this is a completely natural natural thing but it also comes up and in many other uh and many other things never for instance solving PDE linear PDE is only and some
other stuff yeah well and then on
her all or a few shifts onto alternating current here you 1 there to decompose this isn't a currency signal OK so and this is this and this is the time for a sequences zeros and will come back to that and in the integration class action will come back to it all in the setting that appear but the signal is non-periodic so acoustically if you do any non-chaotic sound like this and it also has a uh a decomposition into frequencies however these frequencies are not integer multiples and there but they are continuous 9 and that will be a covered in the source term as the Fourier transform now and it's much more general as is OK this 1 or more 2 or more things I wanted to point out which is what is it uh 1 thing it is on
yeah 1 thing is that actually the proof I gave you for for a serious tells you that you can think of the in-laws serious expansion for a and uh as a for a sequence of the circle yet proof I gave tells you that an what else is there to poor knowledge um well let me do let me do some historic stuff whereas this land New year and so on let me take a quarter of an hour uh in order to tell you why the Fourier series of fully decomposition uh was a revolution in mathematics knew there was actually it's it's 1 of the biggest things which will enhance been so that the biggest impact of any discovery in mathematics and it's not quite obvious so or even a tiny little about the historical impact well talk learning and I think I should have and it actually changed all of analysis so the 1st thing it changed is the sense but the notion of a function the yet what is a function well in X lives to f of x uh this is what we have what we teach you right but uh what did all and say well 1 has said well Bonner said and as function is an analytic expression do and what did he what did he mean by that well he means things like the sign exponentials or x squared plus 3 times x q so it's the means a closed expression and it's all this is no well defined now and because expression also includes uh serious including infinite serious so if the exponential is an analytic expression or not you have on 1st and perhaps only polynomials count but then you realize the power series play such an important role in in order to solve simple unique themselves such as the exponential sine function so sadly on needed infinite serious and so on so far so good but then why worked on and the vibrating string what's a problem of a vibrating string as well the initial so here's my string and I want and I want to produce a sound by uh taking by prescribing an initial values so I would take a string with my finger pull it up and perhaps to such a shape and then uh let it that release it and that the sound what happens well uh is is this initial configuration an analytic expression level it's not right it's defined say from 0 to something it's linear and from something to be and it's also linear but it's 1 of these functions they you'll and are endowed if it's a good function like the sign and the like I have a slide the sign know how like these functions sign of text in and sigma bond sheet which is piecewise defined for negative values to 0 for positive values you similarly so of a vibrating string however once you later vibrate it's a very nice sum of sine functions sines and cosines when using the Fourier decomposition depending what sometimes say it's bent right so actually appointed I think 5 5 or 10 years after giving his 1st explanation entrance 1st definition in quotation marks the he changed to a saying well it's it's what I have here it's at that uh we have a variable and assign a value and words he did this with 9 2 1 slide the you know 8 European Media it yet so his 1st definition was a function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities here of course this should be enacted in and of itself analytic expression is the key phrase here and then after that after a couple of years OK 748 1755 after looking at the vibrating string he said and now it's very vague and certain quantities depend on others in such a way that they undergo a change in the letter change and the 1st hour called functions of the 2nd it is this variable assigned to value this name has an extremely bored in here now extremely brought character it encompasses all the ways in which 1 quantity can be determined in terms of others yet so he's already very general without saying you take a real number you know what a real number is take a real number and and assign a real the real value to it or complex or whatever that this was the change so why did to fully change these change the discussion well fully uh right 48 claimed that he could decompose any given function into a fully sequence which is not correct but he claimed that certainly he realized that you could decompose functions as this 1 and it's a periodic fashionable once I've shown you but discontinuity into therefore for sequence so yeah so sorry claimed so in in any function who any function as a Fourier series of what did College for I guess but you know this is what this is what he claimed he didn't say what any IST but he he said that it's at least at come it that in comprises also discontinuous functions I guess in this this is nice Wikipedia article history of functions where uh this is discussed nothing uh such in Germany and so I think he said yeah