Cauchy-Riemann equation

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Cauchy-Riemann equation
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so good afternoon ladies and gentlemen 2nd class of complex analysis this today all programs will be 1 of the most important fundamentally cations in complex analysis the coaching Riemann equations but before I come to that let me start with a quick review and of what we did last time I hope that was too difficult so let me remind you so we did it the we mainly mainly studied linear maps so there MEPs safe from linear map from r 2 to our tools all and we had several it lend a uh properties namely and it is given by complex multiplication and what yet there this is a complex number and this thought is a comic is complex modification that's equivalent to that this linear map has a matrix representation uh how is or Holkeri hairs by that no Harris matrix my uh representation the a the matrix satisfied that satisfies but they're on essentially only 2 entries a and B and the 0 minus and a minus B and a and in fact in order to go from here to there or vise versa what you do is you set C. equals A. plus I. B. this corresponds to this that's a simple calculation and dramatically a this means that that's the map is a rotation combined with the dilation so it is the rotation on the pets at like this is sortation dilation how an on and what that means is uh but what what consequence of that is its anger preserving and orientation preserving and vise versa so it is angle and go and orientation all preserving they all OK so this will be important this time since we want to apply it to the differential map from secrecy 0 all OK in order to talk about differentiation have uh let's look at a function from say you in C 2 see you're just the plane see to seek it is complex the French ability the all of Mets another city in another and thing we did that last time it's complex differential both if while perhaps let me the little sloppy and not write everything out yet violet difference quotient will limit of difference quotients you look I'm not yet complex difference quotient of course uh or by linear approximability linear approximation and since we will need that I had right it's just the make column and liberty the explicit here that means there exists a constant C such that f office you us age equal to f of sleep plus and now comes the linear approximation C times age plus remainder term never remainder term is the low of 0 so i z i of H over age tends to 0 for h how how tending to 0 OK that should also look very familiar to you I hope the and we also mention but will not need to uh they cover territory the way of writing the hate right hand side in terms of F A C plus a continuous function which is a function which is continuous at the and so that will be needed and the main definition of all the main definition I made was s is called from you to CASA areas where you use in is holomorphic scored below warfare so we know that so can is that on the well if
F is a C 1 function in a certain sense in the real sense namely so there is a C 1 function where considered a function from there are 2 to uh to just in order to say what 1 continuous derivative means uh I want the real the real um derivatives continuous not the complex 1 so it's little that's too strong uh assumption and moreover he actually once you off I guess we want you also open uh missed by the here you opened they should have gone into 1st and maintaining s is complex differential both differential at every x for all of all the things you OK so that's our definition and last time explained that this is perhaps
the twin strong condition that the derivatives are the or are continuous however we will see later on that it's automatically satisfied and it makes no difference in the end uh whatever right there so the main problem now is that which we will address today is characterized characterize complex differentiability the complex differential ability different such In the key and uh also real differential but maps Rio clustering should follow next so the the problem is to say OK we required to up there we require that the map to be real differentible anyway that what makes it complex differential that's something I didn't really and touch in their analysis 1 and 2 so the answer will be given by this go she Rhemann equations the move move them who ought to be very explicit by the co she Reman differential equations so that the in her all cases so what is that while gives the answer to a question to the problem and so uh to to write these equations out let me at 1st a right out explicitly what we'll defensibility means the so and we call f is from what to 2 are to say what is real differential real an after 1 C 1 no differential tho but not yet and so f is real differential totally differential uh as we sometimes called it an analysis tool and at the point say z which we write in vector notation as you read in our 2 the now and it's if there exists this linear maps and linear map not proud deep at which we have uh denoted by idea in sup C. with matrix the on OK so it's are 2 to a true so there's a Jacobian which is a 2 by 2 matrix cell with matrix a in R 2 by 2 at the end the while this property uh which I mentioned before hold such that f of uh moderated on f of that class a which equals if so that 2 plus 8 