Isolated singularities

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Isolated singularities
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in hello Nice to see you to the class of complex analysis and today I think we have a nice um topic namely singularities so this is also no so that's what the um long decomposition aims at uh what we will study today in detail so in so recall he recalled that decomposition yeah a little or a lot more serious lower serious is um given by decomposing a function defined on an annulus would say the a and it's written it 0 was a singularity at 0 uh and and not defined on a small disk of radius and our but don't want to write it out OK this is so good long decomposition like this so where both these are holomorphic functions defined on disks universe here is defined on the larger radius and we will work with an annulus so this G is defined on the entire a disk and the age is defined on the uh disk all uh whose radius is the inverse of this radius OK and in plugging in um policy areas we derived and perhaps is a good idea to write it separately for want to use it for today so I say a some from minus infinity to minus 1 also in the end I to the end when you're not with this negative a principal part of the singularity part of function and this is just a plane were morphic function so it's simple power uh in curve is active yet the OK we want to use this on this unique decomposition and his unique serious the noble serious that in order to discuss singularities namely isolated singularities had um on or NAC Re did something on singularities before namely of a real uh the removal case uh but now we study this in generality so let me uh give a definition 1st which includes the case in which we already know so we suppose the uh with a function with an isolated singularities so say a B so it's defined on a disk without the center of the disk which happens to be in our notation also the annulus of radio I 0 an hour with respect to be if you like anyway this function is a holomorphic on on this time to time all this pointed this problem will all of and now want a OK then B is called the singularity we know of uh so never miss a advantage then B is a an isolated singularities 1 for just a singularity and you all staff and so now I want to distinguish types of singularities and the several following uh all care of it is clear that and now I have 3 items namely the 1 we know it is removal to remove a blue bill hit by the this is the case we know if it's simply can polymorphically extends the function f into a point beat uh some other writers uh s as holomorphic more hands holomorphic and extensional analytic continuation whatever you prefer holomorphic extensions a tool be our speed this is the case we know but then they also poets and essential singularities so a 2nd item is a poll uh so any I will hold of order order the say in in in the uh all came the following case it well what should it be like uh a pole is something like 1 over Z to the end so what's characteristic to it well if I multiplied was said to be and then it becomes removable but if I take any lower power multiplied by said to the n minus something then it's not removable this is the very definition I get so a total of 4 and and uh which I should also say X . be which had an right it's OK Z minus B to the N F of Z is removable has a removable singularity has a removable singularity on this singularity at the but but in the low power by multiplying with that in mind so say n minus 1 times have said and it does not he so what what it just think of this example and multiplying it was said to the correct power the the and the 1st thing the 1st case is that none of the above so it's called the essential on how essential place this wall always the I know but of the OK 1 a singularity which is not removable and multiple we call essential and so for over sisters the technical definition but uh in a minute we will see uh in exactly what characterizes the singularities and way with this gets very nice characterizations nevertheless it's I guess it's a good idea to start with examples so um let's give an example for each case a uh removable a removable singularity uh what easy to detect when you have a power series so my 1st example as I think most standard example for this case will be signed z over Z well since I know the power serious but yet signed z is a Z minus it cubed over 3 vectorial plus ends uh and so forth I divide this by get 1 minus Z. K. 1 this is true for a b minus 1 is tool 3 vectorial plants grown for its place of size 5 and so forth this will be just as
obtained by a power series in immediately you see this is a removable singularity at the at 0 yeah so if you know this so it's clear removable singularity until you saying that I had there by value what so this is fairly obvious uh when the most obvious example for Paul is just this 1 here so that's what I did a large 1 over Z to the end has a pool parents pool by also order in it it's not why is this so well since if I multiply as I told you the uh it is since Z to the end times 1 over there to be an which is long as removable removable groups the removable however well it's like out here uh said to the say n minus 1 times 1 over there at the end which is 1 of that uh what this is for instance what why is this not removal removable of 1 way to say it to see this is to say it's not bounded just take the uh take a sequence approaching 0 and look at what it was then this will explode so this is not when not bounded not bounded so they're not removable it was and so this means that according to this definition this is indeed a pole of order will carry
