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# Residue Theorem and improper integrals       Embed Code
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#### Automatisierte Medienanalyse

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location where come welcome to the last class on convex analysis and here we are laughter laughs um I introduced the winding number which it's quite intuitive I believe and and as we came to the residue theorem and H I give the proof of residue theorem but as since as most important things tend to be it was in the last minutes of the class so let me recall what I did but 1st let me recall their review theory in itself a statement so it's it's formulated here for a simply connected the domain due um as I don't
forget is this is not the most general assumption really anyway if
you have a holomorphic function on this domain with the exception of finitely many points B 1 2 became then end a rule within this domain which doesn't the singularities so this is the curve C here uh and it's uh must be such that you can integrate solace standing assumption is PC 1 and as some would be pointed out last time in the formulation which was on the board I simply said curve but it should have been loop this is important since otherwise psych if she otherwise
I cannot talk about winding numbers here so please Ed Lu who uh inverse statement I had board if you could be the statement of aboard last time yet and then we can calculate
the loop of the line integral with respect to this new uh and it's simply a sum of a winding numbers times the residues weapon residues are uh obtained by taking the long series representation of a function with respect to the singularity and calculating the aim minus 1 coefficient so it tells you that the Louvre and it will only depends on these 2 things the much a minus 1 term in very low all composition and the winding number with respect to the singularity OK and I guess you've proof and let me quickly browser once again from improved if you take and so the idea as and the partial fractions decomposition is subtract from your function f and the principal parts with respect to the singularities which are denoted by H 1 2 uh through HK here and if we do this man uh each of these differences F f of Z minus H. j off that say is a holomorphic function and why is this so while it's since uh each of these functions has a unique role on time-series decomposition but is serious decomposition uh doesn't have all the negative terms
since we have subtracted them and so it has only few positive powers or 0 power and that means it's a power series in a power series 0 always has a removable singularity so that means that each of these differences as minus H 1 f minus h 2 up to h minus h is a has a removable singularity uh or I can say is holomorphic OK so that means HCI can remove all the singularities from G and I have a nice holomorphic function after subtracting all this principle parts OK and now now we use all the of what most of what we know and so now we calculate the uh the line integral with respect to the loop seal of an and we say it's out by linearity of integration is just given by integral of g perhaps and move up and the integral of G plus the sum of the integral over every a minus analyst lessons on our now if it's of the left hand side story in G plus these there singular parts a of sent and so no so now asked to calculate integral of GE we say what this is the case for the coaching into uh formal furan cushy interval fear what does require it requires that C is um simply connected yes he is contractible within uh the domain you while this is the case since you itself is simply connected so each curve is simply connected yeah so when we apply various we use that you'll is simply connected so uh C. is contractible contractible and that means that because she in coaching integral theorem applies and so we get a 0 written out here OK now what about the principal parts and if I plug them in and it looks like this and now the next step is as a probe last you last time is to interchange integration and summation wise that OK well a look at the domain you yeah so this is a picture of you and various singularities bj and there's all loops see which does whatever does were it runs around without hitting the singularities the OK why can we interchange um interchange integration of the summation up to infinity of his principal parts well uh the trace of seed which is exactly what I've drawn in the in you of the trace of C is a compact subset of you'll minus the uh singularities yet in within this set here somewhere the phis defined the trace of C is contact neared C is compactly contained in yeah its values you with the singularities removed the no OK so as a continuous I have a convergent and series the long series converges uh always uh outside I mean the principal part of a long series converges always of the entire complex plane except for the singularity value take serious a so that means I have a nicely convergence serious here which on each complex said converges uniformly and absolutely also had but uniform is here what we need so Tracy is a