times the matrix A multiply it with the vector n each end there is a remainder term are also age and now I'm real differentible unlike the definition of their so my remainder term my remainder term now and needs a modulus with no limit yeah more what our z of age over a over H I was limited equals a world will OK so this is really the French ability and allows to OK and so we can write out as matrix in case of the function is C 1 the member no uh in the C 1 case I showed you last term that then the matrix is simply given by the partial derivatives so if the there is a C 1 function has continuous derivatives it's it's uh see 1 of whatever didn't around it there are 2 the and what we have perhaps should better right again a domain you uh anyway um is it's a C 1 function then various matrix consists of the partial derivatives so then a the then the Jacobian a equals while standard notation will be 18 you came a find my components uh this should be sorry this should be x y to be standard I'm sorry x y in order to have a half uh you class IV which is standard notation of I apologize so this is the you over the X ray at the this is the view of what the y and see this is the UT parity EVA d x and this this DVD what the I the 0 and a standard notation in complex analysis is a an index notation for uh derivatives so let me just write this out since it's much neater to write so that's UX and those of you UTX you why the the the x end of the he why OK so I will keep on using this kind of annotation and what do you the OK so and now the whole point of these equations is is very simple to make the now that we know about complex multiplication well if this matrix product here corresponds to complex multiplications then we know the effects of their that this matrix has the same entries of the diagonal and a new minus negative entries negative of on each of the of the end time diagonal OK and these are the cursory men equations so the that's the named ceremony here yeah it's not really real I differentible only about this
corresponds to complex multiplication then the Jacobian has carry some extra information and those are the Gaussian women equations and I write them out no for inverse form you you can see and the open in order to have a holomorphic defined and S. uh from a C 1 map for you to see where always see 1 refers to the real partial and semantic the caves in that s it is holomorphic His complex uh derivatives or complex linear approximation this a motherfucker IV if if you will equals let's write it out once we have real part of f and the equals imaginary part of the if uh satisfy the kosher equations the groups the the course that is for the from all she had and Greenland tell you the differential equations whatever goes well we say exactly the properties of the matrix stated by us that you sequence by the Y and uh the x equals minus the 1 so let's write them down perhaps I write them all at once on this like the modern in short form you and C equals the why of C on the anti-diagonal the x of C equals minus you why what the ball game and then there's actually a fall OK I should 25 ns for all z equals x plus C. Y Y. Y and you for the end there's a mole of a statement which is that if this is complex multiplication by complex number what then this complex number is the derivative of f a complex derivative so 1 over moreover a right so then if that's the case then f prime of C equals exactly you X + 5 V x i or here which is Romania as of which is what I wrote at at the very top 2nd line yet if the matrix representation is like that better corresponds to complex multiplication was a plus or i but i b this is a up there and this is be down here so this is the same thing and by the course she remit women equations I could can also express those in terms of my so it's the why the miners on uh you why if I want all it's I forgot to write the X X A point C on the right hand sides uh OK sorry this OK so this is the fear the and so let me know and your own you had an all all all lowland in these may slowly and 1 thing I call these solid yes and and the the I will always call see our focal she and dream at the OK if a what were on that you know and so Maryland in Brown his mind if March me the if you will so that it means you get further you the last time was my OK if approvals on and let me say before I go to the proof which is trivial and that basically taught told you about the proof but I will do it and let me point or 2 forms 2 things and so in in short form the theorem I wrote up is short just for you to remember it means holomorphic is the same as real differential bow wrong class kosher women know came just to be for you to be absolutely certain about this fact and and let me also I given example perhaps just to see how this works called when on well my example any problems it's not interesting please if it's about OK let me look at the example of that's squared care what is its credit well is its real part is X minus Y squared and the imaginary part the the but that it's quite a noisy and on Bova at I'm sorry what about you the hi I think you so this step of the function want to consider its just check vector because you remind equations hold what I what they need to cheque well this is here this is you and this is uh well OK the the is this is the without the i so I need to check the I'm sorry it's just too noisy OK so what do we need to check that you differentiated by