and the only really interesting here the 4 shows the essential singularities and in fact at this place I can a reason for it but this would be be an example takes it take the exponential map and plug in 1 of that it's easy to see what the power series uh here just plug in 1 of his in place of Z so it's 1 plus uh one plus Z which means 1 over Z plus uh 1 over 2 factorial z squared plus and so forth and no please tutorial hits it killed in that OK and right now all I can I mean it's sort of obvious that once you multiplied with uh and is not removable since there it explodes 0 so I should like to assess well I claim that this is essential singularity at 0 the OK it's not removal for sure this approaches infinity modernist unbounded um it's not a policy and so OK if I modified was there to be and I still have lots of singularities remaining actually it will only become clear in the following theorem was how it will only become rigorous the following theorem that this is essential made so by next lecture superposition in these here by next propositions are OK but it's sort of visible also at least for me it's just a it's visible directly OK so essential singularities are a rather complicated right you know why pose and of simple kind of singularities so I've now I've 2 theorems characterizing singularities 1 is in terms of the global serious the other is in terms of mapping properties I start with a simpler 1 in terms of the lawn serious so characterize characterize singularities in terms of false long series work so that's the following so each of the singularities has a so it has a nice nice characterization namely and so OK so I I need to say 1st let s uh again from the punctured disk to see people lost their can homomorphic and now I want to have my hands on the most serious so as saying we're long serious graph of society he could uh mu harder rented don't of said here's some cause in and the minus speech we end up with a sum over all integers that all came there the singularity also these 4 rounds go or it can happen these singularity about in at 4 point B is not this case it's removable or if an only if I don't have any negative terms in I don't have any uh undefined undefined powers so that means that they can with 0 for all negative and the analysts and they're all in it well it's a close already a told it will order in question in April of order and is a nonnegative well if the was serious down to the
index n minus and but not any phone over so that means a minus and in the equals is nonzero and at all the all the higher a minus ends are 0 and a k equals 0 for all k less than minus 1 cloaking excel the law serious is than the principal part is finite that's another way to save is finite and exactly reaches us down to the and let's suppose and finally it's essential for a while if none of the above by definitions so what does none of the above the mean uh well it means long series doesn't stop in the negative well with the negative coefficients of a must be a a sequence of indices reaching all the way out to minus infinity cell essential it's an only if all there exists a subsequence of industries sub-sequences uh say nk tending to minus infinity these uh such that well as such it such that a N k is nonzero being and so on just staring in at uh this proposition we see that the example is as claimed here the exponential of 1 over the has the property that and the long series in the negative coefficient is 1 over and sectorial uh don't uh don't stop anywhere and so we reach out all the way to infinity so I can just could just choose and K a equals K from my subsequent each negative coefficient does not so if the theorem holds then suddenly Expo 1 over Z and is essentially a 0 he walking in the of examples yeah you have examples I clear since pole means just I mean 1 of the state is just this 1 equals to 1 and the others uh and all the others also from a minus 1 to a minus and plus 1 the finish OK let's I give the proof and it's all about so the leader of this the 1st case and because the 1st case sexually uh what we did already so actually there's a slide and telling you that it's not quite as both he group if a lot and then yeah this is uh remember um remember be removable singularities and this theorem by Riemann sum actually um them see doesn't really met share yeah there's a automorphic extension so this is uh uh OK and now this gives a characterization of its founders shift to prove that I see that OK so if f is holomorphic let's do the removable case if S is a loss sick on the are as for the more on the other then we know that it has a power serious so as analytic and and let it can go on beyond our pose serious the and well if it has a power series member the principal part of the law decomposition of loss series uh the principal partners manage so the selected here here I ieee or policy areas of policy areas the principle principal part lips part of a was serious vanishes call OK and it's a very claim here that uh all the enzymes a role for the negative indices so this is clear and actually it works we have a way to so 1 is clear on now let me come to true I O K so I want to show that I have a function has a pole of order and if and only if and the lowest long coefficient is exact commands which is nonzero is exactly a minus end from carrying salts in order to that 2 directions let's prove that have a pole of order and then this property holds married OK this comes from the low serious so if F of said that so suppose ever Z has a pole has our living right in all serious directly behind some uh N. equals minus infinity to infinity in Z to the end OK has an will also order and I know it's a world I bet if I wanted to visit B and that minus B to the N and so if I assume this and then OK I have these user definitions so uh it's uh after multiplying was there to be on its removable so so then 1st uh set to the N F of said uh has a removable singularity immovable singularity in 0 so that means that the principal part of that serious
