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# Fundamental theorem of algebra, mapping properties of holomorphic functions

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OK well can complex analysis last class before Christmas let me start with a corrections my I believe uh which is uh which are also in the course notes and it was uh um the best HMM-based of uh who pointed me to refer to the problem so this is the um this is the uh formula showing you that the automorphic map has infinitely many derivatives and it works by differentiating and the co she integral formula and in fact we have to differentiate with respect to the z and not to the state of this was wrong the notes and it's completely wrong so please correct if it's wrong there also I changed the text a little since uh I I became aware of that with what i presented was too short so when we apply the light that's formula which is not present on the slide then we apply the light fewer from we should think of not in applying it directly to what's written here but rather in a sense this is a line integral to the explicit form the explicit form would be insert a parameterization and the and then give the term where you insert you're path interest and have the derivative pretend vector and then you convince yourself that differentiating this expression with respect to say it uh gives precisely uh what gives presented its 2nd derivative and looks exactly like this and then you reformulate sought after you've done this you're allowed to write this but you should be aware of what you're doing and uh so I corrected this in the notes and it would be uh it will be on the META uh later today the OK so this will be the small corrections and so let me the now starting
and the class well and I don't want to is to give their uh use review of the last class but is 1 thing I want to recap at recapitulated which is Liouville's theorem the which is simple to state since it only says that all it says is that F is an entire the and bounded then S is constant the the and here entire means that it's uh defined on the entire plane all of the and efforts homomorphic OK there was much the same as the fuel by Liouville and it certainly makes you should make you aware that complex different stability is not uh anything on the Jacobian of a real Jacobian but it's really a partial differential equation and self strong properties come from the caution equations and this is 1 of them there's nothing in in the real world uh without such a differential equation which resembles a series OK today with the a 1 of the most famous theorems and which is a which we actually prove as a consequence of the of the theorem uh namely the fundamental theorem of algebra so the for a fundamental theorem of algebra the the and I mean there are hundreds of proofs of this theorem but uh uh 1 in terms of units of theorem is particularly easy so let me presented so it's the following theorem if you have uh a polynomial which is non constant so each non constant polynomial the and um let's give formula here of polynomial of said is we address some of the lowest order term the has K and said to the end will the bias term OK there where the coefficients uh a jail a K I in a complex numbers each such polynomial has a route they 1 is 0 has 0 um a 0 well it's a point there the the exists a B and C is such that f of the because it about a piece of things OK I'm pretty sure you know this statement and we I have seen the the number of times um what is difficult about well as usual what's difficult is there is no explicit formula in general it's the real sort of the the uh um uh effect which really belongs to algebra is that you don't get an explicit formula explicit general formula once the degree is 5 or more you have up to degree for or um there are formulas the Due to Cardinal where you can explicitly so that and those formulas led to the complex numbers discovery of a complex numbers uh but from 5 onwards there can be no general formula so this is an existence statement and doesn't tell you how to obtain the zeros statement of pure mathematics not of the plight of metrics in this sense OK so um what you have I should just make 1 uh draw 1 is a consequence of which is uh this is about the polynomial having value 0 what about having an arbitrary values see or that's the same thing since you subtract see from a knowledge and then is 0 office effect a polynomial there will be a point P of equals C of original polynomial so we see that uh consequences consequently he is either a constant I have a constant or surjective the on so it doesn't even even miss 1 point which will be allowed by because OK so here is uh through what most proofs of is the number of complex analysis proved proves in that all use the same effect forced out and so this is uh what I want to uh to 1st but let me just to make notation uh rigorous let me say I can assume that this number is nonzero else did else I have such an expression with lower yeah so but we can certainly assume can assume it and is nonzero for and and in the OK yeah if if I have only a lot of there being nonzero then I'm not I'm constant here so I can really assume and this is least what OK so now comes the effect which everybody needs a lot what everybody complex analysis needs to prove this theorem and so uh what we show is that the highest order term in some sense dominates the behavior of a polynomial in cell highest highest order to the dominates on the behavior I will speak in the following sense that um uh for for line for P and large say and large Z values that are like OK in the following sense there exists and and we may without loss of generality assume it's a least 1 search compared search search for it as um P of z a modulus is what is at least half of this model house in the said 2 to the N the and for z which are uh which are at least our models for all in uh with uh the