X so you x equals 2 x that should be the same as the differentiated by what the by Y will try differentiate to x y by y it's also you do X so in fact this the 1st part of the story this should be the and the other 1 is I differentiated I differentiated use by Y I uh which order to direct them down uh my started with G X so let's do that the x equal to earn to what I and uh I have to show this is minus u y y you Y is minus 2 y so it's minus you want so indeed is correct here just for 1 example what it means to check the course Riemann equation OK anything else to point out and so I let me it's not the case so let me give the proofs on canvas it's an if and only if statement so the Afzal Morphic if and only if sorry it's not yet and often only statements and was the nothing's wrong so far however if and only if well the standard away it's this I don't know if you know this definition which we can now of made this I defined as if and only if will can node and then and OK so then you better right is in only its out but from that's good option might not OK so what prove so now that it's an if and only if statement as for instance I could start at the right hand side I show that if coaching Rhemann holds the if kosher Riemann holds then it's holomorphic right how do I do that and so so I suppose suppose and has a continuous real derivatives continues derivatives thank you today then and then for a real definition the then OK let me go to uh um but the real definition up there OK sorry this should perhaps get to stop so the real definition of so different stability and world can that's of that so what I want to show coach she Reiman implies the complex version you have this is for complex version here and I know this complex version pull and kosher remember and I want to derive the starvation the real version and I want to derive this year with the appropriate um remainder term decays love care so the OK so if it's real differential then it and in C 1 then this holds for the Jacobian of the partial derivatives and start holds for the Jacobian the was s then of that's then OK but then it's a what then the course real is to the that is Jacobian of f has exactly the property that it's a matrix where the diagonal have the same entries and here negative entries so of course you Reman 100 savers complies that z maps to a z is complex multiplication the complex multiplication insect guy to exploits by the and I Q 1 the yet since as we pointed out here if I have the same entries here and and time negative entries there then the matrix of resents complex multiplications OK so that means if it represents complex modification will then c is exactly this number this complex number here and this thing holds since the remainder term decays with flying over there and this here amount to the same thing uh in our 2 and in C. so and red out of it uh um devices complex multiplication and that is that means that the complex the complex definition I don't know perhaps the 1st the complex complex definition while uh different stability of linear differential ability quotes Don when I didn't say listed here is that actually the remainder term there estimate that complies the remainder term here it which comes just for taking norms camp the take the norms of that then this is equal to what's written over there and so the limit is indeed 0 yeah so perhaps I write with remainder term estimates term estimates OK so this is 1 direction and the other direction perhaps I'm right on the other lower portion blogs so this is our men intercom pretty much the same the early all communities of the other direction will be from left to right meaning that estimate the sum of all the data and the course remember equations so well then I go backwards through this proof if it's complex differentible then this year holds well this is complex multiplication if I write it
out in the real version with the matrix A then I realize that this has such a matrix but this matrix is the Jacobian well that means precisely that the coach women equations hold so this is really the same the same reasoning just backwards location is out seems utterly transparent to me so let's see if the same happens to you OK so and column I want to show that if it's somewhat segment which agreement also so as more fake means and fit star star holds and stand still holds falls these on in you ends and that means s is linearized linearized by complex multiplication multiplication with actually ways a but A. plus I. B. or with a you takes I uh here I yet by Ivory description appeared with the matrix of the Jacobian has this property then uh then the limit of fantasy here is you see here is exactly you X plus Y 1 and 5 in order to do I want to show you what so fast you express side you why and that means and that means that in fact 3 complex and and see now I want to show the uh entities the could many equations hold well so that means exactly that these matrix identities here called here correct with C equals this and a and in a I should like this actually it should be out there and a off this form the and uh I didn't do this will and a satisfying the Jacobian satisfying and this year so all harder to use and I right 2 end of the center of the and quotes the a b minus the day that the use of stand for a and and so that means with a equals u x and decodes you why and so I have to you cautionary mannequins with these entries and posts are equal out to sigh OK so see 4 the OK and in any case the
