Laurent series

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Laurent series
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Complex analysis Functional (mathematics) Theory of relativity Einheitskugel INTEGRAL Real number Power series Moment (mathematics) Interior (topology) Integraltafel Holomorphic function Scheduling (computing) Power (physics) Proof theory Radius Term (mathematics) Autoregressive conditional heteroskedasticity Line integral Mathematical singularity Theorem Integer Fundamental theorem of algebra Limit of a function Social class
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Statistical hypothesis testing Difference quotient Functional (mathematics) Divisor State of matter INTEGRAL Interior (topology) Multiplication sign Equaliser (mathematics) Lemma (mathematics) Annulus (mathematics) Parameter (computer programming) Student's t-test Derivation (linguistics) Positional notation Well-formed formula Different (Kate Ryan album) Term (mathematics) Boundary value problem Theorem Circle Analytic continuation Extension (kinesiology) Set theory Social class Concentric Curve Analytic set Staff (military) Cartesian coordinate system Integraltafel Numerical analysis Radius Line integral Order (biology) Mathematical singularity Right angle Resultant Spacetime
Difference quotient Complex (psychology) Mass flow rate Addition Length Differential (mechanical device) Gradient Correspondence (mathematics) Propositional formula Parameter (computer programming) Inverse element Mereology Subset Group representation Estimator Mathematics Sign (mathematics) Plane (geometry) Many-sorted logic Positional notation Different (Kate Ryan album) Quilt Convex set Extension (kinesiology) Compact space Link (knot theory) Power series Infinity Term (mathematics) Integraltafel Time domain Automorphism Entire function Proof theory Wave Order (biology) Mathematical singularity Quantum Hill differential equation Right angle Sinc function Point (geometry) Functional (mathematics) Existence Lemma (mathematics) Annulus (mathematics) Amsterdam Ordnance Datum Drop (liquid) Infinity Theory Helmholtz decomposition Well-formed formula Term (mathematics) Boundary value problem Modulform Theorem Nichtlineares Gleichungssystem Analytic continuation Maß <Mathematik> Set theory Uniqueness quantification Neighbourhood (graph theory) Latent class model Limit (category theory) Sphere Numerical analysis Holomorphic function Radius Logic Maß <Mathematik>
Statistical hypothesis testing Point (geometry) Functional (mathematics) Existence INTEGRAL Multiplication sign Lemma (mathematics) Annulus (mathematics) Insertion loss Inverse element Parameter (computer programming) Goodness of fit Term (mathematics) Different (Kate Ryan album) Boundary value problem Circle Set theory Addition Weight Uniqueness quantification Prisoner's dilemma Integraltafel Arithmetic mean Radius Vector space Order (biology) Mathematical singularity Moving average Maß <Mathematik>
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OK hello I wish you every new year and I hope that the complex analysis will contribute to happiness and we so then what we have today uh is a quite a slightly different schedules since so I want to attend the seminar talk in Frankfurt and couldn't lose control in the class uh due to the break so today we won't have to break however we have the of relation instead and we will have the evaluation at the end so on secretary speckles will come and uh will power handout the sheets and will also collect the sheets in the end but that this will happen as it did earlier than usual so we uh I think you will be done by a a quarter to 4 10 to 4 something OK and so there are some questions I will put this on later the so what is where race uh what have we achieved up to now and well if you look at what I wrote out for the introduction then you see that as much of it has been covered already so what has been covered that is his main theorem quoted here the readable uh that a complex frangible met is at once complex differential members infinitely often complex differential it is analytic has a power series it is uh and locally it has the primitive And so this this was all done in terms of the course she integral formula ultimately in the course integral formula comes from the uh comes from um discussing um what and in our closet came from discussing uh line integration OK so then vendors some of the stuff you know like uh um uh where do uh complex functions agree with a power series OK this is done uh I always do real functions agree with the power series well extendable it's up to the moment you hit a singularity and this is the radius of convergence um we also did the and fundamental theorem of in terms of the unit sphere and and attributes 1 thing I still want to show you but don't show you today which is uh gasses original proof but actually failing proof of the fundamental theorem of algebra uh I would do sometime later and so the only item remaining item here on this introductory this is the uh determination of real uh improper integrals and in fact there is it's not really the goal but uh a more systematic treatment of um so of singularities is