OK so this is easy to believe but them let me give a proof of what is approved uh conditions so indeed with falling and we want to estimate what that's it's easier to write the z to the and to the left and to the left hand side here so we get the P of severity of over so that you can OK so what is this uh when she i want modems outside so that at 1st the divide and so I get to yep absolutely it's just this can so uh what I get different ones I divide well I did a a naught over Z to be an + coming in terms with the right the last 1 perhaps the last but 1 uh to the uh to the Apollo 1 plus and since a N has the z to the end of coefficient is visited in coefficient is nothing that I and now I apply the reversed triangle inequality of reversed triangle inequality uh to separate these terms and can say that this is a mistake in sorry this week 1 M and a a n and now comes sort of all of the remaining terms so I can write rewrite this as in all of us at the end but less class and the last 1 is this 1 and minus 1 was there to be it is a 2 1 OK so far so good and now I really since the 1 denies estimate I want to try and wanted to seperate these terms as well and that can be done by the standard triangle inequality so uh and using uh using the fact that I'm working the sets was models 1 anyway so if I take effect 5 Standard triangle inequality to if applies then I can write uh I want to actually I want to take 1 of the sets out since that will tell me that and then I will use for is to estimate without small 1 I pull out and the remaining terms so uh here and then I have in order over there to the n minus 1 class the parenthesis plus and so forth uh last term is to the plane the term a N minus 1 and since our is larger than 1 this is a modulus is at least 1 so this gets uh gets only smaller when I races you can't have using are least what OK and similarly I erase all the the denominator uh I it's a it's a for all these terms in between OK and so I want this I want this to be a OK now it's almost visible right if I can estimate this by a half a and then I'm done so I wanted this is the whole thing is less than behalf in and wise use true well at the you I in of can so this will be true if and only if OK what do we have to do with the OK all so we subtract uh 8 from this inequality then we get my nose is larger than minus a half and perhaps it's better to rewrite this minus 1 over so that uh times in terms everything times a naught plus plus and minus 1 must must be larger than the 2 minus a half a man you by subtracting minus a in the modernist and so this is equivalent to now I once that on this side multiply by minus 1 some of this is z is less than is greater than I multiply by minus 1 greater than the true times uh this thing over it whoops it was OK and so this is just a complex number here and so I set this equal to our the a if it comes out less than 1 I send that uh I said paths to be the maximum of 1 and this number yeah so on the floor the maximum of 1 and 2 this here the OK and then I'm done OK so this is not really surprising and now what makes the proof nice is how we use a stick and there is is is a torque your various ways we want to use yielded and so and we do it in the following way ways so those problems OK OK as usual this proof is indirect so with suppose that P has no zeros it's no zeros OK so the idea now is um well if this is the case then uh we can consider 1 of the key and 1 of the key uh is a function we want to apply a new it's different to from OK so um then 1 of key is again an entire function the and is holomorphic is holomorphic the have to and in order to apply you it's Theorem Let me prove that 1 over P is bounded OK why is this so well for short the arguments of falling uh on the ball on Z which are less than our here on the disk of radius command and I say well this is a clear nice continuous function so it takes a maximum so is bounded by something on the exterior of this disk so 4 of 4 it's a modernist larger than I have my estimate the what does estimates say well it tells me that the polynomial is larger than a half ANZ to be and while effect consider 1 over P I again I get to this to
the it's less than 1 over the other thing of right hand side OK which can estimate in terms of uh so let's do that at you can sell critics this um and who had arrested for exists how such that it f 1 over p and what you lose is less than 1 is equal than OK for uh on the closed disk about 0 I wise this well by continued glide continuity the and you a maximum right on this context set a continuous function that takes a maximum OK and I don't know but this is clear enough yeah by continuity and by compactness OK and the other estimate is 1 of the p is less than now I have a less than 2 over 2 over a ends at to the end and I can estimate this well since a z a z least are 1 of those it is at most 1 of our services to aid in on to the end which is again uh boundedness which is just a number and so this is the bounded this gives you a bound on the French 1 of the p and makes clearance Theorem applicable who so so that means that 1 of the is bounded at the 1 of the P is bounded so the value of its theorem and its entire it's an entire function defined on C and so 1 of the p is constant and while that means p is constant this world the and we done so the the main thing is really to get this estimate on the exterior of a ball which is not Mozart really um perhaps it's good to visualize what's going on in this city um but before I keep on going no OK that leads to yeah of this point um what you can 1 point and why on his it the and that and we will return to 0 entire finds the latest so and OK so he is a complicated problem the basically right now it does 2 things and it shows you uh for small and the values of Z and for large values what's going on with polynomials and polynomial is this year so this is a so is food source order polynomial and uh what we can do now is OK here you see the preimage of the domain and he's here uh the range no OK so now we can uh take little circles here 1st I want to look at the image of circles ways everything on visibility yes this is good OK and I increase on our home OK whatever OK what what happens we will come to that later the red points of zeros of the derivative and when you have same of a derivative function will turn twice or many times around them OK now we cannot quite see what's going on so we have to change scale OK so now now the setting is a little different and so on the left hand side so you see the uh the domain so good in real size so it will be uh I would increase radius from I don't know 5 to 40 on the right hand side the images get automatically so it infected grows but it's scaled back so that you can see what's going on so if you scale on and on what you see is exactly my main estimate here that the polynomial looks almost like ice coefficient of coefficient like is it to the end if it was exactly is there to the floor what I show you and on the right-hand side you would see a circle and exact circle of radius R 2 4 travel through 4 times the set of a 4 means the little circle uh take being mentioned as that of a for its goes round 4 times and the