Residue Theorem and improper integrals

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Residue Theorem and improper integrals
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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
correct assumption which is what the mathematicians like here it's not anything where you were it's it's meant you have the assumptions of the process's assumptions you need In order to perform the residue theorem of ECO she into the formula or whatever you like OK so this is an aside let me now come uh to the use of the uh residue theorem shall still visible all who OK the so this is about is the classic the top increases the computation of the also improper integrals computation or into the world's phi and residue theorem and as you know there are several a several nice intervals which you want so what what's an improper integrals In this context usually something from real integral over the entire real line from minus infinity to plus infinity and the some of these integrals which a particularly interesting for instance uh in their probability theory you want to calculate E to the minus 6 credited as a uh probability density yeah and OK was the trick going to several dimensions we did this by and we can also be done in different ways but I will concentrate on an easier example here actually so this is very uh is actually I 1 of this slide calculation in terms of the mention in terms of residue theorem and the basic idea is very easy it's perhaps I stated right away but the real as contained in C so let me but if you integrate a function over which which may be real of really exist when you replace this by some other well but it's a limit so say it goes from minus R 2 R and you replace it and the sample we will do we will for instance consider this entire loop which means we replace the interval from minus R to our real Xs all by the integral a circle and we have uh to account for singularities in the domain in between and apply the residue so we will say that the integral of this curve is the integral of this curve plus or minus the residues of minus the sum of the winding number which would be 1 year times residues located that's the basic idea and you will see it in detail below but in order to calculate these residues is useful to have a very simple observation can so a simple observation what is the following a proposition as so it's it's basically lemma abroad since there's no fear and uh and I learned that a lemma always must lead to Theorem I call it proposition so that all cases I want to consider let's go 1st to the more complicated case a more general case which is a told of order uh at most n so symbol of so calculating I should say for calculating residues calculating residues it OK so and what is saying if we have a function that so this is because 1 you know that f from our it punctured this can to see the binaural can so there's a singularity at B and X the singularity and we won the representation sentient f of said can be written as 1 of Z minus the to the end times h of evenwhen with h page page of Z where h is holomorphic so this works for searching for in in in In the Kras zeros and doesn't help anything uh and gage going from the same to see but now holomorphic entire this so we have the so this is all the singular part of uh of F it be and this is uh this is holomorphic never and uh HCA has a policy so h if h is for a little more fragmented has a power whom off with power of serious power serious um while some OK OK or a and what I want this Z minus beta decay Kim equals the sale to infinity you talking then I can I know what the residues then and perhaps this is not done then residue of effort be residue at the of s is equal to well actually the N minus 1st coefficient here in the policy areas which is in fact can also be written and thereby General Taylor series fear and what is the N minus 1st coefficient well differentiate the function and minus 1 times and divide by the Victorian respect patrol the strict notes so it's 1 over n minus 1 uh pictorial times the and 1st derivative and minus 1st derivative of it the looking for signs of squeezing but this is the 1st formula for residue who yeah this here is clear anyway yeah it OK and the 2nd statement of I want to make the grammar manner so this is the 1st case and the somewhat simpler cares and that section 1 we will use this the and 1st order pole so uh path of the band and who came up at the end if you think about it what does this mean that yes since I could uh I could sort of divide by 2 large power this means a that that is sextupole of order at most and by this condition but the real really interesting case is that I choose and minimal as such that this representation in Figures 1 over times h uh because there is such that the H becomes holomorphic at uh be there yet and then it's a political thought at exactly and so 1st on Paul so if I have a picture this is like a quotient will soon there she gage the tool Multics functions and I want to consider the quotient will more figure where book ends I require in order to look if a quotient I need the following uh I 1 g prime t of the 1st of B equals 0 Norman not secure silly I want to consider g e over h so h prime of the wounded no there only on k and the only 0 in the denominator age will be X the midpoint B and age of B equals 0 only for uh as uh z here course for z equals B so if given isolated uh 0 of age then by choosing are small enough this will be satisfied OK and then I consider the quotient then and in get a name to that effect F equals g over age which is defined now on punch it is the OK against a loss exposure uh but it has a pole has a couple of as a whole all also order a well 1st world 1st order ends with residue and this is really what matter