actually um all goal for the next 2 sections so it is uh section is called singularities so of and uh 1st I will do what I call the local the local field the In the final 6 section will be also a global theory uh which generalizes uh line integration to the case of from the case of 4 functions to the case of functions with singularities but this year is just under about understanding singularities so what is the singularities it's where I met the limit of a function is infinity in an appropriate sense never say yeah van explored is a singularity or it is as it is not that it does not exist which is perhaps the more accurate way uh since the uh function is not defined or something OK so am we will what is the goal of discussing singularities a tool called sexually so 1 is 1 will be achieved in class it's uh what I mentioned before calculation of coarticulation off line integrals of line integrals In the case that the domain the line integral of a loop say but in the case that the our singularities inside here when singularities of prison and there's another another good reason for singularities which unfortunate will not covered in class uh cell nevertheless I would like to mention it it's ever discussion of chaotic functions periodic functions What it's a fundamental domain which is that the function is defined and then it only repeats its periodic say in 2 directions in the plane uh if if on the 1 the fundamentals domain the function is bounded and its periodic then OK it's bounded on the entire plane ran by Liouville's find uh theorem it's constant so for at least 4 doubly periodic functions it does make sense to consider what extensions veil uh automatically they are constant however if you admit singularities than some very famous uh functions to another like provide stress P. function and some others and VOs uh those can only be so both actually need singularities an interesting example interesting means non constant in the appropriate setting I haven't defined as properly interesting functions uh when admitting singularities the the no the and that's very classic but this will not in this class unfortunately put OK but decided that we covered so and let me start with some groups Let me start here with and a section on course she's integral formula the so she's integral formula on in you lie the this this what this quality I the 1 a new light on so we will be slightly more general than just um having a singularity what would be a what would a singularity 1 will be a punctured disk say our Pinter domain with 1 punch we will so this will be say uh ball the disk without the point a however we would do it a little more general a see more general and work with uh and you lie so what is an annulus R. well my notation here which is the following i have 2 radial I asked and in the complex plane I have a midpoint b and I have tool radio I say a little r and capital R and I consider this annulus Kiessling here so what is that it's just the set of points well uh where little aren't is less than b minus said let's think of a lot but OK I will admit the degenerate cases salt form so suddenly I warned dissolved in less than are to be worth of this and it was to be nonempty and I want this and this an infinity of positive but I will also admits the true cases that the inner radius vanishes and that the order radiuses infinity and so for instance if you have uh so if I take this 1 to be 0 then I have uh I would have um 0 then I have to just wait here the ball of radius R was thought about the world without saying right so this includes the case of uh uh specific to singularity but it's a little more generally general since I will not uh prescribed for the function does was an an entire disk rather than at a point the and I also will use as football's my convention is that the I will write it in in the case the midpoint is the origin then I would just write a our for it and is there anything else to say for notation now there should be at the end and so on an example of such a form of such a function the will morphic function defined on an annulus uh will be given in the following form um but slower example would be so example slower than they have to say on are they are our to the there are to see conferences and would be that I had a function on the OK here's an annulus 1 way to produce such a function is I take a holomorphic function on the entire obvious larger radius ball and I take a holomorphic function defined on the complement of the inner radius ball and this this and that so let's write it out and and explain it as of Z would be saying g of z + h of I write this 1 defined in the complement in the fall more H of 1 over Z where the the lead goes from say the larger ball the is holomorphic on the larger ball and H goes from what 1 over the little radius the 1 of the little radius to see and both of us a holomorphic the then I arrive sciences and I'm allowed to plug in 1 over c which is 1 over z is a model is larger than the luau our and uh so