modulus is are 2 before but it's not exactly that of for its it is a tool for with some uh well some garbage right and so then is responsible for uh effect that you don't see exact circles here but you see something converging to circles spend the radius gets larger and larger them so we can actually observe 1st yeah it gets more and more closer to circles don't and don't worry about the the little corner so this is just to speed it up a can be made smoother but then would take longer to show so we decided not to worry about that yeah so if it gets closer and closer 2 circles and that means we leave this estimated here you can you can this estimate he gets better and better larger than it is and why so what all do you expect 0 small well think of the disk this curve what we see bounds of this so let me go back to a smaller radio once again so perhaps even start with small radio I and now I look at disks very interfering with trees on little OK so now we see the image of the disk which is bounded by the circles which is so before the OK and now you see once I had the zeros of the derivative whose images are also marked in red so on the left the points you see in the the main threat points correspond to points of polynomial is the derivative is a 0 and over right you see the respective image points actually to agree here so 3 points move on don't don't worry about the signal and if you increase when you see an overlap OK now you see that the 0 values are on the right is attained at least it's hard to tell 2 times for what should be more than 2 times if you follow what was going on that's to back OK
so and forging there's still a lot of effect here which makes it harder to follow right now with tool preimages of 0 2 0 yeah if that value z equals 0 is covered twice a variety of means I've uh I've uh must have 2 zeros of my polynomial to it's OK if I increase radius and I will see more of us yet so now we get to at least the 1st group and now comes a fall from right now the values there was covered 4 times let's go to bigger radio I and how this and to
OK so think of a curfew seen before which I said converges when we scale to to 4 fourfold covered circle now you see the disk which is bounded by what this this it doesn't have a hole right there is no taking the low hand OK no it's not allowed this is a convex function that it covers all the interior well it must have a value 0 also right this is this is the uh the geometry which is going on here outside at infinity it turns around and times if it's a polynomial of degree and ended bouncer a disk well this disk will cover the the origin also and times what OK and this is what I hope I wanted to show you at this stage OK so
as I claims that we have some 1 . 2 not
only in this example had for was arose well up to now we have constructed 1 0 let me and the now the the all the uh let me show you of events like and there was this is no easy I won't go so I can say getting all zeros by a factorization theorem the OK so that's the following to corollary telling you that in the in degree n polynomial has fact and zeros and complex zeros so for each of for each um while polynomials as being polynomial all in all the uh he standard equals the same thing as before in all of plus because a said to be an well let me assume that a N is nonzero the says the and the end members make sure offer that numbers I like this CE is multiplicative constant and uh zeros of P 1 to the end so as many as the degree there's such that can i can factorize yields and equals a constant times and are and that 1 is the 1 time so it's minus the so here and there the these uh must have a need not be distinct right they can they can coincide so you can have multiple the overseer in my it may be that uh some of these agree so I bk other are not necessarily on what necessarily distinct ministers ministers and really the stick OK and but they are not perhaps I should write that they are unique up to ordering yeah but now let's that like this but the colonists only distinct but the unique at 2 ordering the if OK so I'm saying that and degree n polynomial has any zeros is expected and so the main point here is that the the complex numbers that we can actually sector out effect of polynomials something we cannot do in uh in the reals I in the a in and you the that in OK so and preferred so all um the so this will be approved by induction we a split of 1 1 of these vectors uh by the next um so that means by the theorem understand so we have 1 0 uh the existed 0 say and 0 be 1 of p the and now all I need to know is that I can rewrite it to the polynomial in uh as a development with respect to be 1 and get P of z equals c times are so a knowledge of the wonder development with respect to a 1 a naught plus a 1 uh I should have different coefficients alpha rhi used here said minus B 1 in now higher orders of the the what the OK it why can this be done well think friends from the back to the beginning if it has degree and educator and consider this term with the current with the same with often equals and well the distance to a and Z to the N is given by low-order terms difficult there is that is comes out with power n minus 1 at most so then you rewrite the and the term with the degree n minus 1 that taking this into account and so forth right that that's the way to do this is basically the Taylor series Taylor development of the polynomial at a different point also OK so once this is done near then when OK when some well since B 1 is a 0 of knowledge must be 0 a Lewis uh B 1 would be 0 and so no and taking that to call and uh this is not there and we can achieve I presented uh effect of the uh I I 1 factors said minus B 1 so P of C equals the minus B 1 times q of their where q is now polynomial of degree 1 no new we Q hairs degree 1 and minus 1 I in like this 1 so again my fundamental theorem Q is a 0 sector and make an induction proof and this gives the statement here when possible induction induction gives the class of OK so this basically a work to be done in order to get from 1 0 2 and 0 it's easy and let me also remark that if you're interested in the real case it if you have a real polynomial with real coefficients uh looks the same of course but now think of the it a and this is as real numbers then and you can again you apply the same feeling we had well you know with noting that if p is a 0 then also be biases 0 so of the picture of zeros is symmetric with respect to the relates to this then and what you Hautaniemi and is valid uh can you can factorize at can affect arises to if they're in the form but such actually right it takes to make it look real 2 in the form X minus B all and now if you take tool if you take say Bk and Dk by our together and multiply 2 of these vectors out here Z minus p k times that minus PK but uh then you get a term so in the form x
