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# Winding Number and Residue Theorem            Embed Code
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#### Automatisierte Medienanalyse

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so what we do now is uh what's this a quote from the global global complex analysis this the yes OK so what's the problem well as I told you last time it's sort of the generalization of the uh of the co she integral theorem so calculate it calculates a loop integral if of the desired solve a case that you integrate over a domain or it could have uh which is in the domain of a function is not everywhere holomorphic but has been a singularities for f for a if groups holomorphic with isolated the singularities OK this is what we want to do it and the coaching integral theorem tells you that if they have is still more thick and has no singularities that this will be there will and same thing of course is the singularity that removable but if they are not removable receive it you get we will have an easy formula to compute the value OK so what we need for this formula is the concept of winding number so the and so what's the winding number um while it's as often as a curve wines around it is point so to start with an example consider in the circle but this time I want to run around the circle say groups + part to the IT but this time I want to run around the circle uh say k times so I go up to carucate case it's say at positive integer um OK so we know what the integral but 12th 1 over Z descended at the so of 1 of his the miners PDZ the over this curve is namely since since we go round k times the we just have to edit the K. intervals which give 2 power so it's just the in units this is if you want it's k times the integral of 1 of the 2 pi pi Integral DB out of P of 1 over that minus PTZ and this is good OK so here's a so far before at this formula by saying 0 years of French want to integrate and that what's the lucrative will now for the for of the 1st part of today I want to say 0 here it is a formula which tells you home how often that the curve once around the point yet K types of this integral detects you how much how often the curve goes around p OK and now I want to formalize uh this concept and this is for content for winding number so um but and I want to give you a I want to give you a general definition which could is goes beyond complex analysis so for general curves I want to define this number K. and I say that if I have a loop let's see say and this can be done actually since I don't want to integrate right now can be done for continuous loops so it goes to see but it misses the point p in order to be able to decide on how often at times around p um solid be looked the loop and a I want for now uh to get the right to severe a couple of things I want to assume something namely I 1 2 assume it has a polar uh decomposition no political representation which minutes it a continuous polar decomposition because what's in the polar decomposition while this means of writing C of T and polar decomposition with respect to P is so it's p + role of the III to VI of t the and where Our is positive and theta is a real but continuous is is real but continues so where Our goes from tools of positive numbers and the data goes from AB tool on Dutch it's you know the polar and the polar decomposition has no problem with the radius this is just a model of C minus P over uh this is to a this is just the model of the of t minus P but they were the angle the angle has this problem of being not uh determined up to multiples integer multiples of 2 pi so if i 6 continues if it's a continuous it refers here to the the to the fate of the i will be continues anyway of itself this is just the start of the definition then then the winding number winding number sometimes also called index index in German the alongside the index uh I of C with respect to all of the with respect to t it is the following number is the number and all the various letters in use and various notation so if you look at the Wikipedia page on uh give any name to it but on my name is NP C and indicating that it's an integer and what I do is I subtract the polar angles at the endpoints of the loop and since it's a polar representation of a new the angles theta at the end points be in any work on side up to 2 pi so if I divide by 2 pi it's an integer OK so this is the number and 1 to consider and um to give you an example um 1 of the examples I want to give um 1 is
OK if that's a good 1 would be I take a of all of this come OK let let's use 2 different colors OK let this be the curve running around here like here so this is your favorite seal speed dissolute OK and hearsay say is my point P the graph data I don't write it out anyway OK so what's a winding number out I have to look at the polar decomposition of in a minute I would show that any kind of admits that