multiplication is 5 you explores I Y so the my over the statement it is clear from the proof OK so this is this is really 1 of the main things to know and understand and in complex analysis and it's not hard once you understand more complex complex modification of you're behind the do on the whole so I although this seems uh simple perhaps let me let me draw some consequences yeah the which are not so the not expected attack that the the she be yes and some of mean in the the I should add 1 thing which it's perhaps known to some of you since some of the problems stations happen before this class and others have made afterwards so another way to derive the course women equations here another way to derive these equations is simply by looking at the difference quotient namely and you plot in the real age and and in purely imaginary H and know that the limit should be the same and compare real and imaginary parts this is part of the of the of the problem session of this week so that's another way to verify the caution women equations all OK so um let me before I draw consequences let me just ask a pep silly questions namely good question uh how likely is this for functions how likely this is a function holomorphic of is all what and that don't really talk about exact probabilities this is an undefined question but nevertheless let me discuss it if you look at at this question from the perspective of real differentible maps then this question a is actually asking how likely is it that the Jacobian of a given real differentible mad has the symmetry properties that at the same entries on the diagonal and negative entries only enter diagonal while probability 0 yeah I would say not defining exactly what I'm doing but it should be obvious him so so the answer 1 answer is it's completely unlikely unlikely In this since the Jacobian Jacobian must satisfy most satisfied identities namely the course Raymond equations the on the other hand uh most most functions you know all of a little more so is in fact likely for which reason well for the reason that as 1st of all uh see to the N is homomorphic and each yeah that's by will overtake sales of the problems of this week by the same proof as in the real in the real one-dimensional case was as I mentioned last time polynomials all by and then arity of uh differentiation that polynomials of holomorphic and in fact over we haven't done this yet and so by taking the limit of polynomials actually any power series with complex coefficients there is also a holomorphic policy areas yeah power series here means that I take complex numbers and multiply it was the seat the end of our series of holomorphic so this has this needs a little closer but as sort of intuitively it's not so surprising here if I take this limit here and different that's it mainly comes the proof is mainly by beats exchanging the limit of summation and differentiation and as we know from analysis to this is allowed under certain conditions and they hold uh within the radius of convergence actually you know of proof OK so you it basically all sanctions which you can express in a reasonable way of oral morphic for instance the exponential function sine cosine uh whatever tangent to whenever these functions are defined will morphic yet tell me any reasonable function uh from analysis 1 it's a lot thicker so that's an so of that comes really since sort of we are mainly interested not in their arbitrary functions but in good functions like uh flows which have a power series presentation however the many functions which are not holomorphic for instance the simplest case will be at the bar no the and so for example maybe I do this in later or did this last time isn't it example C to see but is not that does not satisfy the smart set cushion romance of them many many functions which are not all of again and consequently which do not satisfy kosher remember if you look at this you will see that 1 of the science it is incorrect for that example OK so it's up to you to have you keep the opinion that that many of you will morphic functions OK so let me give any corollary perhaps I need a little more they saw maybe rented out here look although so let me give a corollary to this fear into the culture Riemann equations I got to 2 statements and 1st the
Jacobian of the the Jacobian uh J. a or what extension the of what more fixed function by of holomorphic function represents so represents a wall uh as a show to you all it's race now it within represents a dilation uh rotation you are rotation dilation whatever the located at every point z the the the OK in particular and that was another an another statement I made last week the if you look at those linear maps while the 2 that determinant you can still see the determinants a square plus B. squared so I have a the determinant is 0 all of this matrix well look at it has rank 2 yeah the determine is either 0 or nonzero and a 0 cases only 0 is both a and B I 0 so that means that the rank of the Jacobian is either 0 or 2 In particular ranking J. has of of z is 0 or to work as usual promote important thing is what I don't say here it's never 1 yeah that's the actual statements never 1 it's always rank 0 0 2 and 2nd 