convex subset so we have uniform convergence of the um of the uh principal part so uniform convergence and that allows me to interchange of integration and summation so here you see them written the other way it of and now becomes a boils down to integrating integrating uh these functions and what's the argument there well as you see we can this entire some here from 1 to infinity goes away and is replaced just by the summand with N. equals 1 for what reason well and each of these functions 1 over Z minus B j to integrated gives 0 except for n equals 1 so integral so this readable perhaps 1 of a Z minus bj to be n so and and these tool and it's banishes how wise the case well when nicest way to put this is and and remember that we integrate Monday and rounded unpleasant curves and so how do we see this what this function has a primitive you can we know what the primitive it's a it's 1 over Z minus B j to be uh n minus 1 with appropriate sector so there is a primitive for this function except for n equals 1 which is why it's that doesn't show up here so is a primitive and if you have a primitive any loop in the will of a function vanishes is this well the respective line you find here yet if you have a primitive of function here as little as the playing the role of the use 1 over sigma minus B J C then the movement it will of little f is um it's the nobody will over f prime and if you applied in parameterization CE into the loop integral of f prime Smollett's as primacy of T times this E prime of T from a to B where uh a C web sees the same C coincides with the loop sequence sizes of a and B so this is evaluating f of c of t by the channel um ed B and uh the difference from mirror at the points B and a and instances loop advantages so whenever you have a primitive the loop integral vanishes Nevis's well what we need to and these functions to have primitives under this n equals 1 and so the only 2 remaining term is the 1 term and equals 1 and this is what's written here and then all you need to do is apply the previous theorem that these integrals here there are 2 pi i uh times more winding number on which we proved bipolar representation the ends of the wire and the definition actually is here that's precisely the residue so if you think about it what do we use here are not much really all we need all we use is um that they she because she interval famine the then effects is contractible and so forth end to the other factors that we know what these integrals I near these integrals vanish at sea of a piece of information and also the definition a winding number and so forth but basically I mean the residue theorem is famous as big theorem but from my point of view is sort of just a summary of what we've done with uh a good phrasing and what's a good definition of the winding number at the residue here but it is not there's no deep new mathematics behind you nevertheless it has some nice applications and these I want to show you by when so open him and the but
who have so the room the the who are you 1 of the nice things you can do when I was there is you've had no 1 shall I say 1 thing right away perhaps yes let me let me just make a little note the saying that the assumption uh of the residue theorem with respect to the domain which is that you it is simply connected and is actually still too strong yeah can be relaxed New right the of all so the connected can be relates and I will have 1st I uh intended to give you the precise off some idea of the precise concept but I would do this is to say but I will show you why it can be relaxed so what really matters it is that if you have a curve look at this is a nice embedded curve C the say target for a moment about singularities and and you will be and what really matters for the loop integrals is that you can fill it in the so we used to that uh that if in order to calculate the movement it will over uh overseer of s time we use that we have a domain which is bounded by the loop and where the function is holomorphic or has singularities of a given residues all residue or whatever right so it really matters that this curve C can be written as while this so in order to calculate this it's important when we know uh harder right it the yet we have to have to say specify specify you with d u equals OK if I write trace you then it's not quite correct since I uh leaves out orientations yet in an audience way yeah I 1 namely I want this curve C to be uh and such that the domain use of a to the left right so if you think back Weber's problem turned up its inverse real part and the in the potential theory I did the 1st half of the class and where what we did was we use a homotopy H of St and homotopy this curve to point and we integrated over the entire Horn of homotopy yeah there was the key lemma hold key point of the real theory for uh for and the full potential so he had to to see that the curve a curve can be contracted and by integrating over over the entire homotopy we saw that uh uh but the loop integral vanishes and we we obtained in complex language the coach integral theorem OK in general this is really what this is really what you need here but you can write a in your domain of integration as a boundary off something and this works in more generality for instance a simple case will be this you where my terminology is not sufficient essence as you see on the domain the winding number stool uh this is sort of you need an immersed domain with which comes up twice here you you need a concept where for writing out domains that he say this this is a domain bounded by discuss right and down even more complicated cases which I don't want to dwell on the back of the blackboard but we need to generalize this to him and so I say this how here this can be generalized to 1 can be generalized and also it's nice that you can you can actually ate some other curves something like this in addition and then OK this would also be ultimately name has would be truthful to cover here and this can be done but the concept of philology and so on and I want do it that way that a so HCV and the really correct their correct hypothesis for the for everything we do is not a homotopy is I pretended but it's homology which is a little more difficult to define it's it's not any magic but I don't want to do it but the reason that it was a nice section in um In the this In of the bosom on the homology version of the co she uh also residue of fear and yet they you can read more details and this is a concept needed in several other cases related to integration and what some actually uh yeah this is sort of pleasure right onto bright topology the all of your grade OK so I don't want to go into more detail all I want to uh to state is that actually you can do it in a little more generality and then it's the
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