z itself is I did it wrong way right uh harder save so if if uh 1 over Z is less than how is larger than 0 wall been little off of it and uh then z itself is uh I mean less than 1 over here that's it so this does exactly uh what I wanted and actually will show today that any any functional maulstick on the um uh on the annulus can be decomposed in such a way so this is this this example is sexually uh general all so the 1st one I have 1st in order to achieve their goal is a generalization of a cushy integral formula so remember the coach integral formula on the desk for the days gives a picture of a slide to remind you of 2 business that here is my OK so recall that the gaussian integral formula represents a function within the desk within the this is uh by the sorry OK by a boundary integral uh of the boundary of the disk and various this weight factor factor which produces a singularity at the boundary but not in the interior so this is uh so for that and the body and not here so this is a well-defined ice integral OK and they have the same thing now for all functions defined in the annulus B can represent the values of such holomorphic function f within the annulus by a boundary integral which in this case would be given by integration over the outer and the inner uh bounding circle so this is the following lemma now so it's a because she integral formula for any the and it says that if f Cecil and um missile figure on say and then you lose that they are to see what Wall Street ends OK and I want a slightly a slightly smaller annulus contained within this set all let's try it more of him and so this would be is the origin so this will be our and our and now I want to consider a slightly smaller annulus of this form so as to so over radio I and just to make them look similar call the 1 the larger on capital P and the small along the road on OK and so I suppose I have another and say a role T. compactly contained in a ah ah which means nothing that this is necessary Cohen's is a logical than 0 and role is less than a little our and and which is in turn less then she which is less than R which is uh any number including infinity OK if I have this when I can ride and then I can represent my value z and this will work for all said in in the annulus sorry the and in the annulus namely in the smaller and units so i have a point this the so here OK and I want to represented as a boundary integral although there Red boundaries you red circle boundaries so I write it as 1 over 2 pi i and it's the same formula as in the co sheet into the formula for tests room the we also but now the same integrand associated Over zeta minus Z the same time and by definition I mean was this boundary integral uh the difference of the true circle intervals saw 1 over 2 pi i integral of the larger radius t dbp also same integrand half of data from the dominance it uses that it did they tell minus and the minus accounts for the fact that a sense of the orientation of the in the boundary is in the opposite way yet if the domain is all there is to the left of the curves then uh I data or in my in an interval the by integrating around the circle this way and this means that if the write it out with a circle interval I uh I need a minus sign so this is the deed the uh role of the same integrand as of zeta zeta minus it that's the way the stator this is for all said India small animals OK so essentially yeah activists orientation business which of course makes a lot of sense is the same formula as a the quotient interval formulas but it's for disks make sure proved this also in a way is also the same but it's a generalization solidly gift the proof of the formula and so the idea here is and Pollack should prove is also there for the standard formula and there is a decomposition of the integral we're interested in into a concentric and it will always be 1 of the other I don't know which 1 is consent again and the difference quotient and it will and I start with this difference quotient interval in the in the following way I said it uh I sit um she from the picture I can now take the larger and by as the difference quotient office of this function f with respect to z in variable zeta which is not correct here time OK by spontaneous I think the difference quotient for of the case that zeta it's not equal to z and now I wanted to uh a continuous uh an analytic continuation to the point to the singular point at equals that said well uh this is just too many derivatives of its F of state state OK so why is this good well it's a what so to states said to say it step by step at this difference course the limit of the difference quotient makes this a continuous function so our and GDP is continues what by the French ability and this means that by some by the um uh removable singularities theorem of the man did I get my hands slide for this remember that if you have a function which is defined except for 1 point and holomorphic outside but bounded
then uh we uh we can actually continued into the
singularity and it would be think and to prove what prove was just given uh using analyticity her this is as a 2 annoying for use OK OK so if I if I apply vis a theorem to this function here here it's since the every the derivative is defined to be the continuous um continuous extension of the difference quotient it is a bounded function and so it's actually holomorphic by the by romance the romance removable singularities fear and the which was number whatever 28 and we have she is what think the OK if I just make an excursion on low-level newsletter and compare that since we did this this step we did a different way when we prove the standard Gaussian integral formula but I convective is inspects bit away OK so em what do we have well it's it's horrible is a total Mossad when we know that the line