and I look at what stated that so here's a video 0 pi over 2 pi 3 halves of pi and here we and at this point where it turns uh well I did the wrong know this is correct there it turns to Brown OK it's it's too high and then I'll go on to pi + pi over 2 2 pi + pi was free pi and so forth and I end up here after having turned around twice twice year right and this is very nice is very nice through little uh animation and a wikipedia which I want to show that to come get the thing running right now so you follow so look at this under Wikipedia winding number in the English page German doesn't have that so it's really tells you how the how you getting a solar consider race uh yeah this is a racing track consider calories follow a cochlear here whatever it does from you find and the number of times you have uh that's precisely and PCs yet it's not a it's not a delicate thing to see what it is here so for instance so here and and c p PC is to yeah outside here it would be 0 the and in here 4 point p in here it would be what yeah why this 1 the if you have a curve which there's some stuff here and here and here yet just think of what the polar angle that it goes fast and back a couple of times here but also to give a transponder once well so this is again 1 inside and 0 outside and a nice example would be In Figure 8 where the number would be 1 in here so 4 point P and this component and please see equals 1 4 point in here since the curved lines around the other way round it will be minus 1 and for any point outside it will be 0 OK yeah so it's not yet clear
logarithm of our and the boundary the and uh this obviously is the primitive of theta prime sulfated itself the plus lysate OK so if I have a loop then the report the IR models of a curve is the same at the and day so log out of B minus log of a finishes so for loop this goes away and wouldn't have well I have i times i times Fatos B minus theta of the minus theta of and so since this since this is uh 2 pi times and PC it's 2 part times and PCA by definition yeah you see the difference coming up to do right way yes I think so so dividing environment but I left out the eye to try so I hope it's correct that and now if I divide by 2 pi i get exactly what I think him so the you see that indeed it's important to have theta continuous since the otherwise I'm not allowed to to do such a thing and actually then there if if it's discontinues member a whole business or you what work yeah so it's really essential to have a fatal continuous infect frangible but this is what I assume groups so the yeah for curve C and P 1 actually how OK uh you can say with a polar representation which is also piecewise continuous piecewise dirigible vessels of OK so and you see in another way which is simply a not homework but will uh and the what problems is said to take this integral and use the real representation with capital X vector fields covered a wide vector field and write it out of course you get the same thing but it's also worth so what 5 doing that in order to see what this integral really is yet so what is the integral really is while it it gives you the complex logarithm and complex logarithm is good the modulus which is why we get a 0 here for loop and it's bed in an argument uh since if you go once the complex plane yeah then there's a step of 2 pi i which is exactly what we used here in order to measure the winding number OK so the so what is to be done well with to check that each curve admits a polar representation it seems pretty straightforward actually but uh we have to be a little careful OK so that's the following lemma people but stress that each curve so that by way this is according to this is not the uh this is not in the course notes by the put on the net at the beginning of class but this is the change so if you don't find his literally in your notes that some purpose here I change this part OK and it's a little more work but I find it would doing so I'll take a curve and for now let just be continuous in a minute it would be uh back uh differential in order to integrate but it works in the continuous setting so let me do it in the continuous setting so it's same assumption because doesn't it 4 point please is continuous and of a claim each curve admits admits a polar representation and he group indentation with a theta continues uh do where 1 to write it out uh and where theta continues and determined and determined uniquely determined what and uniquely determined up to ending 2 pi times an integer up to anything uh models its integer multiples of 2 pi an integer narrative consists of to . good and moreover is fair moreover if I have a assumptions on the French ability of so uh on C they will also goal into fate and are here so dosages conveniently is moreover if C is PC wall and so are our fate assuming the continuity and the same same c or OPC K or whatever you like yeah so if you if you could if it's 17 times differential than the fate of the polar angle theta will be so as well OK so um In fact this is a concept uh it if I view this very abstract is the concept of topology so let me just mention different apology the to 4 G and we call we think of theta it's very refer to and there's no nice German word for this whole he belong there is no and so sometimes also lived in German is the lifts area off 3 the of V say 1 valued value may over or C minus