squared minus will plus whatever um in real terms so we need to it takes plus the yeah and you have terms of this and that formula you need indices now and I don't want to do everything have but it's clear what I get to the so in particular and but for or degree I must have a least 1 of these sectors in the vectorization of my political since the other 1 once even degree what and solve this what had the fundament theorem of Viterbi has a long history and um I wonder in if I should tell you about the original approval of goes from 1799 uh which uh was prepared a few slides of but perhaps I do this at the end of the class depending on however time works out the and the self is a long history of proofs and actually the 1st proof of goals there was not complete OK let's come to that last point later and let me start with um the next section which um basically tells you something about mapping properties mapping properties of holomorphic maps holomorphic functions but I don't think that so 1 thing I want to start with and is the discussion of maximum of extrema well and a complex function can have extremists instances are to value uh but what you can uh consider is um you can consider can consider that the modulus of a and discuss extreme yeah remember that and then standard analysis real analysis the discussion of extrema is a 1 of the main subjects while here it's not a main subject since we can only consider 1 what have but very is still a nice theory monitor which is the following put so called maximum maximum principle actually on red uh I sort called maximum modulus principle which is a nice name principle maximum conceived I so it's a the principle of the maximum for more than that so let me 1st write it out and then discuss how can we that QP of Maine which important here so emphasis emphasizes once again then f from you to see the holomorphic you should and know what's the statement the all the salmon is that the modulus f cannot take the maximum so what does it mean mod it uh attains a maximum the the the and maximum and at some point within you say it and let me call at a and interior point of all this right now it's meaningless interior point since they domain is uh domain is open so each point is an interior point and initiate point of being you know what you want to call it clean people in blue OK and then which means that that means F of 1 this it is less than what if of the for all those that did you the yeah I'm not talking as usual I'm not talking about strict maximum for extreme Abacha's weak maximum saw it's the largest will you which is run and then is constant on OK so that means it's impossible for F interior maximum and just show you how Wikipedia it's phrases which is OK which is the OK so what you see there is simply the unit disk and the graph of the function cosine z on the unit is still not visible on OK it's too large to much OK and I if it's also hard to see but what you see is that actually there's no I mean it's a saddle shaped surface right and it it on a settled you always have maximal at the end of the boundary right In fact for this function here it looks also as if the minimum is attained only of about name of minimum would be obtained while probably only there maximum ideas the maximum is inventor but this is not true in general so in fact this is not the best picture if you have a take a larger disk this is the unit is but if you go to pi over 2 then I mean since the modulus of the father of its function is always non-negative and cosine has a 0 at pi over 2 then use would see a 0 uh as a minimal so uh and all of this this so I guess for a better functions actually to show that due to lack of time we didn't get it done really in Andrea visualization projects project and but the main thing is the think of modern this is a function which cannot have a maximum a set of points in a minimum of 5 and and if some which were actually J I guess the best this if we have to break right now since the proof takes a little time OK so let's make a break right now from OK I would like to continue with and say she this there's never statement missing which forgot and products perhaps I scribble at in here to OK if you is bounded if you are crack now this is not enough sorry let me back it was so let's write it here but if you is bounded by an then and but and I f can still be defined on the boundary of few years and In extends continuously how to do you particular perhaps is key more think of honesty you mean and where slightly larger domain and containing as you by and then I then we have then the we can estimate that the values of modern by visu a post modernist on the boundary how i which is exactly what you saw in the image of the maximal values are attained on the boundary is not a material on indicates the functions constant that he said that 10 everywhere but basically attainable boundary OK so that I would also be forgiving
proof make 2 remarks on I mean someone uh wants the sadly this is not true in the real case area meaning that many real functions so certainly certainly not true for real functions more in real cases Neil setting we ever many functions which take the maximum in the interior graphs like this and so we'll christianism why can't it be true what's the what's the Magic here what began its equation and then as so I can say for instance however it's true for harmonic functions equation the Prussian equals 0 and it's true actually was loud and modulus man and harmonic function is there 1 workflow plashing u equals 0 so this is a scalar-valued functions so it's really true that you've said is lessened the supremum of you all that if you tend to make small interior point bladder when you have said this list and then and then you itself is constant i OK and actually there are more equations than uh which you can come up with the maximum principle to for instance the heat equation that I've mentioned before so I could say how or or for solutions of the heat equation the yeah if you have the heat flow in event it's not true that suddenly at if time 5 4 and the locals and 6 you get a maximum of heat but it spreads out so will not have a max is the maximum on the on the boundary so so it's once again you should think of the course Reman equation