integral over uh of the boundary integral of a boundary set the 2 red circles uh must vanish news so to do this step by step I would say that if it's more on the right on of course is only horrible thing on the uh many us in R our so that means that that's homomorphic there then I can say that the boundary integral of the larger radius also uh g of Sato data it's the same interval as the interval of a smaller radius use data using staff Weiss this won't remember this comes by there was this 1 lemma 22 I believe where we made in included these connecting curves and said that OK obvious said here the French so all 6 of the boundary integral vanishes and the true connecting uh line integrals cancel 1 another since they uh the direct the sense of since they they are in a different uh parameterize the different sense OK so that means we have uh this equality in the 2 intervals must be the same as they need not manage but they have the same right they only count the singularities which are at the inside here inside of a smaller disk and if there's nothing in between you can just change the this is what what it's saying so if this is so then I could also realized that 0 is the difference of the 2 world case and think and and now let me uh write out what g is and in order to save space I use the annulus notation remember here's a minus so because words so this is role of capital row all f of and by definition since we are on the boundaries and to singularities in the inside I'm always in the applications of the definition itself I can 3 right this the s the difference quotient integral and now I want to use the same trick as we use for the cost interval formula for the case of tests I want to seperate the numerator and right integral of so starting with f of turns it means minus is z using data overload the annulus will blow um minus OK I've now I have to write rewrite the F of that term and let me do it's a separately for the inner and the outer circle so and also know that isn't in that integrate over d zeta uh this is a constant so I can pull it uh out of the integral to get as a of z as of so that also rely on a larger radius ball DB Capital wrote a 1 of the what's remaining is 1 over the state of Minas zaitech class and now I have 3 a circle boundary with the other sign that same same integral along I was a student that the zeta integrated the little boundary OK and no I know what these what the letter terms are so this is this here is what I want to get remember here this is this is this integral here that's the same on OK now what about the other 2 integrals well 1 of them 1 of them a is easy and so the letter 1 is 0 is why is this so well I integrate over uh um um over right I integrate over the little boundary but the singularity years said said zeta equal set so the singularities outside the disk right this is a function which I can view is defined in the interior I also have this boundary here and it's a holomorphic their senses singularities outside so it's a home and I need to go over the boundary um of a morphic function so this is 0 by the Gaussian integral formula yeah a cushy integral theorem if you like OK now what about this 1 here the same the same kind of argument applies however the singularity is now inside uh the larger radius uh boundary circle so that means uh this interval is 2 pi times yeah remember that we had a little lamb are telling us that uh uh have if I know that the the integral of over a concentric circles say is 2 pi and here the functions almost then I get the same result for the largest circle just by the same devices here I I can connecting go forth object right and and so that tells me that it's no harmful this integral to be to be 2 private the singlet that cigarette it is a set in the center but it's somewhere off center OK and no um have a look and what we proved so uh so this here on that side so F of z is equal to this and divide over 2 pi so this gives the factor 1 over 2 pi and which arrived we wanted to rush working so using a little of the fury uh well before and then mount 22 at least OK so this is cost into the formal and let me
just compare out of curiosity to uh to the proof of the no this led to the proof of the course the integral formula for this I gave you what's the difference well except for that the terms appear on other sides of the equations and the main thing here is here is the difference quotient term and we argued in a slightly different way you we said that this difference quotient term here it is again can be estimated by the length with estimates and just by Ivory as soup of the difference of the numerator since this is this here corresponds to radius so that goes sort of goes away and then we said well by the continuity of this is bounded this is less than upsilon and so the limit is 0 and what a give vanishes so you could replace this part of approved where is it to it as start and right so this part of the proof you could replace by the same argument I give you here but uh but you have a continuous extension of the difference quotient by the limit and then it's a little quicker to argue this way but when we prove the formula we didn't have to remember removable singularities theorem at hand so that an so that's why I get the proof of a kosher formula in a different i in a different way in some the and actually