vanishes friends of full of pilot of this led to the end of of said law serious vanishes uh that means precisely the before that means pre happening what is the principal part what it starts with uh was the original coefficient a n minus a minus end uh minus 1 here and so all these negative coefficients are 0 so that means any k equals 0 for all k this then minus end and segments of that segmented means that by definition that if I multiply with the lower power if here and then as I don't have a removable singularities so uh let's write this out what what is this uh so this is uh a so the uh taking this already that for granted then it's only a minus end has 1st of all coefficients but now we modify was added to the n minus 1 so it's not there to the minus end but it's just 1 over Z which is highest term and then I started with OK now I go up when it's the minus and plus 1 times that of 0 plus and so forth you were OK so this is not an removable singularities what that means that this coefficient must be nonzero is not removable and 0 so any minus and must be non-zero otherwise it's removable working OK so this is this is not hard to see it and and the other direction this and the so of tool in this if I have this property then if I have this property here on the wall coefficients then uh I must have a pole of order n well that's basically the same thing so if I take the law was serious moving in and sell what if I have this them below 1 serious is just to uh that's right L and let s for it it's L of said is a minus and to the end of class and now come back to the n minus 1 and so forth in England lets anyone said and fault and if just use the lawn serious uh how safe for or the series with this property here we obviously have said is I multiply it uh and uh wouldn't want to show that this is the case yeah I need to show that if I multiplied with say to the uh if I multiplied exactly n right if this is the this is most serious uh satisfying this requirement then obviously and OK and this is nonzero who knew you and then obviously modifying it was that to the end i and up uh with removable singularity and multiplying it affords any power less said to the n minus 1 and I end up again with 1 over which sector 1 of its it's like here and so it's non-removable so but the then I solve the claim follows said to be an analog said this removable groups clearly movable who while in low-power said to the n minus 1 else said it's not and so actually know it's the very definition of a pool of water and walking so what am was remaining in this property uh would be essential for the essential singularities small now in fact there's nothing to say things all sins sends um in my theorem proposition uh have 2 equivalences and the remaining case on both sides and if I have proven that 2 equivalences than the 2 remaining uh cases must be created when the level also small groups of various examples the MIT from the more the better the crux of this which we are in the other be yeah so noted by definition or the essential cases remaining case but also in the propositional and In the proposition uh if I don't have this and I don't have this well then I have this so this is also this is also the remaining cases so if if I actually have this being equivalent equivalent and the remaining case here must be the remaining case that obvious it's just a matter of logic cannot training content the new so for the proof of the I say here's how it is that the remaining cases of In many cases cases on both sides so for the singularity prose singularity definition as well as a form of a loan series discussion loss in its so if you like to be more exotic for a long the 1 series coefficients uh so given I've proven that 1 and 2 are equivalent I must have that also frees equivalent as remaining cases must be equivalent in a cell by the truth of 1 and 2 this means that equivalents of
walking you can solve this that this time gives you an understanding of the singularities in terms of serious and now let me and I give you in the characterization in terms of mapping properties what's a good place her perhaps up of warrant colors so now we characterized singularities at singularities in terms of mapping properties have local mapping properties in the neighborhood of singularities the 2 and that's not so obvious think of it but so it expects its more explanatory cell being singularities the singularity OK be of suppresses holomorphic from the of the but without being a hand are to see me we will see you can all it all K users at the on the OK the 1st the 1st of the characterization of uh remove abilities is clear but let's included year removable is clear from and now I'm sorry but now I should cite the the money back consonants and so it's removable if and only if f is bounded in s is blonde and on on where it's defined here so our actually I could the way I wrote this out as a solid thing uh let's do it on MIT's to use the boundedness on the slightly smaller it's in a cell if I have a capital R here when I say there is sufficient vector s is bounded on a little on the little on where the source and the online positives right that this is a key it's not equivalent to be bounded on the entire this since I could have a blood of the boundary only on context is CEO uh would say singularity I could just take the same radius here and that OK but I hope it's clear but no now I have the interesting cases are the uh at the poles and the poles and the essential singularities so I have a uh poem the ban true but it's an only if is the modulus of tends to infinity when the approach to singularities not for z tending to be wooed nemesis almost exactly what you have time in in score at high school this is really what you would say singularities where it's it's of it's a poll function approaches infinity OK as stated otherwise by the what does this mean uh and that means that for all a constant positive constants there exists an at sign on uh such that the values of model the values of 1 s on a larger than this constancy on this ball of radius and silent that's like this out there exists English there where's and z Italy's least saying and when I restrict to z in Uppsala enable so all say in words it's might this 1 can and now interestingly enough essential singularities is 1 is 1 very this does not hold but actually it's more when I say now actually I find f of said arbitrarily close to any given