P E I should write to the minus Pete right so this here this is just a the direction of a point viewed from here's the here's my kind of so this is this year's this for 40 is just the unit vector pointing into and that direction so it sits in the unit circle OK so this tells you that a map which takes value in the unit circle and which you cut it 1 point namely point and then after that it lifts yet it if you don't counter to that book right also this is clear if you go uh you cannot I mean you cannot have a continuous angle uh where you turn once around you cannot uh the there's just no way to make it uh continuous and is 1 where value to just then if you go once around them where's the to by get show so and this is a general concept so so in this case are where theta is in Minnesota takes values of freedom is the the R is called D. covering a covering a covering should say a covering of this 1 so you think of unwrapping a circle into real life and unfortunately this is an in this is not the a theory for its own sake but angles of frequent so this comes up but it this concept comes up in many other respects also in physics for instance if you think about Spain and quantum mechanics here this also this concept also applies namely the uh the group of Euclidean motions as so free yeah if you take that rule and do a similar thing you follow a path and its closest pair then unfortunately uh then unfortunately you cannot be then also you need a covering namely to uh in that case to the free sphere and there is a double cover and it has 2 values in 1 is plus and 1 is minus and that's sped and it is for mathematicians if you look it up for physicists explained you know a lot about how matrices and so forth and you know all these things have you seen them well and if not you will see them I'm sure yeah so I cannot I would love to explain a bit more detail and they achieve nice physical experiments have telling them illustrating that but whoops I refrain from doing this since this is complex analysis but this is the reason why I mention it it's not only useful in some parts of mathematics but if it's a natural concept also in physics our community um have this everything groups and would you OK so what's the idea for getting a continuous polar angle theta rather than 1 in the interval 0 to 2 pi or something will be the concept it is easy and if you follow the curve for a small amount that if you look at the current look of racing trickier for small time yeah band it has a unique angle that we've and know this is the angle I don't know 130 Friedman then if I move the little then it's 133 plus minus pi over to see here then I know exactly what to do in a continuous way OK and now the idea is simple piece this information together so prove the means and the idea is idea the cut cannot see into pieces into pieces where unique what can be uniquely continued right and then please please with different pieces together but overlap uh this the polar angle will be clear so as I know that here I mean in the interval 133 uh say for the free 233 plus pi over 2 veterans of around them in the matching parts and at the same angle and I can continue for over so this is the idea now how do you how do you implemented in a formal proof and the various ways but let me let me do it by concrete formula ambient OK so 1st there are a few without loss of generality so namely uh the Our of T I know anyway what this is this is C of the minus by there's no uh there's no ambiguity right so solved he must be it must be this and so on so it is unique and remember that the curve doesn't hit key so the French ability of a model is also no problem yeah if C is the frangible then this thing here so is differential as many times as you like yeah so is unique and and differentible whatever this means him timing errors and save the or piece PC ones which were right click here you and so forth OK so there's no problem also uh certainly of the problem doesn't come from piece so bad ending P curve uh I do everything with respect to the origin which states that space also without loss of generality P equals zero him by translation what edition all can solve all I need to look at is a curved so it
which takes values in the unit circle so and I want the angle uh to determine what theta is for that so for the a b tool S 1 unit vectors and see some OK can we now prove OK let me do this step by step can prove the existence existence of theta are continuous the with c of t equals the Korea effect of now emphasis what I remain within the unit circle case where P is a uh this follows here in 3 steps as follows 1st um if c takes its seed takes values they use in in the right half plane say so um in sworn intersected to win the square the real part is positive yeah so what is mean I have occurred uh which takes values here indebted semicircle segment the the I will end there will give an explicit formula that Arkansas architect and will do it and 2nd if c takes values in and now I take a general semicircle as 1 intersected with the said and say OK let's consider CD is tool and so just consider the scalar product with a fixed uh was a fixed effect of the that this is positive so what am I doing I'm doing following I take a vector VII which is assumed to be nonzero the and now I say I take the semicircle around this picture of the unit a circle right this is this is this semicircle here positive scalar product with the some arbitrary semicircle a specific 1 arbitrary and now 3 will be the general case general case OK and I will come it that and the so I will use this I would do