as the formula for residue pairs the residue at the midpoint of the desk of f of the quotient is given by G of the over h prime of it no OK so this formula that we will use later on and to prove at most as long as this statement so that proof that start with part 1 you know how do we do this well uh we use um a OK we use a use Taylor serious so if F of said includes some they can minus lead to the k k equals 0 to infinity and OK uh um no not if if if h equals red then f of said is obtained by dividing by Z minus B to him so I can just write this like this yeah this is the the uh Taylor series of automorphic functions h divided over Z minus B to the and so minus and so this is what I get for a grant by using perhaps I should they are using using current policy areas positive this of NH and uniqueness of laws here is if you like the opera and this is the unique uh 0 6 expansion for this and now if you look at this and then what what does it mean what is residue of this long where does the coefficient where's the coefficient that to the power of minus 1 more you have to plug in n minus 1 in order to get a minus 1 and the solve for k equals minus 1 you get the residues of the residue of f and the is equal to 8 and minus 1 and this is what we click yeah so you get this formula up there perhaps they need a numbering so you have so this will be tree is slowly er er who know to a new case study uh it chain tool and not sure perhaps I need this I'm not sure but definitely I need it the loop OK and the other at a OK so this gives Formula One and part so it's really appear pissed and now I am for too long they do something we use power series and fall both G and H who in case Israel must have power series expansion so I can write our world as opposed to the 2 which is she lives and over H of Z and now whatever power serious well and using the assumption tool here uh for GE I start with a nonzero coefficient g of the plus an outcome higher-order terms and I summarize them in large show of Z minus B is standing for anyone 1 times said minus B plus a to Z minus p squared and so forth and what about age age starts with h of B is also so age starts with a linear term so what is a linear term 1 it's prime of B and now you see what I wanted this to be nonzero so this is again a month and there were a number of of the leading term class and now I write higher-order terms um number should be actually actually what's this is the right way I don't know I don't know what my notation is insert the correct notation unify higher-order terms OK and now I can rewrite this as 1 over Z minus B times yeah sector out the that defect on 1 of these and right inset minus of b plus tho of say minus B and h prime of B plus 0 of Z minus B and now you see that we can take the limit of this if you like so this after 1 to write solid pepsin right this quantum separate line so if I multiply was set minus B at obtained it but this function here has a limit the all In this function here has a limit which is exact precisely given by what in the limit of the Spanish but remembering that these are nonzero I get to the very quotient t of b over h prime must be here OK I'm not yet done but this is of course this is what I have my formula my desire formula so why is the uh why is this very residue what I computed brokering um its this comes from considering very uh 1 serious flaw as loans serious flaw on poor by saying that 10th of z equals some old for a and Z minus B to the N than equals minus infinity to infinity is OK and this means if I might apply with now I'm trying I want to multiply with z minus piece of the more in the times this equals the sum of end and now I have Z minus B to be N plus 1 who hold day and this is what this means that is what is residue of this function well this is where this becomes minus 1 uh so this is a a this is uh what I want to say no I want us to look at the limit so I want to look at the limit as well the amendment z to be honest that minus B as said what that is precisely given by the term but this is the order of so this is 0 order if this is minus 1 so it's a minus 1 the the grenade on a minus 1 is the residues so but the residue of its it the cell or together we showed that what's here but the residue of f is a minus 1 can be expressed like and limited and this limited is g e over h time and we're done focusing so this this calculation I had to do 1st and now what were an well I can come to some interesting examples will I early room who word so here's the rooms how firms but actually before I do that speculation of an example it's too let's do an example for the sphere to see how word for the proposition to see how it works and then I come back to calculating improper integrals so who of can selling awareness the in examples of the propositions namely uh ACA function where In both cases can be applied so it must be a pole of order 1 an so what's my example here uh F of s z equals uh the to the i over since then this 1 no OK so why is this a 1st why does this have 1st order folds well this of course is that minors i times in class II so it plus and minus sign its 1st order the poll so let's look at for instance the case that my it it aII so there's there's that minus sign factor so all if you want to look at to use the method to which is fairly obvious that it's a quotient It's a quotient so uh let's uh let's use the quotient rule in there so this will be this would then be g and this will be H and so all the assumptions are hopefully uh true or g of B G of i is nonzero is any exponential h prime end points i is not 0 since this is just primes to at so to that it evaluated I and so I want to look at it I did a b equals i yet then h primers tool I b is nonzero and it's uh it i i have i squared equals 1 0 and and this is the only 0 in