if you look at the proof um you see that it will apply to much more general sense than you lie here for instance you can admit any any topological annulus any set with the whole and provided you can integrate over boundary so if you have topological annulus with uh say piece wise piecewise differential the boundary here they the yeah such as it would also yeah would also work provided am I can integrate over about units and then all this theory applies OK so this is and this is the important formula and now let me come uh to representations of um of functions with singularities the sex so-called long in the hall serious this is strange the 2 words was serious an and the 1st thing I want to do is uh I want to show you that the decomposition on the upper blackboard left side it is valid in general and after that we will come tool a generalization of power series uh which is actually what uh this year stands for so 1st this is the 1st thing is a she kind of Lemma that mimic still college or I could call proposition 1 the this proposition and it's the so called wall decomposition the the often automorphic function F into such a g and such an h well cancel cell is so this also as a number of 9 you no change um so if I have an answer say from an annulus um on um to see the hole more often the OK is will freak and then such functions uh GNH exist then there exist um or logic functions FIL functions F and G E um so for a g and h GE from the larger disk to see and age form 1 over the smaller dose of the disk disk of 1 of a smaller radius to see the way the and lot of of the type described before and this holds for all I said in the end user and this decomposition is unique In case what we would therefore free to place Constâncio of their cell um so we must as said say 1 function in 0 2 0 in order to have uniqueness units so the additional requirement the tuition the requirement the um that h of 0 equals 0 I could also say g of C which is your own any value whatever liked uh uh makes it unique him many Cs how she age unique the case In the case of my function is holomorphic my function F is homomorphic to start with yet then I don't need M. Ben H. would vanish so H captures the singularities of a function f defined on the annulus news so so I would say think of think of f as the homomorphic Carter later will be call principal part but still savings holomorphic party uh not s the GEM and so if G E N is holomorphic part of this year and and likewise H 1 as singular point OK unfortunately it said yeah I mean you might you might miss examples of this point um I would give you 1 later but um it's not so easy to ride out simple examples here which is why I um I presented this theory to you 1st OK so um let me give you to prove so um we won we want to um have it contains 2 parts 1st uniqueness and 2nd existence of course uniqueness sounds as all we assertion existence but there it
will be used to prove existed the existence part so that's why I want to stop uniqueness so we don't know if these functions exist but if they exist they are unique in all cases all suppose I do we show uniqueness well suppose you have 2 different representations suppose or say on the annulus may have had an the answer has 2 representations say is G and G + H of 1 over the g + stage and also there's capital between plus capital H 1 of those that but both are defined on the appropriate sets and re rewriting this equation by putting the genes on 1 side and the ages gives me she gives the minus g of z equals the difference of the ages age of 1 of his Z minus capital 1 h mobile of 1 of resent so why is this good well uh since now all have different domains on both sides remember that the genes are defined on the larger on the logic this and the ages of the complement of a smaller the or not complement on the inverse of C smaller disk right so um so that means we have here we can say that this is defined on on the capital In this year so is defined uh to be careful I say for once said larger not OK so this is a nice uh situation since on both sides um since of both sides and we now have the functions so so that means I mean that means that this is 1st a drop picture so the 1st of the left hand side is defined on the uh on the stick with the look at a radius of the right hand side gives rise to defined on the complement of a smaller radius so I can actually combine those functions to get a function on uh the entire plane see in order to show uniqueness uh this is a good idea since that the only way to show uniqueness which weathered hands is here so I want to produce an entire function and this way I have a hazard so if I define a new function this the key differences the G difference on a small on small are on small sets and the age difference but the for would Langevin our then this function becomes defined on the entire planet the yet remember the here's he an overlap right hand question a you know to probably intimacy uh and then there no so if I write g minus g and this is H minus a couple H minus H. aligned hurricane OK now she work I'm sorry realize clubs thank you OK that's a monument right no problem so we have the same as the same function yeah and 1 was an overlap with and it is a matter of due to this equation here it is in a and the quality of the annulus uh it is a matter uh in the overlap of these 2 