this 1 after the other and um so I start with 1 and 1 is clear it's there's an explicit formula right here if we have our z here yeah and or angle associated there then it's just using sine and cosine here we see that new cosine uh cosine and sine receive that uh the aka attend of Y over lakes if theta right so let's write this out is said is the to be isolated then writing this real she was real uh mutation x y equals cosine of theta sign theta OK and remember now would be and this case where this the 1st entry is positive OK so if the divide this Y over X and we get cos tangent theta equals Y over X and this is positive so no problem and if you have this uh if it uh so awesome now what does this tell me about uh data while it tells me that the data is Oct 10 but I have to be careful arctan of Y over X + adding integer multiples all 2 pi right so I need to say there exists in the such that the data is this why is this so well this comes from the previous discussion 10 note that the 10 is pi periodic so on the 1st side you might thing it's it's 2 pi times and here however since all angle is since we are in the right plane 0 angle is in between minus pi over 2 and plus by over 2 so it's in fact uh it's checked up to 2 pi multiple here but uh prior over to minus pi over 2 is less than the intention is the pi over 2 so I don't aperiodicity only only gives me values up to it 2 pi right OK so that means um that means I can just set um I can just said so uh I have equivalent of c of t equals to be i theta here to uh to state of T writers once again so let's now look at different points in the park at 10 all of the 2nd component of the 1st component care plus 2 pi and off T so this just writing the previous formula here as depending on on the right and here have with her so I should say and of T where enough to business at home OK so this is equivalent and actually did this when discussing polar angles in the 1st term look it up here we use the there but it's the same thing here um but now that this formula is to get rule the
ambiguity we see that uh theta can only be continued is continuous if and only if n is continuous right so C is continuous so
is the quotient so is a continuous function of a continuous function so this here does metaphor the continuity discussion so is continuous if and only if n is continues yeah then Satan continuous the continuous if and only if n is continuous well what is it what is a continuous function was a discrete target while it's constant here it's and this constant welcome so in that case also I get all the higher the French ability claims easily the fact if I can differentiate sees 17 times you have any kind of prejudice and so they could do it again equivalent that faded is session was 17 times and In this constant here similar and then and then the today in P C 1 and everything else I claimed here OK so all this just to make it clear that we I mean I'm I'm very carefully and I know it just to make sure that nothing goes wrong OK now this is basically the case now I want to pieces information together and let me do they do 1 thing and then make a break at the break here OK so segment of a general circle here what do I do well I rotate it since it's it's like that I use this formula and go through that have rotated I must subtract the rotation angle minus 5 at the end of the argument of this vector the so this is all I need to say here so reduced to 1 reduced to 1 and by rotation my rotation was angle by sigh equals any the argument of the and then use the same very same formula but uh but subtract the you use use of star star the but he was just about to clicked but subtract see and then and then infect uh we have of the same equivalence OK and then also also uh the to the IIT C of because you to be isolated after they're both doing this explicitly OK and the last case I need to piece all these little pieces together very very curve rotates but most mingle of Part and I will do after the rate as so here you see cover winding number 2 and a it's Rivas solver is it's a bit fast that they're watching it several times as he could tells you that indeed the winding number is too yeah it's positive to actually since it turns around the right way yeah the OK so then well was a question yeah cell and there was a question regarding um then I proved so and actually regarding the question regarding the problem do we need loops in order to define this polar decomposition winding number answer know that have this will be the only implication so here are 2 I see I did this for where I wanted to do this this is fine so he I did not assume it's Andrew right just any curve admits a polar decomposition of a continuous angle full stop find actually the winding number could also be defined in the case of curves and then you get fractional winding members yeah like a curve with little which goes as 1 and 1 and a half times around here and if you like 1 . 5 times uh it so I haven't i've never seen any implication of this but it there's no problem yeah OK so let me continue with the proof of you seen what example long enough
and so an well wait was proven the statement in the simple fact that we on an arbitrary semicircle now we want a piece of information together to get to the general case what do we do this well there's a nice as a general analysis argument at which i which is a