some neighborhood timing missus the plus and minus i it has singularities sorcery this would be a neighborhood where I is the only 0 so there will be satisfied from i equals 1 the can and then I can just copy be careful ministry then I get the residue at i of s is equal to what I have to do I have to evaluate III to VI i t to realize I Scranton over uh and I must take the derivatives and i which is 2 eyes so this is so this is a minus 1 so this is 1 over e so it's 1 over uh dirtier sitting correctly 1 over a tool i.e. or have written more conveniently uh was the i in the the numerator minus sign over to eat don't and using Method 1 for the same example if you want to see that yeah issues whole now here we must OK here we must effect alters the minus turn right so we write f of z equals factor alter 1 of those that line of sight and has remained with the to the i z over the other sector which is z the last slide but yeah so this will be my age death in the power here is obviously 1 yeah in looking at this representation here him OK so all I have to do now is there to do in order to calculate the residue look the the residue and I of S hopefully I will obtain the same results is what do I need to do now I I can um I can be differentiated 1 to do that forgot that um it's just 0 here N. equals 1 means I have OK there was nothing to do really n equals 1 means the zeroth derivatives of h itself over 0 over 1 sorry so it's age of I but what I have to compute the age of science and what is age of I was this function evaluated at i which is the same thing as you see it's the III of each of the mind yields minus 1 of size thread over tool I which also is minus or liabilities to the this to give you an idea how you apply this when it comes to problems here OK and let me know that OK now it's before I introduce the example like it's better we don't have a break locally square then let me come to a final example was which is inside change yes and and if if OK so I want to compute and compute uh an improper integral and the into want to compute is so example it is that I just want to do 1 example character nation and coarticulation also integral of 1 over 1 plus Z to the floor from uh or T 2 before let's be real from minus infinity to infinity and remember that the remember how into improper integrals are defined we 2 years ago no 1 year ago yeah if if you have the yeah to an improper integral and a tree and here the case that the equals infinity matters and load but a matters infinity also there is just defined by limit where you take intervals approaching infinity but when it comes to an integral there you have minus n plus infinity there's a convergence counseling problem yeah you want to make sure that and it it so if you take an arbitrary limits of this uh this boundary approached and that boundary approach then you may have consolations by doing it 1 way and uh you have explosions uh by doing some other way so it's defined by 2 inserting an extra points seat in between the natural choice would be 0 here and S. splitting the to go into to a house and requirement each of these limits the single limits exist same problem as for laws serious where the sum from minus infinity to plus infinity of so yeah so this uh this a keeping keep this in mind now here there's no problem Rachel Wiseman no problem since at each of these intervals from 0 to infinity and minus infinity to Zero converge nicely uh why do they converge nicely well since a one-dimensionally can integrate anything was a power at a larger than 1 and the denominator and here we have so how do I say this size say 1 over 1 plus T 2 4 of is less than 1 over the t to will follow right and so this is interval integrable uh from 1 to infinity near or from minus infinity to 1 year for the so that means that means this integral integral uh converted into exists to be the British short him he remembered the the critical power his 1 for the interval from 1 I mean yeah I'm I I can consider this kind of integral minus infinity to minus 1 yeah the critical powers 1 so if you are larger than 1 you're done yet by achieved by what we did a year ago at x just do a calculation using son a theorem OK and so in particular this integral is just 2 times and here types just write it out it's just 2 times and the same value for t and minus T cell I get 2 times into of 1 1 plus the into the 4th dt from 1 to from 0 to infinity if that 1 can snow is no problem with convergence so Heider exists and converges absolutely so now there are actually um 20 pressure I deleted sorry I cause I want to do this so I do it's the this is correct but I don't need this I'm sorry a what do we want to do we want to we want to write this as a limit that's the right result minus infinity to infinity of 1 1 plus the 2 will form of the teeth equals the limit part to infinity and now it doesn't matter it does matter that I the strength of the at the limits and I have the existence of this interval uh also 1 over bond plus t to 4th OK and what do I do here well known now here's the idea which I sketch before so we integrate over the interval from minus all too often in which I will actually as a curve I simply call it this end we replace it by a a semicircle so this is so I have here by semicircle and 1 to replace integration over here by integration over there plus or minus 4 residues uh according to the uh residue theorem so this is the idea um and perhaps I need a curve In this way I quote age of t is the to be out of our times the to be and parameterized with T and C with 2 pi it the yeah then this will go on display around and this interval I consider oriented like this so that's the whole thing becomes will become a loop and that's the principle idea here now the the OK so in order so calculate this by EU the residue