2nd and is the test uh test is while this is actually the definition is weakly conformal In the nonlinear sense and orientation preserving and orientation preserving so this tells you about the geometry of holomorphic maps and which by definition which the by definition means that the differential has these properties the him variant of the righteous that what information uh S prime of the uh of multiplication so that CU maps to as prime of see is uh uh I should really write out the linear map so much application with Jason uh quadruplexes out that W. maps tool f prime of C times W it is we conformal and plantation preserving so they linearization this has this properties and orientation preserving some of the properties we stated for linear maps hold fault the differential that is based in is in the definition let me give you a consequence of this and then we will have short break which is good since I want to show you also some pictures that no so what is it what does this 2nd property means what it means that then so consequences both truth it is take 2 curves in the complex plane here for instance Haydn I C 1 and another kind of see tool which makes an angle such that C 1 time and C 2 prime of the tangent vector at a given intersection point um the the that the OK now take any 1 or more segments at and look at what the images well he is an angle uh no colored chalk is a certain angle of alpha will since I can since the tangent vectors the map of attention vectors requires only did to information about the differential and differential is angle preserving I know that perhaps limited completely different point f of C. and the lengths of the image curves may be quite different uh order 1 to draw them it's all caps draw them like this so this is as fast as C 1 and there's another the other curve groups this neutral point this we have occurred the answer as fast as the true and the tangent vectors here and which perhaps I should draw them longer since they don't have the same length but they do have wall the differential is a uh rotation dilation so dilation means both are extended by the same factor lambda set and the angle is stays the same this is also l 4 of the angle between the 2 curves in the image is the same I will show you some computer pictures and after the break but perhaps I leave you with a problem namely so why is that if this is clear to you then I ask you if you look at these and 0 this is not true OK this is C 1 this true risen angle beta at 0 and then if you look at what happens on disease stranded well meant all angles double the so these race here these curves here will go the we still have constant angle but as we know pretty well is an angle to beta it's 0 so think about this in the break why how can I claim that the angle is preserved in the simplest example does not satisfy this property OK I will tell you after the break it OK solid me go to the money in France in and the light
setting in OK so what I tried to the
the what it did here the geometry of holomorphic maps is something I want to show you uh here uh with Jill work so here you have the most and most common function in complex analysis namely the exponential and on the left you see the domain on the right you see the range and while we do on the left is the mark certain grade and since angles are preserved in particular 90 degree angles are preserved so this you carefully look at the right hand um uh image you see that each curved line crosses and the other is a set of lines at a right angle this is what makes holomorphic was which is equivalent to as being holomorphic and also as I told you when you go it's basically the exponential map is polar coordinates with a stretch factor With a stretch factor in the radial direction like the real exponential that is also something you can see here and you can we can make experiments with larger and smaller domains and so forth here like OK so if the imaginary axis has you know goes to the unit circle the to we I t and the real is should go with a real exons should go to the real exons parameterized see it in the exponential that way but if you have an area and you see these intersecting grid lines than the stable for the mower that's precisely this property of the angle preserving property here I have some of the some of the uh mappings this is actually the uh map from the homework from problems with a factor of 3 wanted on the denominator but of enumerated and just in order to use the same scale here yeah so this map maps the unit square well if you take a large portion and this occurs moved to this actually OK maps in a nice angle preserving way and in fact if you look at what happens to the upper half plane which is what and is in the problems then the image uh why this is not the entire upper half plane on a certain portion of it so actually this is a certain portion of a unit circle and that if I use the entire upper half plane then i it will fill out and cannot show it I will it will still out the entire unit circle what is nice is also that you see uh here's how the um the boundary of a boundary angles are right angles so on the right you see a cursed quadrilateral with right angles yet since in fact the map is holomorphic also the boundary so it's angle preserving on the boundary so the cursed sides of the image for the literal must meet at right angles and then I have a mapping which is not holomorphic units perhaps go to the smaller uh in its 1st make reset OK Labs OK so here's the units there and its image now if you look at what's happening here you see where this is the map Haydn of can you read it is it's z squared over mod C plus 1 it's not holomorphic In effect there you see the curves meeting at non right angles here images of curves and when angles which do not meet at right about sort of a general map model of Moffett map slant angle preserving talking and she like you can play play around with this and see larger larger parts of the of a complex plane filled and the non angle preserving that way OK this is what I wanted to show to some were used to be in 1 of the what