as being responsible for the maximum principle to be there to be valid actually it's and there's a very close cut connection too because she interval formula uh he once again because she into the formula remember that once you plaque this center of the disk into the uh cost interval formula then you get the mean value formants insomnia f of B is equal to 1 over 2 pi interval from 0 to 2 part as of the to be i t which is exactly the uh which is exactly the mean value also essay on a circle OK taking this formula in 2 columns so it's no surprise I mean just the this folly lantern colored f of B is less than uh less than its average on a circle but it's no surprise that um the maximum principle holds so the and so this is into to the key idea of the proof so let me just start without the technicalities and say that s of the IVA and mean value formula and is 1 over 2 pi well perhaps I don't need to write it up it as it's written there but I can just estimated by the modulus of f of B class a are to be EIT these on little loss there I will work needed with Continental IT later dt my and so if a quantity going or are there is a quantity is equal to its average it will not a and if it coincides with its maximum at an interior point will then the only way uh is that is constant that's no surprise but let's make a math mathematical proof of so the proof FIL technical so for 1 let's do the following consider now so in 1 we say is interior maximum of a function you want to show that f is constant the idea here is to look at all these points that act as takes its maximum so let's call this set of end for instance consider and the set of points you are such that f of f of is maximal he create is equal to the value taking and b and B is the largest value OK what what do we want to show what we want to show their M. that f is constant so m is all of you and what's the standard cell they also say that this is a non empty set since B is obviously in it right so what's the standard way of showing that M is you for you domain well show it's open and closed then it must land in small that's a nonempty it must be everything so why is it old minimizer close the easier thing is in a way is a closed well this is a closed condition conditions being equal to something who so this set is and so the set is close to and right away so as what f continuous the that means that M is closed now formally continuity means preimage and Close said is closed well this is a uh uh a point is a closed 6 so this is the pre image of the point the real number 1 all of the the inverse image so it's a closed set habits of former for easing why is it open well to be shown in him and all manner is to be shown how all the end if it's both well since user domain use domain and M is a non empty this shows that and is you know which is the same as a right to claim is her so we need to show is that this set is yeah so each point in the set has little neighborhood that uh so each point that f takes its maximum like the has a little neighborhood where if also takes is flexible who and this will follow from me the mean value formula dS of who can she so of so we have on larger than 0 such that the all of these areas are contained in you a share 1 compactly contained in you so in the notes they also is always missing I'm sorry OK so and you within you the highest will point as the he shall have an arbitrary point z actually I 1 let's no than than than that show this 1st full B and then do it for every point and stance ABI's arbitrary when on OK so if b is in the center and then I want to use the uh mean value formula for circle well can sell I pick the energy in order to show that it's constant uh in some neighborhood I take as an arbitrary cycle so pick our uh pick a little ends in such it and uh no not such changes to it and then became a then the claim is that um on on DDI was being I guess it is also what F is also makes no so the good news so on the entire cell this is ah and so on a little circle will be our uh dysfunctional coincide with its maximum value f of the this claiming want to
use an e-mail you formula will carry cell let's assume this is not the case so we are on a little circle and is 1 point of the circular purposes bed drawing sorry there is 1 point on the circle there s is not equal to its everyone if is not equal to its maximum value well that means that 1 is a little smaller Marie so well suppose Monte suppose smarter than and then there exists a keen wanted in 0 2 2 pious such that mod F and the planets are to the EIT knowledge is strictly less than half of the 12 OK 10 well if it's actually strictly less while by continuity of this function as a function of cheese uh bamboozle neighborhood on which this is also the nest then but as 1 of the rights of this this corresponds to the knowledge I have a little neighborhood on my circle that this is also smaller will carry and by continuity it will be like ingenuity same same inequality holds on and in little neighborhood saying all inequality the same inequality here who wants in the neighborhood of the neighborhood the most you know what and that means looking at my and my mean value formula and then I mean value formula mean use formula which I'm all Gibbs man or all or 1 hand we know that f of B is less than the average or a or 1 of the 2 prior integral of f of B plus the are to be I T so when he teed who will carry on the other hand we know that this on an open interval of teens what this is all that's less than the half of the search and it's a certain points t in a neighborhood of T not it's strictly less so in fact the and the interval must be strictly less and then you are strictly less sorry this is strictly less than the interval of F of the but that is so I could write because Our below the in it yet using this fact here as a integrate a function which is strictly less by a community of about monotonicity of integral that strictly less of this value then it's strictly uh and then I get the uh strict inequalities of interval dt but this is as of the and a modulus and it's uh since the average so it's a contradiction right so it cannot be true bed on a circle a bound on a circle contained in you about my given point B in the Senate that on the circle of friends HC it as smaller modulus then f of the it must be equal length cell and since this was for arbitrary our I get I get for all that allow hours in this interval you know I get that m s also this year and from the circle is equal to a full speed so no 1 in the location direct is out so I had on