number infected image uh is dense so this is the so-called the theorem also because over T and minuses he group can sell its essential I is essential if and only if OK no matter how small take a neighborhood of speech so let's take on a little Lyra neighborhood in if the electrophorus is diamonds in C for all of and so on and so on no matter how small they are now which means well what does it mean of uh led to um cell so each point in the MIT draw picture should this uh where does it go here well clearly he is the domain his range were not exactly the same point and my OK so what is it mean for the image uh of the image to be dense it means that all of which ever well you might take in the in the range I find it well you close by so an act of the news say is uh well actually I guessing as for speed I signed a value if of Z close by for some so it's here is be uh and you see a loss I should radio bias so this is a little if for all of our instance so I need here and EPS say that
could actually say and make this asylum close it take any complex number w choose any small neighborhood of radius Eps either you will find in an image point s of data uh which is arbitrarily close to you given point the right so restricting too little disk of radius r you find is that which maps into there to be of all OK so that's the situation for an image to be dense yes all over the place they're coming image to be dense means all of the places you find something but here I consider is a kernel gets there this is take any Absalom ball near you find values there perhaps w itself is not 10 that's a problem because they maybe that w itself is not attained but I would really close you find values so if I write it trentorise then I would say that for all w in sea and hips Cylon positive it Cylon I find that said in B R ProSpeed such that the distance of ever to w is listening to sign on with what ever of that minus w it's less minutes as well OK so that's the characterization and perhaps the floor OK this time for that anyway but let me say it before I there's 1 more thing that's so this is found away from just being a negation of his property here it I mean that in the limit of a model is is not infinity but in no way means that the images dense In fact something stronger rules yeah in fact for aid in fact uh for an essential singularity a singularity it's not only true that the image is then but in fact all values except for 1 hour to yeah its limits and all all values example but no 1 except so long that I uh attained in any small neighborhood in no other trees In absence any in any neighborhood of the singularity neighborhood well singularity itself no matter how small you choose your and always I attained except for possibly 1 uh and what is the exam all its race now in our example of expert of 1 over you see right away what this value must be in which is not attained yeah we know that the value 0 is not attained cell uh by Vince characterization which I'm not going to prove in this class and all the other values are intact contained in any small level yet so this is called the great because great she Garcia and and it takes a while to prove and to actually get a get a reference like Mets book number 2 but to be honest haven't checked improved by myself in detail OK so maybe this is uh a good place to stop for the break so I would like to continue with approved was the run the the so as I told you have 1st case the coincides with the of removable singularities here and so this is done log on from so 1 where is by Sharon that Theorem whatever it is 20 8 the world at all uh OK so let me come to the mapping properties of poles so or she had 1 1 to distinguish direct she's and 1st proved that if I have a poll he modulus approaches infinity OK so suppose that and so suppose I have a Paul what is meanwhile if I multiply this function with air uh is said to be an then as a it's removable Saul um the edge of the z the multiply-add OK let's keep the minus the Z 2 minus B to the end times as it OK that by definition if S has a pole of order n the then this has a removable singularity of the right itself parents the removable singularity at the end and and actually a we have a h of the by the definition of that Paul since a multiply apply with n minus and the same thing with n minus 1 the power to a power of N minus 1 and then I see that actually the value the uh is nonzero yeah there are in each of the it is non-zero since otherwise it's a poll of 1 model on I and so on if I have if I have h of be nonzero then and I can say and it's and it's a bounded away from 0 and modulus in some neighborhood so this page of society say is larger than a half and a half of the age of these I want a say in some neighborhood or whatever there being some neighborhood you boss be all came the so if this is the case then we have for all z uh in you led into the following and I want to recover an the modulus which is OK s is h over the power of so in this age so had the power to be so when so this is an now using the inequality that H of Z is larger than a half edge of P I get that this is a large of an age of the of the tool times Z minus B to N the and now look at this what does it do when a z approaches be while tends to infinity because tends to infinity you s z tends to be and this is a very claims