theorem in order to do that we need to check that the residues OK and since this is an easy polynomial uh you know he's conceive is directly so all and then dp equals 1 over 1 plus Z to the far guy right now complex notation since he of the numbers of complex uh is the holomorphic except but exam 2 example x well at the full places where z uh 2 before the equals 1 when is that the case well nicest way I guess is to magically itself on the unit circle consider the phone numbers that z to for the is a minus 1 his minus 1 so 1 way 1 number which does it is here yeah since the 45 degrees the other number of nonsense about 4th policies exactly minus 1 and the 2nd way is here yeah that's was 135 degrees since what if you take the 4th power of this you ate 4 times the angle 45 degrees and plus 4 times the angle 90 degrees so actually you go once around to minus 1 and similarly so on the the diagonals here so it's not a false roots of unity since I have a plus another minus but uh it's said these nicely symmetric numbers which are called P 1 P 2 the the for and uh the different ways of writing them out I could say it's 1 plus I. overrule tool or here I say it's easy to you uh or expert also I a player will form year and this year will be explored I I I I know I take 3 times this free force of type II this would be a explore of 5 fourths of type I and this was the explore of 7 falls off my power OK about those I will I will meet since I apply the residue theorem large applied in the upper half plane but I will be large applied uh if I go blue below so the singularity is by hearsay over unit circle if I'm assuming that I've large which I 1 yeah so if I take a domain you which doesn't include B 3 and B 4 goes up here I'm fine near so this is what I want to do it um so table OK so can u equals just to uh to everything in a clean way I said said consistently with the it larger then larger than minus a half care services everything up after his life and that will exclude that will exclude beefy and before yeah so that the only singularities I will have our B 1 and B 2 I could equally well actually assume use the entire complex plane and include those singularities bubble winding number would be 0 you that's that's no way of uh uh attacking the problem OK so no um the now I want apply residue fear and it tells me that have a single the climate residue filling now put it on once again but Canon of some sort so what does it say what it tells me that the integral over this closed curve our or need a name for a need and name for their then from candidates C. R. B. this interval run through in a positive a plus H R which means I start here go 1st there and then In the of OK so now I am the integral over C and which is so they should be I could also write directly when the CIA it's like that a who came to me if I integrate over this curse my function 1 notable one plus Z to the false and so what by the residue theorem is 2 Part II at times some here I will include a limit imminent uh and how many singularities I have to uh unfortunate across nodes has a k uh but it's it's have to lose singularities assuming that R is at least 1 so long for then I have to singularities in my domain here no and bounded coast so this can also be uh J of C R times the residue off while I didn't compute for residue yet can't and here this is something I missed I still have to do yet OK residue of the change uh also my function truck ever call s therefore the F is a that includes wonder work that key forced us 1 OK so this is the residue theorem and but if that's true for how I can take the limited and in fact on the right hand side word of 0 here's an hour it will not depend on it since they only to fix singularities and closed and that in the domain as we will see OK so this I want to computer the and so the 2 sides to it let's go 1st to the left side left hand side meaning that I will replace the straight the straight line integral by the the semicircle the line integral caring so I want to say it so the left hand side of staff is what what's really interesting is we to go over H R and I claim so wise why is this a good approach I forget I forgot to say this well this function here this function here go to 0 would tend to their old man as that goes to infinity so on this semicircle the function tends to 0 yet so I want to show that integral while H R 1 and 1 set for the z that this is an a tends to 0 yeah that should be true how do I do this well I do this by the standard estimate for line integrals which is the as the estimate by the length of a curve times the soul of a function the sooner is 1 of OK on the appropriate uh steps I'm she and let's plug in H R page 1 of the to false here and now I have to and 0 2 pi for instance OK and now how can estimate this wall the other 1 for this to be less so that means that the nominator I I can enlarge the denominator uh I have to find a smaller denominator solve this will be image copy this time swooped all of 1 over and now have uh the only way to do this is reversed triangle inequality and I can say to gage that are to look for for the the let's do it like this minus 1 OK and since Oslo to that 1 this is mostly defined and now of this will come this is just an hour to 4 so well let's write it so this is uh um pi r the so what's the length of a child's pi times on more OK and this is part of the fall of minus 1 and is clearly converges to 0 as r tends to infinity OK so that's effect which sort of you can almost see directly from just from affected his decays rapidly enough to make good for the length which is 1st order not here 1st order versus fourth-order here OK so that means a here I don't have any contribution all the only contribution to this interval comes from the residues here now that's that's the conclusion OK