different definitions which 1 I take OK so I can say that the this function psi is and entitled his entire and in order to and apply the use of Theorem I would uh I would like to and say that it's also bounded right so why is a bounded well this comes from I mean what does this function do uh on convex sets well it's continuous so that's fine what does it do outside compact sets what is due at infinity while there is a limited since is a z tends to infinity 1 over there tends to 0 and age of 0 and H of 0 hour defined right here they have they have values Due to my definition OK so that means that p of a sphere and will apply here so let me let's write this these arguments out uh then and so far I in this founded upon I mean it's continuous so stranded on the complex sets subsets of subsets the C and its hopes quantum and and also the and at infinity we have so that uh age uh um so the limit it to the tends to infinity often alright in the correct order that order the In this age of 0 minus h of and so this means uh so this means that it's also bounded uh in a neighborhood of infinity by continued to new and so also bonded outside an or I say at infinity to use sloppy notation so the values in the NAPs island in an at sign a ball here words bounded correspond to the value uh no I should say the waves of the year of NIPS excitable corresponds to the complement of the 1 over excitable cell to make this precise OK and the of that we here the man and the the the so all remains to be done is to appeal to the of those very in and say that the function defined must be constant In due to the additional assumption we have that the ages it's there all vanish this constant must be 0 so the 2 representations must agree so of fly it is equal to a constant like and into the additional
assumptions will be uniqueness that age should be value of age at the origin is eros and Ms. constant is 0 so M by the addition of the SOM channel and so that means that's that means I have unions half of of my came and so this is this is the 1st font and let me now come to existence what span features his palm will depend on the parameter and by uniqueness uh who will see that it does matter so what's the parameter that we show its so instead this will be cased are lower case that went point z that are say it is contained in a smaller an annulus a rope and contained in the name of our are compactly whose name will pay assuming this book why do assume is low then the boundary of the small annulus is contained within uh the sets I can evaluate boundary into will so here was so there the science is not defined in these boundaries fits a difference so that's why it's good in SL now I can apply the cushy integral formula which is sorry about it's gone in my hands how she will inside of her she's integral formula innovation uh so that's the previous lemma who use Lemma shock and I have very uh I can replace N and F of z s the boundary integral of the uh of the annulus he mean uh and units taken er it's taken apart right away so it's the writer against uh and 2 of zeta zeta miners then in zeta became minus who of the 2 pi i a smaller radius boundary in the gold and uh I wonder OK users are also and serves a turns into the miRNAs eds user talk and I want to send these as my son she's she's and H. Song who I will say 0 this equal to Jeannie's ends I will how can now he I cannot work directly write H share below at the age of says R H of Z since I need to consider the inverse value of said the inverse so in order to write it in terms of the inverse that's just uh multiply this by 1 over Z 1 of his there and then this becomes 1 near the same thing no difference and now using the and the yellow versions I could say uh vectors is then I call this just to be clear conversate W here and he signed his is each of W 1 and so on age of w equals now I said this is WS of Satan over w zeta minus 1 using 10 more appropriate boundary he the role and I forgot to minus 1 or sorry the age of w equals minus 1 of 2 I I I want Putin signed in cheap OK so that I can write this is g g of z age of 1 of the red car so OK where these functions defined while am these functions are um assessment of who ends where I explained that have how who and claimed that she is whole the people will will while being in order for these fine she's a she tool as satisfy requirements of acid during unfair uh wouldn't have any Nagy is he signed on uh on the entire disk also repulsive of modern so I explained to usual Maltese on bees are not trolls but who OK N a H who will say again on my uh 0 when they're needed beach 1 of the partners all at who OK why is this song well look at what we know what we're doing here uh we have a z tell Saul and who and and you use are McCain in the 1st term i have a zeta on the boundary of the larger and uh at my as a test on the boundary of sees a circle of radius or and I have the time and I wanted it it defines a function so as a it's within the disk of radius r wall no problem right safer singularity happens at the boundary but but air if I only want this to be defined for i z In the BIO loss in B. you uh roll all I'm fine yeah so this is but yeah so this is OK whose use or came and similar argument here now what's the danger he and the danger is that are 1 of the z are made 1 over Z is an ahead as modulus remain low near but if someone might sanction age to be defined on the tip of the the uh also balls radius 1 of the roller I'm also signed In a salute to do it step by step here for h who are candidates went this letter exaple is in B 1 overall have prison in the circle of radius 1 of oral as its use no it's not uh zeta is in B wrote who'll came and men whose mean squared 1 over zeta than than and so and they were in this means that and W 0 yeah right w which is of if w is in beer all of its means that 1 over 1 over Z which equals w's are equals 1 over z because w is in the role that means that they model Z so 1 of the more is less than