instead so all I can say OK so given a poll and I've I've shown that modernist approaches infinity and basically it comes from but what you like to think it comes from the long a serious for H of layers and all so um now it's in place of the uh doing the reverse direction 1st let me uh let me postponed this to later and go directly into because about viruses fearing and prove once again only 1 direction OK let's proves that the image is it's in the image of is essential any small neighborhood of an essential singularity uh has a dense image in see that's what we want to prove well OK this is the hardest part here so we do this indirectly on indirectly so most posts all the we suppose that OK and suppose this is not true OK so that means we have to we find a w number and see and an epsilon recalls attorneys such that uh what such that there is no F of said in the neighborhood so such that a b excitable so met um but such that f of b r of PE would be removed an intersected with the upsilon ball about w OK that should be empty her current he all came yeah so this is this is a point where neighborhood this inc excluded in the image so that's the negation of uh the the images dense OK now let's so this is a situation very similar to what you I believe what you did in the 4 in a problem with the of sphere at nite that you exclude and Absalom both of the image and then you consider 1 over the function and so we do here there it's still there and and then we consider or 1 over the sanctions S 2 Mary subtract W and call this G OK so this is now uh this is now well defined sends evidence that doesn't approach that doesn't take the value w nor does it take any values in the neighborhood and that's to this falls in small was was small neighborhood that need the our neighborhood of Beit from can then dysfunction um OK what is it is now defined well defined that yet since the W's not attained and it is uh homomorphic world War figure as a composition of holomorphic functions and its non-zero nonzero senses of the kind 1 over functions OK and moreover I can estimate the models of GE Margie also said is um given by 1 over margin and and at z minus 1 W booking and so as the assumption is that this is that the function doesn't take any values at sign-on close so this to the denominator is at least upsilon in modulus so if this if this here is at least upsilon then 1 over it is at most 1 of its size so what does that mean well with means which is bounded In this on this disk right well OK Iine g is bounded on the net same idea s uh and so in variant of this problem we use delivers at the hoops share on on chaos song although although G is not defined G is not defined at the very point B if a singularity as is a bounded function it's singularities removable who or cancel on saying that and all hands and the singularity all of G at the ISP removable who by the remember uh removable singularities here in OK so um that means I can actually write by Edouard g of z in terms of the uh the police's uh sources there is removable meaning that I find right and that I can find a power serious for g where as a power series of all of some herbs and 0 but in general if I insist on this being nonzero who then I must started good yeah plus and plus 1 and so forth the minds the the unit cell uh there exists a power series for G of 4 G worth leading coefficient 0 n n and n and gain in that not because it is a what James so now and let's look at uh let's distinguish 2 cases namely in is 0 and ends nonzero in order to derive a contradiction that this is an indirect probe we want to show this is impossible right now has information on gene we want to and go back to information on that right so what do we do all the right uh and look at the look at the definition of G. If I wanted uh to results for S then its efforts there too and it's rented out then efficent equals 1 over G plus per cent plus W might miss an because in the s now if she has this policy rules and it starts OK for n equals 0 this here vanishes and this is a naught so it's a clue n-order constant class something some powers uh we see directly but this a singularity is removable here at the singularity of f at b is removable just by plugging in the serious here it's 1 over a naught plus in the it's
1 over a naught plus and 1 senator plus and so forth plus W and so in the end I miss out there be some so class will uh so it's the edge in particular at z equals the all these terms go and I 1 away a naught plus W so that its removal OK so no I wouldn't say this is where some niches yet so there's we movable immovable singularity had at the ways with 1 over a not you know it's a speed is 1 away in all of that that the bay to remove it so contradiction since there since we assumed that we we wanted to show so we want to show that it's not a name is an essential as singularities so it's either order was not contradictions as have a removable or it's a pool that's willing to show here OK so in the other case we must end up with the uh well and yes creative and 0 that's correct yeah and so we have a case we must end up to the pole right solid see how this goes and well then doing the same thing but now we have 1 of the In a all times in history times Z to the minds between N plus and so forth just like it in plus W and and uh clearly uh by my we see that this is a pole of order n since if you multiply this with Z minus B to E and then this year goes away and these other terms uh I no problem in the neighborhood since this is non 0 you then and the denominator we don't have a 0 so it's it's uh removable singularity after having multiply was Z minus B to N and if I only multiply was about as B 2 n minus 1 then it's not sufficient time remaining with will this term here to 1 and so I see that this is a pole of order n is this cold also order and it's he I missed out the same in the right so if the conclusion holds if the conclusion does not hold if images not dense then I'm in the case of fiber removable singularity of told meaning has proven that they claim corrections will OK so this so this tells you that the right hand side properties that claimed and the remote in now that it's a bit easier to prove that the converse statements hold an so both for 2 and 4 3 we need to show that if if the mapping properties then we either in the case of a pole or in the case of an essential singularity so suppose the mapping property in case to close efforts said tends to infinity and modulus ways that goes to 1 1 K Asada Cisco and then all OK events that delivered the singularity is not removable singularity and B is not removable a it since then I need to have a bounded value namely what I can remove that will and also also it can some 100 service and and also according