so we need to check what the residues are uh in but not winding numbers are so winding that start with winding numbers so on on right hand side of stone so 2 things to compute the winding numbers the the chain of C I I claim rail 1 why is this so while various ways to see this 1 way I mean 1 nice geometric way it is to say to say that curvature is gone is OK I'm say B 1 now I watch out to my so what's the polar angle near what it's actually an increasing a strictly increasing function along the curve going from whatever it is to to apply plus whatever it it's right so uh you see directly since it has a since the polar angle comes can comes from projections that the uh winding numbers 1 and similarly soul if you go if you go around here yeah if you you recall from that point yeah it's again the 2 pi 0 angle increase or it's by an embedded provably various ways here so all the way I say of this is here it's synonyms of C our heads I and can be parametrized that's a nice way of saying it can be parameterized referenced bipolar angle problem angel yeah and and alternatively i could say that you can check what the integral what we integral over 1 1 over Z minus B B B 1 or B j is and use all over the on and use the previous theorems and you also get that is 2 pi so and that by by various theorems and believe it's 22 and I'm not sure various statements yeah this also works OK so and if I had included into you uh the free and before I uh code of similar said that the winding number is 0 since the integrate this uh if I have a 3 year and integrate over the I see that the integral is 0 for instance or I can directly say of winding numbers uh 0 achieve when I didn't do it in detail as the homotopy invariants and if you use that it's clear right away from the geometry whatever so this is this is clear but what the residues have to compute them right so I want to compute the residues and have way I want to use to do that is I want to use uh that formal ever G is 1 here so use of self or residues you use uh and number 3 was g equals 1 the who came here with 1 over uh a function f is 1 of the that you to evolve plus 1 so the H is the Z to the fourth plus 1 contains so this gives that any residue of pj of uh s is equal to residue of PGA also uh 1 over H just efforts who uh today ever say what pages what H of Z for agents said equals 1 places that thought to came and this is by the pheromone it's 1 over H prime yeah it's 1 over H prime all of feature this so this is really by 1 by create came in and so this is uh what what is the derivative of this function was 4 times Z to sell its font times 1 over the simplest 1 over 4 times bj q on no uh and that I have to calculate groups so these numbers they be on from this get closer to them being you communities so want to have um we have that there is a dual of P 1 of f is equal to OK 1 over 4 times and now I have to plug in my values e to the IPO overflows here in this war on over for the access to all of the from while of lizard priority will fall off to the 3 make it like this and that is and can rewriting gives the to remind us from so this is the false keep that the to remind of 3 fourths free of 5 of Part I the yeah and what is this well this is uh minus 3 over 4 times pi i so I go 3 quarters around saw on the unit circle I'm here so this is what is this number it's of groups 2 times 2 minus 1 minus by yeah just since the length uh the length of this vector is true so this is altogether length 1 vector and it this is the fortified and this is the correct angle here the OK and supplying this N means I get a however of this a half times the quantity a turn through times tool times the minus 1 when something and similarly so if I ask me residue of B 2 of the f what's the change uh it's frequent it's another because another three-year yeah so it's for it 3 from the power 3 times 3 quarters of the time we can make so the difference is I get here in 19 the dXt willingness mine quarters Part III whom carry so what does that mean I have to turn I have to turn around a 2 year so this is this uh quarter fear have through 2 plus 1 minus pi this the point here OK I A X I could have used the half food tool 1 plus I. expression light of a then perhaps that I but then I have to compute for it cause I don't know perhaps this is the better way yeah self what together this gives again in h root tool also and this time it's this time it's 1 minus i and now I have to take the sum of the 2 so the sum of these 2 lines the yeah and if you look at these things it what is this some more work here we have different science so warm minus 1 1 doesn't contribute anything but here we have a minus i minus I so this is to so I get there are some rest you want as cluster is believed to have is equal to 1 and all that one-eighth goes away to a quarter to a true n minus I the minus i him that should be and that should be the sum of the residues came so why don't we write minus i over 4 times with 2 the OK and so altogether so on together I get from star the and we the it implies that OK on the left hand side what's remaining implemented it's the interval I desire to compute the the the can overwrite so I have now 2 pi pi throughput at times uh 1 OK times 1 times this plus 1 times this so I computed the sum it's and this year so it's minus uh so my interests are I over let's continue to group true and so this is minus i squared goes away 2 and 4 remains a tool so this is pi over tool route to or if I cancel another route to its pi over root OK I think I have to do 1 of these calculations in detail yeah cell um was computed as integral and it's was no way to do this directly in terms of the fundamental theorem of calculus can be we don't know what the primitive areas of assumptions OK so the there is a large number of of 1 less