roll call z quadrants so that that's larger than 1 row and so again I have a lot of time she's essences singularity here is et and the point that 1 over z equals uh matter uh w is 1 of the zeta OK so that means it means I have automorphic functions on the on the sets I need to the end so basically I'm done exact full effect and have his restriction year bed and I am on the way to have slightly smaller slightly smaller in you so why is his no problem uh well some of the 1 2 weeks 10 or extend the obvious to all her Haas Z in the annulus walls larger anions or and so and this and what is the annulus of the halls larger radius is larger read EIA it's the union years it's the union of the X small Arabia 9 where role is larger than R and OK well perhaps I don't want to write this article entirely on 1 of the since the mid of space of a so are nest then overall less than captain were less than are so if I take the union of all NUI contained in and my fixed analysts I get I get my fix analyst and so each of these smaller and you II have where sanctions G and H G and H define right so and what I can say now it's and so the as the are in row on the road use compared the contained in r in the annulus a R I can say that man hi man whose 8 2 while or is i if if I have a um their arms I X Yang tool Rainier adjusting earned man well 1 like 1 and yeah and who wrote to rule to word cocaine I say I have tool any lie in Role 1 Role 1 able to low to then an and say 1 is smaller than we other to make it easier yeah then the I get tools such decompositions went by the uniqueness theorem however and these must degrees then assuming that I have the initial requirement so let's find cheap age men will gained then an Cohn-Kanade and the 2 decompositions or FIL who who's machines stack of mean into of she'd flask that age and mask we agrees in a provided by and provided I and taking constant always into GEN provided H of COI mass and this effect here from agent under the age of 0 and an entity in order to be unique since now and was used not only of the hands and c l Corvette of yeah uh that I didn't see his carefully through analysts not and so on the other hand these 2 integrals raise these 2 integrals agrees um blooms something who's not required cheating where now and solve to decompositions must degrees minus sling since scientists all came what can and any to say something about this constant which probably has not been widely as it satisfies the but the age of 0 equals 0 and then and must include an argument which I don't see right now but so provided I just say provide the constants without but and let me do his who say were to this then makes time guidance reduce the now how when she had she argument slightly different than what the had mind right now and so then I can say that uh 1 of the function of the larger and on the functions on the larger annulus on an analytic continuation of holes of the smaller use and so on the small annulus in some of those must achieve extend can extend in a unique way to the entire Union Nam soak 0 so that means therefore are meant for humans an continues back whose all vector X you and I don't need to continue icons just saying therefore for all these representations investing science is a function that takes you have to take care of his agent 0 he's in this which I don't see when I was a wouldn't you see it for for SL here so on any smaller on any small any of those I have my friend tools and famous and resell the HTT sine functions on the of the Union OK so cell chain are no km unique function it doesn't matter which any of I considering here and can anchor and then obtain the same unique son she's need to positions whose issues it on means higher in units at OK I'm sorry are you still get which and she was not see who why this is so it's all came and then I have in terms of these boundary equals I have sees uh I have general case I have to see composition and 2 G and H and now I can use policy areas