OK and let me return to select records this and the and and and the but the so to the um the the the the the so anybody so we've seen the angle preserving property in the uh and the computer pictures it's does anybody have an idea why and Z square doesn't preserve the angle etc. you no OK man I must tell you since I have an idea well z squared at the origin has derivative to z so as prime that is to Z so that the region it's 0 now angle preserving was only defined for linear maps which are nonzero and we know that rank is either 0 or 2 but in case it 0 with a 0 linear map when we cannot talk about angles so it's sort of amazing that never listen angle is defined here but if you uh looked at the tangent vectors that move it up suppose you study tangent vectors here what would you see what uh it's 0 this is 0 so the dilation factor is 0 so the end of the um tangent vectors have images which are point so attention because I'm not defined so they don't have an angle uh so this is sort of surprising at still an angle these rays defined but this is not the angle discussing so this just a word of care if you have f prime equals 0 then then this is not
defined its weekly conformal so it can be 0 and you may either say if the derivatives nonzero then this angle preserving or in as I did you say something about weekly conformal and understand that angles are not defined a because the derivatives in OK if the so this more to say as consequences so this was sort of the geometry now I have some of the consequences more energy I and so again sort of continuation of a corollary 5 of but written separately but 1 it is that the confused if s assume all fake uh in and In the that's right it's new class IV understanding that these are real valued if this is a C 2 function say from a couple you would toss sees her so 1 more derivative available that is f is a C 2 function and holomorphic then the real and imaginary parts then you end the are harmonic meaning the that's in the passion off you and the envelope fashion of the the initial the someone makes a real and imaginary part of 4 was 6 functions are a and 2nd consequence is that if the other section this is more like an example of consequences so if F is more freak out and real valued and for instance real-valued the in a valued which means that the is identical to 0 say located in right C 1 of you to see then actually the map is constant the now and then and In this guy locally constant and and here I have to explain what locally constant means and you will find out what happened you in the so locally constant means that there is a neighborhood of each point where the function is constant for new cell here so this much about the corollary and locally constant I defined as here on the side of this uh from Europe to see is a locally constant so it is so in each for each uh is for each z in you very exists a borrowed be R of such that it as restricted to br uh is constant we this is not the same as constants and so OK so this means take a point z and there's this ball where f is constant however if a domain is has 2 components so if both of these constitute you yeah then you could tell if u equals 5 year in Europe was 6 here and nevertheless the derivatives uh vanish but the function is not constant so this will be here if you have u equals u equals 5 and here you equals 6 then nevertheless you have UX and UY the derivatives equals 0 and so of kosher and hold for uh imaginary part constant 2 0 so OK so this is a counter example here this is a non constant non-constant small example f locally constant but not constant of and we need terminology to say that if the domain is connected then OK a locally constant functions constant we do this later OK so um so let me give the proof what 1 is simply by calculation let's calculate the low passion of you say so what is that all this is the square dx cramped postes squared p y squared all of you and now uh um i li I used say uh while that's right 1 more I write it like this so that you see the make to see how to use the coaching women equations mainly well uh namely this is so how you but by 7 2 same as keeping the x of dealing d y of the minus OK this year is minus DD X of the the DT stays there and say to function so um we can to develop defined and now for the 2 functions the shots lemma holds so this is 0 you remember the lemma of slots no matter which order you differentiate so the result is the same but it uses that f is course f meaning U and V the R C 2 functions and this is that here and same thing for similar for Laplacian of the yeah the you can do that by you're on OK same calculation end for a the 2nd item of fissile morphogen real-valued there and you finish