k equals l since then I made the foot cables be here so I should say this is l intervals so an error is assigned a number this is all point of it yeah finitely many intervals yeah and then I'm done if I hadn't finitely many then is obscure will be going up here and it's it's a compactness incitement you here OK and now I I use step 2 and yet now I have these intervals that's picture them a a anyone may to and so source here that they could they go like this here have to be on a quiz OK this is what they look like camp and all the idea is to use that tool on each of these intervals and would and iteratively here I can choose any point no I lied there i choose the polar angle which admits but which them fits to the 1st pole and 0 on the common intersection of these 2 intervals so then on this interval it a continuous extension of this polar angle this you need right you have a choice many my and for all the others since they overlap I don't have a choice I have to adjust my m which is my integer n which is a formula by the fact that there are these intervals overlap will change saw let's write this down in somewhere it's defined he finds Satan so data on OK know it up to like can be iteratively that 0 yeah cell all and I know and and when I can uh step to step tool gives continuous continuous data and there I can choose for any n n z yeah it's slower than choose choose uh and instead the and as I go from a k i k 2 a k plus 1 you know on the a can a has 1 the what I do well again I use to to Google obtain a continuous fade however I have to adjust my and in order to match the the choice on the interval I k yeah so to Gibbs as you need can act out a unique in extension continuous extensions continuous the and quote it at that continuously extension also Satan saw that means just and so that it sits on the car here on the commenter sections so this is due to you to i k + 1 intersected died K is non empty all came and so this is uh so this gives the uniqueness statement so here's the 1 thing to choose and that's it yet so the whole thing is determined up to this uh integer at which I can add to the angles and effect I think which is intuitively clear anyway right here I mean this is just makes sure everything works out there the really works out by doing the arguments carefully in order to do them for similar cases in a similar way their self if you have local information piece it together this is the idea piece together to get a global object but make sure and loops make sure the way works of art and its topology OK cell the have entered different ability via the statements following media OK so now M who bear that perhaps lots of things to say let me only state 1 thing which I don't want to prove uh the cell just for since I'm that sign here let me let me do it uh the following say important tragedy important property which we don't will not tools and and not used is the so-called homotopy in by variance but since I've introduced the how they let me say a word before I say this cell where we so we have proven that each curve this has to be no can be occurs each curve has a polity composition for curve with polar decomposition we know that the winding number n PCA agrees with the integral also uh 1 over Z uh 1 over 2 pi i in the line integral 1 over Z minus peak right so uh now we know that the common definition of winding number is correct right the assumption
in that statement of good number assumption in this statement as that it the curve has a polity composition is always true OK so now let me say what this homotopy invariants is it's it's I mean pictorially it's the falling has a point P and say a here's his mind worldwide stuff of Figure 8 use musculature here I can deform it as long as I don't hit P E I can deform it as I like and I won't change all the time that date when the number peace so I can do things like this here here I can make necessary and here will be gone away and the winding number will t will not change as long as P is not hit by the homotopy so actually in the home whatever you of loops they had a fixed base In this in the definition but here as you see easily even that can be is not necessary so that could be that there could be some the sense of how much of the uh where this is true is uh true more generally itself continuous deformation continuous deformation of C in of cause must be done in C without P. uh doesn't change Disney change a uh a NPC and in fact there is a tiny little on-demand yeah he's the 1 continuous deformation that you can do whatever you like can basically it's used for unwrapping it could make it simple yeah and there's 1 thing which I claim before which is that the end this lining them as constantly connected component that's easy to see using this property by just a sliding around the curves uh that translating the curves along a given path yeah if you if you point P and Q get just think of translating the entire curves along this path as long as you don't hit a it's the points then PC is a continuous function as an integral in the integral of 1 0 was there so a continuous function which is integer valued is constant so it's the same value at P and Q very nice problem
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