to what I prove here this is no uh why have done this in the strange order so now I want to refer to the party uh to proof of party uh so what direction right to this thing here so according to what i've proved as when I have this the singularity can lobster cannot be essential right since since the image must be dense here so if the images dense than I can have this but also according according to Part Three put forward to who uh yeah this property means here that the images not dense so it's not a it's not essential so for so career singularities essential is not sensual on her so and so since the polis remaining cases so must be appalled so you must be Paul down Pierce so and take care of done almost nothing here more this logic and is part of the proof and similarly so the remaining cases the backwards direction of train red if the image is then really singularities essential all that's um similar so suppose it's supposed and that has been and is a in essential singularity and these and I want to prove that the mapping property holds and not so if I want to prove I'm sorry I'm confused myself I want to prove if the images ends then I have an essential singularity itself it is supposed that hairs the dense image so it has a dense image up here at at the singularities at singularity some can be so OK then because many I see directly that it cannot be and 1 supposes severity with dense in Japan and Europe carry 0 yeah then it's impossible OK is impossible there if of said approaches a value intending to be instances has a dense image of will not converge here sure this is impossible and also we have a case also the dense image it's not and not possible that all values of model that our efforts that are larger than our SA so this is also possible and are impossible to OK came and so he can only be essential yeah so singularity is an essential it's again I'm arguing again this is that this is remaining case the OK so base and really the key thing of this proof is what's written on the left hand side because a lot of us OK so I believe this gives you some feeling on
singularities and and I will come from the next to this the the the next 1 expert 1 class last class and but he is for remaining 20 minutes or 25 minutes let's do an application of this but I Ch look at the the he so the problem for the next 2 classes to Innisville study was studied general uh integration of integrals over general functions with singularities and will come back to as the the characterization of uh singularities in terms of uh at least in terms of loss but for today what I want to do is uh he discusses an application namely the so-called partial fraction decomposition the partial fraction the decomposition of the all rational functions of the rational functions had solved it may well be that you know what to say this but here is a a rigorous treatment in terms of lost serious so and what is a rational start with what a rational function as well as the quotient of 2 polynomials as you probably know so if you have P and Q say uh the polynomials or nomials no can and on C you on on C and uh this becomes the denominator we won't be better wanted uh not identity to 0 who OK 2 is not uh identities of the 0 polynomial then the by the fundamental theorem of algebra Q is a discrete set of zeros and the cell phone a fundamental theorem algebra tells me that the time give it a name she who his disk 380 said all the zeros said enough q in the complex plane focus meaning that if I divide p by q then is well-defined except for finally many points so now p over q which I call OK let's call it s going from CD power was the uh singular set of the denominator removed f of said the coolest cumulus led to public you've said who or where is the rational thing is called a rational function it a rational function but I think is so this defines a 10 and what do we know of form of for rational functions so this ecstasy Silas said I have an ivory um the function f is removable or it has a ppl so poems and tolerance also in q uh s has removals all has the movable singularities all pole and so why is this so well sexuality say take a B and N of Q factor out the beads the look at sectors Z minus B in the factorization of P N of Q right uh by the fundamental of fundamental theorem of algebra we can factorize P and Q into the effect of such as this and it's a denominator 0 B also turns up in p then it can be that the singularities actually removable actually this is the case if the order of the power so here's some some policy is a power of in the numerator is larger or equal than the power in the denominator all the relevant sectors OK so this is the case if removable singularities and if the order of 2 is and if a 0 if the order of of the 0 Mb of Q is larger than that of p even with a poll also uh well of ordered the difference of the orders of the polynomials OK so 1 important factor 1 important fact for additional functions is a they form a field so the set of rational functions I'm a bit sloppy here rational variations my in my hand uh since I do not tell you where the defined very defined so threshold functions defined on some subset also uh some set of which is C minus a discrete set here that's very defined the set of rational functions it is clear that seals the company the sector rational functions have inverses when it's not not so hard to see how will be inverse there should be man and you can add and multiply and divide looking yeah so it's quite a brightly it's a nice as today and I said and in fact isn't it did we do this exercise that uh you can make it actually ordered field which is non comedian many I guess he did this in the 1st term and no does anybody remember what did we do this yeah it's it's a nice example of you can take it for so you haven't done it by yourself horrified did and said it is a problem then you can look it up in Wikipedia there is an English version the you how the orders to find the order is just given by the leading coefficient and then this becomes an ordered field but it's not a comedian so it's so that the simplest counterexample with their properties work at the OK but I am I don't worry about in order of what I want to do is decompose get to a decomposition such that we can easily integrate uh integrate these functions which is the partial fraction decomposition so so that that's the following composition than just you and says that each her own rational function each rational function 1 C and is this some out a of a polynomial of any polynomial people want to and terms of form and same as a linear combination could will always mean