the uh and it's the gradient vanishes then you can recover the function you by integration and this is what's we ask you to do in the problems so what else do I integrated OK I'll use out that step but you should be able to do that what we did an analysis tool note that the OK and so this is of a economy and is the number of similar statements the which I didn't explicitly address here which is if is purely imaginary value so of you equal is constant at 2 0 man when then again after score locally constant and actually also if that models of is constant when S is constant thank that the modulus cannot be constant f for non-constant map it is easy to see for fact that rank 1 is impossible but yet modulus equals constant means you on a circle of radius R on a circle of radius R and map has rank 2 of them by the inverse mapping theorem it should cover some area and that's not the case so it must be constant OK so the this much about always 1 more OK 1 think here to mention of all I will not prove it right away will follow from what we do later but the natural question here is OK so here's asked composed to you because I The then U and V are monarch what about it given you cannot find the v such that F equals you plus i've and the answer is in general is positive so this remark it's you know this is the sign of the plane it is the 1 so then we can end up uh and domain you is good terms you know it's good for what I need to specify various uh for instance convex doesn't so what have any holes infected men uh then we can signed the defined on the same set you not tool the harmonic but such that you because ivys holomorphic all is holomorphic so that means that the harmonic functions and the automorphic functions on a set of sisters of 1 another and a monarch functions a quite important in physics um In this law potentials for instance however uh this is a purely planar domain theory in the complex plane is the plane and not space and often physics you need this for space so in the course notes and the next thing you find um is said on power series the next subsection while I decided not to cover it here the current loop so we can go on and so I will actually move on to the theory of the the path integral slide right away now the the if you carefully look at the course notes that section the small print all do you know what know FIL and what do and who are wish now the the OK so I move the I will not talk about em power series right now also and the Florida to later when we actually deal with policy areas and the the so I will move on to path the of not long really and what before uh I I start the linearized uh the videos since this class is the videotape uh has anybody made use of a video is it accessible lectured check mice by myself not not accessible yet OK OK so I will check that end suddenly this week we will make it available I'm sorry OK so path integrals and actually I will look at that real serious I and so why why are we interested in path integrals well the goal is well I've talked about differentiation what about integration right M what's the use of integration while in in real analysis the main use is the fundamental thing is the money comes from the fundamental theorem of calculus so uh namely that allows you to calculate intervals by uh guessing functions with a given derivatives the goal here is to generalize the fundamental theorem generalized a fundamental theorem + catalysts connections to the complex case yeah 2 functions from C to C never from fundamental theorem tells you that say f of x is defined on an interval from a to B of F that's a continuous yeah complexity and it's uh satisfies satisfies f prime equals f on this given interval as you hopefully no and so how can this look like in the complex plane well the in in C. we don't have I mean if there's no problem for integration of complex valued functions on intervals yeah that I actually covered in term 1 of analysis however if I want a statement like this it must involved in invoke the entire complex plane and so if i have a point a and I 1 2 so what did I do here which fx is not a is is not good so I should I should know all the phenomena fear amp OK maybe this is better the the fact sorry this so if I want to integrate from a given point a to some other point um say x of that disease in the complex plane I have the choice of past and it will turn out that we can make come to similar theorem here like the fundamental theorem by
integrating over past love is a choice of possible In the end will see it does matter which 1 we take uh that's the that would be uh where will take 2 or 3 weeks and uh to get to this point the OK so this is so this is some of the final goal um so 1 1 thing is and I will uh use I will tell you but the real theory uh and I will say so I will present the real scenery here and make it perhaps a little more complicated of the society me a group presented the say more general you real case 1st we'll theory 1st the uh and I I think I have good reasons for that too for 2 reasons but I want to be explicit on those namely 1 is uh when it's for mathematicians and 1 for physicists and 1 for mathematicians is that the the complex case case is a special case a special case of the general real theory them general real serious and to my I mean this on its own is perhaps not