db are also being we have had estimates as well so that said equals to so all I our training for all of our in 0 2 times so that means that we have on the entire disk we have this and also that by the same argument we can replace with B. we can replace the year by any point in the set M where f is maximal years same argument and show that it's all so that tells us that the set and so are you are saying that for any said in an um by and so are her in place of being in place of the end so that tells me that and all each point in M has little neighborhood content and and and and that comes from the mean value for OK so this is sort of a little technical use of the mean value for me lessons uh since he of the problem is that we want to extend the point inequality for function uh really to an interval inequality and then we need continuity and this is this is the tricky bit here OK and it's also a tool to is no easy so you use you know about is contact meaning that s what s takes a maximum snakes so 1 I mean I require that the effort extends continuously to DU so actually also a lot of uh it's 10 is a continuous function on and the closest domain on the closure of the domain and so f takes and the maximum her bike by compactness as takes a maximum all and you hooked on the complex if you buy off and now I'm let's look at the 2 cases says I need a point that say at these no for it and I is the very will uh junior and why not be in you all carry so now I have analternative either B is in the interior of all B is on the sits on the boundary while both cases I'm done I How wise decyl over b is so when is in the interior while then part 1 tells me uh part 1 tells me that insect and S is constant and constant and then a constant functions so that the takes its says soup at the boundary so they're done all DE is you all redone and anyway now then uh the maximum is taking of a boundary well this is the very thing we wanted to prove it walking some this and gives a proof of the maximum principle let me remark then um bad actually leave for so this not a similar principle for the minimum of modern however I had so little to soon more complicated problem that you can show and it's not hard to show just take 1 over the function then if d is an interior the interior minimal mental all of and also also models I mean either answers constant has default or we are it is 0 of its and constant or a minute precisely it is 0 are so the only zeros of the homogenous can occur at zeros of
function and subtly there must be it it boasts its zeros any 0 is a minimal most-wanted clearly all can so now for remaining part of of the class my goal is to show you them more automatically what how um how complex functions look like there's some theorems also out but they are the main chain you she is the to get an idea of what's going on in terms of the so I I have to see what I can achieve a high AUC B for the boy who are are all she should OK so it can be let me say that let me know right down as a heuristic principle it's actually true but I don't want to prove this which is easy well easy to stand out for its entire so let's ect I can look at the if function so morphic functions on domains but let me just concentrate on entire functions although this is known as the but it's easier to see what I want to say well then uh my claim is then long cane then I claim advanced has used the same number of preimages all create images whoever this number may be infinite so all I say it in in 0 union infinite team a for any n so at any point in the for any point and in the range except when I hit when I'm mad zeros of the derivative so same for any point and rearrange the order same w in C such that the the uh for all z in the pre image I and I have uh and if prime is non OK so only for critical points critical points that the derivative is 0 I don't want to count images but an outside I want to think of a polynomial we've seen outside the red dots are at the same number of preimages was for the example I showed you this is the state and I want to make here from arbitrary function and M. also um I want to say that it's not OK or um perhaps I can say the following and OK and but achieved as follows the it follows shown there implicit mapping theorem images of mappings hearing why this cell so that gives you the idea for this time so here's my demand was his mind a part of my domain In order to do or something like that and look at some while perhaps this is multiple E I don't know this is my please uh and some some set of the complex plane it's and magically cavity some state is same the cover art and some of us it's I'm not covered so this is not for if entire but let's just look at it from you to see right mass so this is and this set is f or g you for using a common you with complex color and would say under closure or OK why is this so will take any point here this is a point of a user my assumption is that the pre image content small values which uh derivative uh as zeros so I may have said perhaps I want to take take it here in order to to preimages z 1 z 2 while I know I assume that I'm not end points that the derivative is 0 how does the image look like well but implicit mapping theorem little circles mapped 2 little circles here so if I have tool preimages a w i have to images in some open neighbourhood of w same thing with and 1 was 0 preimages right OK so this is this is where this principle comes from and it it points home care and tolerance and rich as prime as points z with as crime of CAT tools no and B. Ristic ideas right out a Taylor series men and men we have an s also OK I need an ACE model let's called the min and f also z equals OK then if ever is order ever derivatives and 0 of derivatives member Tennessee a serious doesn't start of a constant term but it starts with order but science and a and Z 2 mutant plus and and plus 1 said to me and and so on and so forth and so people came the floor in and and that means OK if I'm sorry if said man be high should right there's a minus b bonsai since B is that point and I do my a and Z minus B to the N spent much to be great if I'm close to be when I can hear a significant forget about this man my function looks like this how does this function look like well it have roots right so if I look for instance at z squared yeah when I see going turning round original chanting ones from the original sorry lending once from the regions I can't all I turn twice around the region on any little circle I can do this and so that that is the fact that I have uh and this is also solar effect from refinement theorem of either by if you like it if you forget about all this stuff that the here you get em roots of the equation so and maybe points here with a single image but also the points where uh where S prime so I may have points z such that I have only 1 image here but then within the tool but then f prime of sin is 0 and the Taylor series would start with a 2nd order term so think of the values being punished fasted edit at 1 point and it's time to tell you the geometry let me do the following let me show you show you some images of a computer and then I will see lect Hindu formally in terms of theorems will be all in me 1st show you just an image there so they all