that that linear combinations of cyanide here but some we have have a wide out a slightly version slightly different to what is in the original course notes well I think is this is superior right assigned a finite linear combination of terms 1 over the also yeah the 2 also form 1 over Z 2 minus t to some power of a full B who for B and C and and if you want to see the entire thing F equals here so what is does that equal levels there is the polynomic cannot have said plus OK this terms with possibly different orders and with possibly different beast right so there is the Nobel sort of uh the many terms of this kind and these are these terms of and easy to integrate for instance and I will write out f and full form uh course of proof my OK so how do we do this and so on why would have said about us and reason for actually if has either removable singularities of poles and so all its I write so if I take the being 1 say I label them B 1 to B L led to the post them and uh was where's all orders and call the orders N 1 to N L and 1 to L the respect and so these are the numbers in that in our respectively but on the OK so suppose f has so we know about the best you no essential singularities of X so that means you must have a pulse and perhaps as I told you a 0 said of the denominator can be larger but this includes not removal cigarette but only the polls B 1 to B L tuning up with water in 1 2 and no OK and now I want to use the all the composition were 1 serious uh then I go and look at what then I want to look at any of these and poles and use long decomposition OK the principal part princes tool Todd while you on the composition the long to serious and she of Paris and the change in this world we have a poll so we can write it out as a h j OK wave this was done and was that the principal part always contains 1 over Z minus the 0 j here and so this In this what's gives rise to the umlaut serious and writing and out of positive and the positives power so now I need a different letter case and will carry OK and the coefficients and since I have a J. index here honey double indices and now all case will complicated whatever you like and this is a sum ranging over k equals I have 1 yes since it's only the principal part his nose in return uh up to at most an and effect including NJ yeah this is the most serious of 1 of these I take a appalled at the pole I have on serious and it's it turns its terms go up to N which is the order of off respective Paul OK another is not using this using using the 1 here is there's not much to do in fact so this should actually right for some numbers in see for AJK complex numbers then OK and no um yeah I know I know I have and I know a bit these h j uh rational functions known to man and his the Jason clearly they're rational functions 1 over yeah Her rational functions can be seen here so I can write them on a common denominator in and see that I can write the sum uh is a sum of 1 over Z minus B J to is uh largest time and then as is a rational function Mountain and so that means that um are OK so now I want to subtract all these principal pilots from uh from my function so not to do this I defined 1st the Lord of semantic and P. not have sent my polynomial as ever said I the subtract called the principal parts how and when it's the minus B 1 who plastic plus and the last 1 is H. L. off 1 of the incident minus B L and and that should be it OK so that means from I s I subtract the singular parts and then I should be left and was something that was something polynomial really here if I take a poll in a particular function and when it has on the poles so this is the rational function uh out so if I wanted to say it is not their BHJ irrational function and so on a so here is also rational the will carry yeah it's a field I can subtract uh is a rational function but I've uh I've subtracted to all the principal parts and uh this is a unique decomposition so uh all the singularities here of penal it must be uh must be removal right since isaffected the poles blueberries all but but the singularity is uh singularities and also the knowledge that the most my polynomial penal and removable he are removed so that means that pin or it is a polynomial is a polynomial regression loss function and we the removal singularities is a polynomial locate so if I have a polynomially of I just write right all this stuff of the left hand side In order to obtain that and that answers that had equals the node of said in Maryland we need to add in the ages and if you like I write it out an explicit form so that the
sum so whatever the sums well if you like it's a 1 1 over Z minus B 1 plus plus a what's my index system 1 on 1 l over Z minus B 1 2 to the N. and no it's not else it's n want 2 and 1 past and now I go to this 2nd singularity and I have to was here up to the last 1 when have the last indexes L so used L 1 over said minus B yellow plus plus a what is it l and l over Z minus the and to be an outlier all came so this just writing out the beings these principal parts of the left hand side a to the left hand side and then on right here and this is the desire to a rational fraction decompositions and it's important it is desired to uh representation I can yeah so basically it's the it's given 1 decomposition and nothing else to do that note that this is not how well this is not true uh over real so why not well since I cannot as synthetic cannot decompose and my function so I