interesting intervals and you can compute like this and perhaps 1 thing I could could point out is that other approaches of approaches uh this would be the following you could use and in order to compute this line integral of our you could also do it in terms of right angles and Our minus science and some fixed height h 0 1 or whatever and let go to infinity or minus infinity respectively or if you look at more complicated cases Leadenhall you can I mean you find a large number of examples here in the the fight oppose them book I I don't know if you can if you can see in this kind of curve but it can become more more complicated here you can see for instance if our singularities of the real line you make little loop the you go around a little here like this and then you I don't know what to do perhaps user rating or whatever yeah so this you would use at a singularity in our and this will be done with some radius Eps Ireland and then you it uh here's our minus or and then you let uh silent to 0 and R to infinity things like this these are ideas to the you can use I don't think that everything will come up in the exam yet it won't him OK so this is actually um this is actually uh at the end of this section so um you know that I don't like to close and the early so um the rather like to close later and the so let me just say 1 thing about what we what we have learned BM hopefully hopefully and and so if you look back on the uh on on that on complex analysis yeah what what is it actually what links to everything what they might have said is often enough I believe but last class and those too I realize that the things I said multiple number of times the were not observed so I say once again yeah it's it's really this scene arity conditions of the complex derivatives yeah that that is the key point here that the bit f prime is the linear means means that the cautionary money equations and visa partial differential equations and of course the Riemann equations here we we to show that they are uh integrability conditions and infect for line integrals for loops now in the complex setting these are the Peace our integrability conditions the now it's basically by writing out interval F of that is that in real terms now you see that you integrated what is the best part and I have bar the he c-vector uh that uh year 0 rotation for these intervals of provided provided the sold and then you get to the er co she interval sphere yeah and and once you have a co interval theory and what did we do then well the coaching the formula as a key to everything how do you get from the cushy integral theory into the coaching integral formula was if I remember correctly the only piece of information you need is that the integral of 1 over Z over circle is a 2 pi I hate him and some tricks self so it basically we get yeah by this equals 2 pi how OK you know what it is that if you don't know that don't go to the exam um then you get to the course into a formula and the course integral formula it developed into the integration formula or into the integral representation for the value of little off a uh um of civil morphic function that is the key to everything here is so for instance we plact invade geometric serious and of course the integral formula and obtained uh and elasticity aspects that should have started once the barely a week differentiated because she integral formula and obtain smoothness of 1 of more of one's differential and who uh complex defensible maps we and reduce the and the 2 metrics serious and we obtained of necessity and then there are all these um other large or small theorems like the identity theory like the and maximum principle 1 has to be of principle and so forth of and as I showed you at the beginning of this class not not anymore then because she interval well because she interval fear in together with this form well together with the formula together with the formula integral of 1 over z to be and is 0 over a loop and n is at least 2 yet which goes to residue fearing never so it's but it's from my point of view it's not that much energy which has the strong consequences which are not true in real analysis and the key point here is a is this year here we have it's not that complex to French abilities death a little version of um real different ability but it is really the French ability class a partial differential equation and as for other partial differential equations that strong I mean the better of the partial differential equation is the stronger the properties you can derive from it and this is sort of the trick of complex analysis and I hope this became clear my what I try to do is I want I gave I try to give you the concepts of complex analysis in the generality to be applicable some else yeah which is why did the and also I I thought I do good thing for physicists but now mathematicians tell me that the I shouldn't have mentioned rotation well uh I could have easily but uh it is by using differential forms so what but I don't think that anything is gained so it's in fact not physics I would tell you but it's used in physics a lot and have many applications of applications of complex analysis also to physics yeah think about the the state space of quantum mechanics yeah from are free to see their scattering problems and so forth where I don't want to mention everything and there are many Avrim applications mathematician to mathematics so many that I'd better not mention them uh but perhaps I keep a note in the notes and that because at this point I hope this was interesting for you and I will see you next term hopefully uh in the integration class the integration class also makes sense for physicists and least certain parts of it I can be more explicit it if you ask me here so thanks a lot for attending at the