it is so he's visualization of z squared actually what you see there is the the points in space or our hope also form sets granted 2 continents in real part of the function all and you may see how it turns twice around with a with a point uh was just 1 preimage in the middle of the right to see the same thing for the event q then the function turns 3 times uh around originally and it's punched at 0 in the actual function you don't see this part so you should think of a squashing everything to be the x y plane and you don't get a trafficked on the right-hand side you get it should cover all the uh of the complex plane except for this 1 point where uh which corresponds to a value of 0 and same thing here you get a double cover so in general you get an n-fold cover around the points where the derivative is 0 and this is the intuition you should have also for the polynomials year Sun its goal once learn through various cell and perhaps the style and mode with the this is amplitude little and forget about the cold bare this is not correct so if you increase the years man and the hate sorry this is way
too fast and so on so
our let's OK so this much here is where I don't hit any zeros of the derivative man I come to point uh where 0 the derivative is a 0 now I see as little as spike where'd Irish bits wide receiver and halting go off the stink once again of Z maps to z squared and take for instance being the open up the last in the complex plane I mean him and map it and as its grant to the complex plane what do you see what it's it's the old about half plane and then a ship and the angles doubles men you get everything except for this year so you get the exterior of the uh you get all those except the positives that line and 0 so if you look at under microscope and what you see is what you see at day 0 are at the 2nd order 0 here many will see exactly such a spike however is nonlinear so will open up or do something the so now if increase radius now this is this spike here you that what is it easier to see yeah and the in the sucker mode OK if you increase the radius now beyond the 0 over that then you get a double covering up to the moment you hit the other 0 and you get the same behavior right and then you get
you get a multiple covering and you should see number 8 or the image of the right hand side and several sheets which are punished at parade points and you know that should give you some feeling on how recent Harvey Haugh morphing maps and she look like and let me see what I can tell you about variance of no erm she OK that no words he so what I try to tell you heuristically now this part of the following form of statements slowly I mean it's actually the theorem guilty mapping theorem which is has different names in English and German the and sharing and in German it's usually referred to as the sets ready to each time so what is a town in an it tells us that we had a little more sick function in a homomorphic functions but it is all I ever when is either a constant so if you don't work on the domain I should say is locally constant yeah I'm not assuming connected this right now or it maps open sets to open sets In particular instance uh no offered functions continuous uh and um connectedness is preserved under continuous maps we have the world mosaicking will morphic image so this should be a in particular In particular FIL Mosaik image now will mosaicked image of the domain is a domain of men incident make the OK so um this is something you've seen and read um and the computer pictures but let me show you a just to remind you let me show you a map which is not so in the middle of the slide and you see a real polynomial or whatever it is and talking about this image here it has a ranked line here what's the rank 2 line here so that means that the person is a search there is a set in the domain like I don't know like so the the ball which is mapped to this thing here and so the image will be a non open-set here it's now I've open or closed right so it's really a prominent property that holomorphic maps so we also don't get confused with continuity the prima showed sets of under continuous maps is open here we talk about the image of a tour of an open set OK so this I mean it's it's hard to see that the kind of information you get from this theorem uh on on to start but it's it really tells you a lot about her holomorphic maps and that's exactly what we saw on on the computer with the polynomials that the boundary is always on the outside right here the boundary of the domain is not on the outside a doesn't bound and this is a summary of the square this inbound all of the image this triangle he is not bounded for a month and map it would be is was OK I was I can see there will be hard to prove this so let me just tell you the results of and then we used useful additional entirely about this yeah perhaps it's it's worth showing so so let you uh be simply connected connected so this is telling you something about taking and fruits and from you to see below maulstick who lost again you with that but I'm assuming that S said is nonzero so look at if I don't have a 0 well then the statement is I can take and fruits so actually what I showed you with the tennis serious words in general so then then the website only state you state what to says that are willing to have a picture for 1 that's to both cases there the there exists but I G to you to see the search said or Waszczuk who offer a uh with and such that f of that can be written as the exponential 2 g of that so this means this means GE insulin is of s marriage she's logarithm if it can be written as the exponential or something and now I stayed the same thing for the nth root the outside is 0 so for all n n we can write exist I think X is a little more function will shape how who cool fake h from you to see uh such that h to the n equals F so what this tells you is uh S S H is the n-th fruit age is and mood of that but of course we are not allowed to do this in general yeah there is no well defined and it's only if sanctions were taken to the end equal set and let me know and show you so let me show and looking in order to prove the theorem this is this is the key step really and but I will not be able to prove that today and probably I will skip the proof and I ask you to have a look at the proof but let me show you what it means and In the following picture the world period minimize its so Lord pool of we hold an or OK as so sorry you need to the Merrimack uh are you'd ask flu it's we