who uh and in order to the them in order to it it is serious composition our series for G and H the kitten I couldn't quark on so now has 2 functions g definable in H. Simon a ball or this and I can right out policy and since they are Marseille Nelson then they take a side can use posteriors representations minutes long are news talk How are sulci gene chief age who all loss in so therefore in undertaking well trained self can it solve various who has power serious Hatton's policy areas have since representations of 1st namely I can write ontologies same as high as the sum of the ANN existed to be an hour and I came there and is actually used convergence for z creates an hour and I can write out a movie in ages said I think different different coefficients and due to additional assumption in this series will started 1 going to infinity and this converges Fuller said this in 1 over and now let let me do a following right in place of the and let me write this as a minus and so I'll write a minus and a negative source for negative indices this defines and follow S for positive integers and 0 yes slow the sign a 8 index something for negative indices and now I can put these 2 serious together and this will against credit ethyl Sontag means which is this is the overall order h of 1 over Z + g of z who and now age of 1 over Z has he's a minus end and a index z to the negative and a power series of presentation so and I have terms like n minus 2 in 1 of his it's created in minus 1 more example class so this far is fired from negative infinity to here extends the power series of age of 1 of of z and now comes the provisions of the of G so a naught plus a warnings there plus a 2 traits clarity and so forth so this is just a serious set of a policy is representation of her ranging all meantime integers all of this fall and again you see that h in H collects the singular coefficients in G Chrome then heard collects fees or maulstick ones well can solve an and so this in general this object here is called a long series themselves are green all serious laying low serious and there is a a in ah well is the term of expression so I come out onstage his fault in people's minus infinity infinity in the city and and I will say that it converges yeah since this is a double limited I need to be careful as for improper integrals I will say that it converges if both the negative part and the positive part converges 0 provided uh that uh are the sum from an equal 0 to infinity unions and and we have a some also alliance infinitely the when is being in the uh the haida it's N. equals 1 to to infinity of in minus ends at a minus end converge converges so both musculature OK and X I can give you an example are which is not slides and this will be the last thing I will to to change or not all and what would it you're right so here is an example of an considered function from C or without 2 times 0 and 1 have tools given by the and snared equals 1 over who uh z times 1 minus this not to singular points 1 and 0 it's not in not in
science while buyer I can rewrite this an angel song just as I write this the interval minus the cluster Walters said times 1 minus Z. I see that this is this is 1 of this is 1 of a very yeah class and the other term is 1 of the z 1 of them was mine descent who OK and now uh I can use sperm so uh hexene Oliveira tool distant laws serious representations sigh of singularities is 0 and 1 so I could consider the annulus any 0 1 0 but all can also consider its complement yummy outside any 1 infinities so on both on and on both and you I have uh uh are asked to use whole Maltese and so it I get to distant and all serious depending on which at any of those I chose men itself and how do I get loss use 1 case plug in the geometric series once all I plug saved for sure z how missed when 1 has me so this means sort z the depths of the the right z in our in 0 1 for that which is the same thing and what would I get I rewrite this as a geometric serious and but so I keep the 1 of event and I rewrite this as in as a standard geometric serious and we saw is all of how this will be my long series of the H H of 1 over Z has just 1 to near near Chris you a false statement learned and see who practice of ACP yeah so this is it OK so this this growing at a strong kidding yeah thank you so this is minus s er 1 over Z I should like is alive it'll making United stated just and then wakes up sorry had that have OK who and so on the other end you this on the annulus I get and I want to rewrite my geometric serious inform using 1 of his ex salt and let me 1 of Z assigned her and then I error take also the 1 over say it here and I remain with who 1 over 1 minus 1 over Z and so this is a sheet Mr. just putting out 1 of the there and now here I can use the geometric series as well so this idea here I will obtain OK so what's the geometric series here is 1 plus 1 over Z plus 1 over z squared and so forth so what can be won over uh the 1 over Z T councils with the 1 and I get to be 1 over z squared and so forth terms in small numbers it's cute this is correct and it has the wrong sign since he again reminds McCain but then you have continuous and this is valid for the annulus someone to infinity so here the G coefficients vanish in only have H terms yeah with n H C range all way Higgs indices just as the BEG times you here OK so this is the example and next time I will continue and and now it's time for variation but there's a Christopher's and who she felt moved on a lot and was correct OD 0 dear and the yeah which is correct yes so that here and you're right and now the next 1 will tell me that is incorrect what have I got it anyway you get and the thing is is used as you said it's at plug in 1 of 8 here in and then I get a negative I get a negative terms here here thank you I'm sorry How confusion How OK is any do for you to attend z slow the somberness which I don't understand OK thank you very much and now it's time for re-evaluation salt aah Kelly Backhouse will class all procedures and correct them in 10 minutes of self and he's hey