enough but it it I feel that um I mean in the traditional way of explaining this only in a complex way like you find of the book uh by fighter uh was um and will not give you a feeling that you really understand what's going on here so uh sort of gives real explanation gives the well known real in the other sense of word gives a genuine and a genuine explaination of what's going on CPU so that's certainly for true for mathematicians also small better known them all for physicists uh I say that the problem we deal with is the for problem of finding potentials and the path integrals are quite important in physics anyway so the In physics so uh and physics it's not the complex case but it's the general case physics path intervals Prof intergrowths potentials mentor mentor and so on and so it's good to deal with and so we'd need better includes the case include all 3 cases and not just work in the plane yeah so I think I do a service to physicists by doing what is in this current section OK so let me let me come 1st to potentials on OK and what it so when if you want to if you want to integrate uh in higher dimensions and provide a path integrals what we need to integral grade is and vector fields here cell will will who claims that the functions by vector fields or and integral and vise vector fields carries and so and what we have who cares is as say is a C 1 function of you lost in and now to our uh to the and this say is a function uh we want to find this distance here for the capital capitalist we want to find the derivative source and then integrate the derivative to come back to and so if you have an answer the N so cool you is open this is my if I forget to save his his my standing assumptions for the entire section the In order to be able to move uh around 4 in 4 point of view and then OK M. so then we the the gradient the gradient the the same great equal say X sorry grade deafness that's where the lectures from you 2 our is a vector field defines vector few so this is how I get from any scalar function as s to vector field say creditors sphere on the gradient you know the direction of steepest S uh I explained that in Moses to and so so on and er how factor and now vector fields can be integrated along curves and this is willing to do this and what will be the vector field of a complex case and I will only tell you later on lips the number of the graph it yeah thank you of the prior in what yeah who talk the this is what we had for me so you see this so 1st more minutes some of self no a natural question is can I integrate so given that I get the gradient as as a vector field what can I problem is can I integrate takes the vector field X the general that too few x 2 functions near social problem and you can we integrate the vector field f x say x from you to our and the use n-dimensional to function using path integrals yet we know that B has path-integrals I will state that in a minute battle lectures achieve in many minutes next time but and this is the problem and this is the problem we want to deal with and study in more detail well I the appropriate definition here which is useful as if X is can continuous vector fields if X is a continuous vector fields when and the S N C 1 of you to our is called a potential uh it is called any potential world potential function potential that is grad f by the gradient of s equals 2 x 1 year OK sigh I don't say I can do at all I save it if I can do it I call it a potential enter of course the uh potential what we don't know if it exists but if it exists
spend many potentials exist since I can add constant to my of and the tochange credit it is as as a potential who so it is F + C force you not the that's too so if I have 1 might have a word for real numbers many off potentials and I mean this problem here uh will be surprising if it had a positive solution uh all throughout for following reasons and let's make let's be physicists and make a parameter count and a parameter count tells you uh s is 1 parameter here defined on of the plane that is 1 real function and here you have any real functions yeah so and 1 real sanction I x and real functions so it's unlikely that for each x to get an F since then you could encode the information of N variables into 1 very unlikely and actually know the answer to this problem and he's been all a necessary condition um and perhaps In the last minute I write this out fear of a 2 OK is a necessary condition OK If tanks has a potential so yeah perhaps I should this x and should right out as C 1 vector field F of X it has a potential yes then M the the Jacobian the Jacobian and also j j x y of x is symmetric or you call this in linear algebra cells the joint no yeah on you on you which means that actually and that's d the exercise of xj equals the of the exchange of it's on the that all points of the p and you know and we did this entry in analysis tool we prove that the process simply by differentiation just just to this 1 line yeah its if I write the for various then D I A X i exchange indicates that there is a potential is D. I also DJ of f which is 5 the slots lemma gage DI of which is by definition of a potential DJ of X I TGA also XII and so privacy especially if in OK and so that's the proof and I get that led in the last semester also and we're a little over time yes I'm sorry this is then the next week thank you for this