located so here you see his the a domain and you which is simply connected and here you see what this theorem actually would tell you for this for this and other reasons as a function of just take function Z and express Z is easy to something so how does this function look like well it's the logarithm so that I can cannot display the love of but I can show you what the imaginary part is imagine part as the argument and then you get this uh a thing like this so this tells you that on simply connected domains uh would simply connected domains you can take a logarithm which is in exactly this statement here so it's it's highly nontrivial although it looks like an bike um statement OK so M. I see that that I cannot explain everything to you and that you can let me a you can let me know if you want to see the proofs then next time but in any case lies which allow christmas and doing good start of a new year see you in January thank you
Rechenschieber
Arithmetischer Ausdruck
Klasse <Mathematik>
Integraltafel
Derivation <Algebra>
Bilinearform
Vektorraum
Funktionentheorie
Term
Ausdruck <Logik>
Aggregatzustand
Einfügungsdämpfung
Punkt
Hyperbolischer Differentialoperator
Extrempunkt
Gleichungssystem
Eins
Zahlensystem
Arithmetischer Ausdruck
Einheit <Mathematik>
Latent-Class-Analyse
Theorem
Existenzsatz
Ortszeit
Funktion <Mathematik>
Parametersystem
Bruchrechnung
Kategorie <Mathematik>
Dreiecksungleichung
Reihe
Spieltheorie
Gleitendes Mittel
Konstante
Arithmetisches Mittel
Polynom
Menge
Sortierte Logik
Rechter Winkel
Konditionszahl
Beweistheorie
Koeffizient
Tourenplanung
Ablöseblase
Ordnung <Mathematik>
Aggregatzustand
Standardabweichung
Ebene
Stabilitätstheorie <Logik>
Komplexe Darstellung
Gruppenoperation
Klasse <Mathematik>
Zahlenbereich
Term
Ausdruck <Logik>
Multiplikation
Ungleichung
Ganze Funktion
Gammafunktion
Schätzwert
Fundamentalsatz der Algebra
Erweiterung
Linienelement
Mathematik
Stetige Abbildung
Differentialgleichungssystem
Funktionentheorie
Numerisches Modell
Punkt
Extrempunkt
Gruppenoperation
Zahlenbereich
Derivation <Algebra>
Unrundheit
Term
Gebundener Zustand
Spannweite <Stochastik>
Theorem
Ganze Funktion
Analytische Fortsetzung
Funktion <Mathematik>
Schätzwert
Zentrische Streckung
Kreisfläche
Kurve
Zeitbereich
Stetige Abbildung
Polynom
Menge
Rechter Winkel
Kompakter Raum
Koeffizient
Ordnung <Mathematik>
Verhandlungstheorie
Arithmetisches Mittel
Erweiterung
Polynom
Kreisfläche
Gruppenoperation
Gruppenkeim
Konkave Funktion
Geometrie
Unendlichkeit
Varietät <Mathematik>
Faktorisierung
Subtraktion
Punkt
Statistische Schlussweise
Komplexe Darstellung
Gruppenoperation
Zahlenbereich
Bilinearform
Term
Komplex <Algebra>
Reelle Zahl
Theorem
Vollständige Induktion
Jensen-Maß
Topologischer Vektorraum
Abstand
Gammafunktion
Leistung <Physik>
Fundamentalsatz der Algebra
Erweiterung
Dreizehn
Vektorraum
Teilbarkeit
Polynom
Rechter Winkel
Koeffizient
Heegaard-Zerlegung
Ordnung <Mathematik>
Nachbarschaft <Mathematik>
Einfügungsdämpfung
Punkt
Extrempunkt
Wärmeübergang
Gleichungssystem
Ungerichteter Graph
Skalarfeld
Trigonometrische Funktion
Einheit <Mathematik>
Theorem
Schnitt <Graphentheorie>
Analytische Fortsetzung
Große Vereinheitlichung
Funktion <Mathematik>
Obere Schranke
Extremwert
Viereck
Kategorie <Mathematik>
Stellenring
Gleitendes Mittel
Biprodukt
Ereignishorizont
Gesetz <Physik>
Arithmetisches Mittel
Konstante
Randwert
Lemma <Logik>
Menge
Rechter Winkel
Beweistheorie
Konditionszahl
Harmonische Funktion
Projektive Ebene
Garbentheorie
Trigonometrische Funktion
Ordnung <Mathematik>
Standardabweichung
Wärmeleitungsgleichung
Blase
Klasse <Mathematik>
Abgeschlossene Menge
Term
Physikalische Theorie
Ausdruck <Logik>
Unendlichkeit
Mittelwert
Flächentheorie
Reelle Zahl
Gleichgewichtspunkt <Spieltheorie>
Indexberechnung
Holomorphe Funktion
Analysis
Einfach zusammenhängender Raum
Fundamentalsatz der Algebra
Erweiterung
Kreisfläche
Graph
Mathematik
Zeitbereich
sinc-Funktion
Komplexe Funktion
Vektorraum
Energiedichte
Flächeninhalt
Offene Menge
Dreiecksfreier Graph
Innerer Punkt
Nachbarschaft <Mathematik>
Länge
Punkt
Extrempunkt
Gleichungssystem
Komplex <Algebra>
Eins
Theorem
Explorative Datenanalyse
Wurzel <Mathematik>
Analytische Fortsetzung
Funktion <Mathematik>
Sinusfunktion
Parametersystem
Multifunktion
Heuristik
Ruhmasse
E-Funktion
Gleitendes Mittel
Arithmetisches Mittel
Randwert
Kritischer Punkt
Menge
Sortierte Logik
Rechter Winkel
Kompakter Raum
Beweistheorie
Übertrag
Ordnung <Mathematik>
Geometrie
Aggregatzustand
Ebene
Algebraisch abgeschlossener Körper
Wellenpaket
Ähnlichkeitstheorie
Gruppenoperation
Klasse <Mathematik>
Zahlenbereich
Ikosaeder
Derivation <Algebra>
Unrundheit
Permutation
Term
Ausdruck <Logik>
Überlagerung <Mathematik>
Spannweite <Stochastik>
Ungleichung
Mittelwert
Inhalt <Mathematik>
Ganze Funktion
Schätzwert
Kreisfläche
Zeitbereich
Komplexe Funktion
Primideal
Mehrschrittverfahren
Kette <Mathematik>
Unendlichkeit
Integral
Taylor-Reihe
Skalarprodukt
Differenzkern
Offene Menge
Mereologie
Kantenfärbung
Rangstatistik
Innerer Punkt
Numerisches Modell
Ebene
Komplexe Ebene
Polynom
Punkt
Menge
Rechter Winkel
Mereologie
Derivation <Algebra>
Ereignishorizont
Raum-Zeit
Überlagerung <Mathematik>
Funktion <Mathematik>
Ebene
Subtraktion
Punkt
Momentenproblem
Ortsoperator
Zahlenbereich
Derivation <Algebra>
Bilinearform
Inzidenzalgebra
Überlagerung <Mathematik>
Rangstatistik
Theorem
Offene Abbildung
Wurzel <Mathematik>
Holomorphe Funktion
Analytische Fortsetzung
Varianz
Funktion <Mathematik>
Einfach zusammenhängender Raum
Exponent
Kategorie <Mathematik>
Winkel
Zeitbereich
Frequenz
Dreieck
Stetige Abbildung
Konstante
Rechenschieber
Randwert
Polynom
Lemma <Logik>
Menge
Differenzkern
Verschlingung
Rechter Winkel
Beweistheorie
Mereologie
Ablöseblase
Ordnung <Mathematik>
Aggregatzustand
Parametersystem
Komplexe Ebene
Arithmetischer Ausdruck
Logarithmus
Beweistheorie
Zeitbereich
Theorem
Mereologie
Funktion <Mathematik>