he here it says uh had a general conception of a function which includes functions that row neither continues modified by an analytical analytical expression OK so there was a need to go to a good definition of function and this is the definition of function there we still use up today and which was I mean this definition actually was made clearer by um dirichlet during play an Lobachevsky by usually Lobachevsky 1 of hyperbolic plane uh these are usually uh quoted a it's making this more rigorous although oil already had realized it but if you think carefully about it this is a real big problem and this is considered this is not I mean why do you make definitions in order to conclude something so unique properties like 1 of these 1 of these words which change after the Fourier after 4 years this a discovery is his sense of convergence and
what is the problem here well you have a so always I was happy with His infinite serious since if you sum up single-finger exponential if you a sum up the x to be in 1 0 and vectorial and x to be an nice functions you get 2 nice function exponential is continuous differential those moves and entity whatever yeah however with the 1st sequence something actually happens right if you think of if you think of the rectangle functional sought to function and think of its fully decomposition and when you have you have a fixed say make it easy it and signed a and and and these are these are the values Nice analytic functions so up to infinity so here and this is where each of them I should say each of them not the sum sorry each of them this is sort of key point here so it makes it up and the whole thing the sum is not intended any means not marketing it's discontinuous yeah and so this completely I mean there why that would have been shocked here and if you got known to this although he had some glimpse of it with a vibrating string already got this is really bad yeah nice expressions here in something I believe here how can it happen well you need to study you need to study more carefully and this is actually what we did in the 1st term yeah we had we distinguished uh here and so what you need to do is distinguished point wise uniform convergence on it uniform convergence and actually you need also I mean the same same issue arises live continuity and remember these variance now on a compact domain a continuous function is uniformly continuous and so this all so what made all other mathematicians busy in the 19th century and up to last us uh uh and led to the if you like and related to island delta arguments in that that some which are mostly by a stress and made popular uh this actually um so cavity uh work of many mathematicians in the 19th century and end it actually led to rigorous analysis here since this example shows him if a power series everything is fine yeah right hand sides analytical left and also but for general theory not so inverse the Fourier series of examples and issue here is uh well it it's the sense of integration right in the alpha and beta bands identical so whenever is defined so on the big question is what is an integral or which functions undergo and M I mean it may be the it's from physics point of view you think you're creating all the functions I will ever account as if this is a good functions in that is these examples and so I do need to worry about 1 of those pathological examples mathematicians come up with you know like the characteristic function of q or were actually was examples but this example shows you let a least once you're willing to take infinite sums you you can get easily in into trouble and so the integral she took 100 years to get a properly defined from the point of the story on itself so basically I mean the Riemann integral is fine but is the so and their 1st term will question is whether the functions for which the fundamental theorem is true right we said it's true for continuous functions being integrated sulfide but it's too much more general what is the good class to work with uh yeah this is not answered by remember and actually uh this is you need a debate in theory so good theory of integration of square integrable functions is nicely coupled by what he did and this is why I don't know but how about the same is this is in between 1900 and 1920 or something yeah so this led by uh uh the Fourier series are in the 1st the 18 and 10 20 what does this mean to be an entity Dinesh no year OK I don't know it exactly but it's the 1st decades of the 19th century and so and actually before a sequence is a serious um uh make clear that mathematics could not be continued in a way it was done up to that time which is sort of uh the physicist style of mathematics avatar just do computations that everything works out nicely and well but some rigorous ideas must come in OK so this as a historical remark a is 1 more thing I want to tell you about it says look as if it's a good idea to start with itself again and then again let me a closer look at earlier thank you for your attention to this fact the quack
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Metadaten

Formale Metadaten

Titel Fourier series
Serientitel Complex Analysis
Anzahl der Teile 15
Autor Große-Brauckmann, Karsten
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Deutschland:
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DOI 10.5446/34036
Herausgeber Technische Universität Darmstadt
Erscheinungsjahr 2015
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik

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