cannot um uh and in particular I don't have a the long series decomposition so um in fact if you do the real version of this it gets more complicated the essence of factorization will include not only linear terms like so here I mean atoms I mean and and and and and bright and orders and what I called this pure terms that that's the same 0 and and and possibly Apollo but you need square the polynomials like x squared plus 1 no Soulier irreducible components of every every reducible sectors of polynomials over the reals and so in the real version this becomes much more complicated yeah to write out the complex version is much easier and it's good enough for and for integration even in the in the in the real version uh practically um I mean this is a typical mathematicians fearing that tells you something works it doesn't tell you tell you how it works my how do you see out and what these coefficients are and will polynomial is well as the best idea is actually plug in the given mass and as a result of that for the coefficients nearby then by planning in values right OK so this is something I don't tell you how what the best way to do this is actually but it can be done now this is the message OK so all of the last 2 minutes do you want to see how how gals proved his proof or thought he could prove the fundamental theorem of algebra of the how let me show you is just to use up the last few minutes let me show you this nice idea to prove so this is about proving fundamental theorem of algebra doses original proof of 17 1990 which is not approved um but it's a it's a great idea so if you want if you want a uh in value be such that the polynomial is 0 right have you look so Mozilla of a polynomial what when we can write this as a real party or speed is 0 and the imaginary part of the the key to this that's the 1st idea OK how do these sets look like yeah let's look at sets Z the real part of peas uh P of z equals 0 and similarly so imaginary how does this look like well these are subsets of the it's sort of 1 condition these records and this is what's and of the slide shows you uh where the green ones say ah well I wrote something out of forgotten save the green ones are real part of s OK I wrote an essay instead of P is 0 red ones are imaginary part is 0 what do we look for well we at a
point both things happened so where these curves intersect right so what we want to locate is read the green curves meet the red curves OK why do they intersect while agalsidase idea was the following and he claimed that if you go out at infinity and you can see it here nicely at least the simplest curves come out in a nice pattern always alternating imagine impaired 0 real part 0 and so forth and actually at the very same angle in the limit where does this come from well look at what z to the end us well 1st of all look at z squared i here's a little the picture I don't know how much you can see of it so this is where the real part of z is 0 and this is and so I want to look at this next letter has the conditions that the real part is a real part is 0 is it the fault green dots where guy II and minus and there had to be imaginary part equals 0 which is where know I confused myself uh which is at that you must be a direct points of direct ones are 1 and minus 1 the pre image if you plot is in the image which is what this picture desk you see a direct the regular pattern of we'll part of some polynomial z squared is 0 imaginary part is real part and imaginary part now if you do this at higher power become nicely what nicely space and it's that uh at the same angle and the way to see this is just plug in the to be i t to the end and look at where uh it travels n times around the unit circle so it it equal times you hit the real particles if imagine having a 0 and we'll particles in so this picture is accurate at infinity so that's a point 1 point 2 is why do if it's accurate why do these curves need well this is so obvious here if you yet imagine the imagine you have these red curves here and here's the green curve OK how can the green curve not need 1 of these neighboring red OK if records go like this and this goes across while but then consider the next curve here the next screen 1 and it has to then this is basically his idea and you see the many zeros here so I who OK I'm starting with the over time and that'll please give me 2 minutes it's not much so Dallas says 1 thing I did reason for as these curves don't stop so yeah this is what you find illustrated here such pathological cases have to be ruled out of course OK that can be done by looking at the polynomial locally and now here's what goes says about his own arguments when if you know a lot of letting go to follow paragraph if you don't and use the English
translation it seems to have improved with sufficient certainty but an algebraic curve Conniver suddenly break off anyway as it happens for example With the transcendental curve was a creation is 1 over the log X think of this curfew no itself as so to say in some point after infinitely many quotes like the logarithmic spiral this far as I know nobody has raised any doubts about this should someone demanded however then I will undertake to give a proof that is not subject to any doubt on some other occasion and this is a footnote by Dallas in the original
and so he himself expresses is a good mathematician himself in press says I don't believe my proof yeah this something else to do but his as well I don't worry about it asked me to do the details and it took almost 100 years therefore these detail to be details to follow to be filled out namely what would you need to say is that if you have a curve from the boundary of a disk to the boundary of this it's you to regions which are disconnected it disconnects it disconnects the disk into 2 components the Jordan curve theorem proven in 19 10 hours 15 or something and so ghosts courses original prove was incomplete but he nicely acknowledges for this vector of funds is very nice to read it OK thank you for the time and see in a week from now