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# Identity theorem for holomorphic functions, Liouville's theorem

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Erkannte Entitäten

Sprachtranskript

00:05

OK so let's let's 1st of all going from 0 to and from Oh global of the title of the the book I will go OK so last time around because a quick summary show last time but this will what do not so to lead to the consequences of the formula so it's valid for holomorphic f defined on this ball and the BI of the the draft left in the area of the overlord of the and and who consequently the of In the 1st and then we want the next are the only ones that principle but it is the author difference so I hope really along the romance the with the the you who is the head of the instability and there are now and entered the event that they can be developed into because so they're analogies or about and world she were OK so today we will actually work in little more in this sort of thing and also some all of his head and who is 1 of the things to things you will see Theorem hopefully but it and so let me 1st start with as a result which is me well basically preliminary for it and it has to say on the removal of the 11th but if the people see them in but for all so whatever these these are the other the the hidden german what does it mean a singularity of that is the point where the function is not defined a little isolated point solve isolated quality the the you say function functioning it's not designed of the so the goal the predictions of the efforts that 1 over there the thing on the file that was little but it may also be is sort of the artificial and the fight this like a think over there the constant function 1 written this way will not be defined example however this is not a problem as lossy OK so we want to distinguish between the 2 cases and so this case we will say that will see you know that the singularity and they also so here is to use examples of singularity and there should be available because this singularity as as that it will still is removal here by setting function equal to 1 that is not removable years since there is well the collar the poll later on her it's OK to so this question is is to

04:02

define this since I can and I feel like the way and say that I have defined the final ball without a single point so this is the feeling of the singularities that as form of all the of the without this little more in so it's a punted

04:29

ball for ITER this isn't the mentioned tools wanted this to seeing I'm at the losses that will OK and now if you had 1 more some of which will be satisfied here but is satisfied by many that the function is bounded then we are in the same time that we can have some refinement income and could still luncheon analytically broken warfare and bound the and then I have an analytic continuation into the singular point the ends uh we can say that there exists a and a consultation here are some of the action and the relation to from the from the entire this the of the of the of the but to see all as before all and this little the and I think it was found really and continuation Quentin notion here means that I have restricted tool the previous delay quote prevented the agrees with a

06:10

given X so that the continuation so this is what yeah real situation where the I you know many fantasies I like I don't know it's meant the uh what of all words in him and you know exclude signed its Saul minus 1 of the

06:41

negatives the really it's easier for a 0 and 1 for plus 1 and which are uh which are defensible outside of the origin however there's going to continuation of this case not continuous 1 and this is on the basis of so this is a the little different here and again it comes from the uh quality in the formula all actually follow the validity of the in the lumen planes and the trees and their contents are hundreds the proof goes well I think that the life of a policy is a presentation from the left or way we do this to make it easier for the love of generality please you'll be moves in the book they will and we and so we consider how of the function namely the um you have to here is the difference in the fifties known be our of the of the defined by a set of books know we might allow by this the so is the so long as the or and uh we wanted to these little for the people to suggest that this would be if I were to ever have to do with definition quote here so now we claim that this is major so all goal is actually to derive the policies and the presentation for age which in turn give you good 1 whatever and the policies of um that for f will give the policies of continuation so in order to derive a power series we will we will appear to be having the analyst is until the last time and in order to do so let's prove that at age is all the more the a ball in cases so in this case the immediate successors along the sense that of all the products preserve full this the soul her I want to prove that that age is complex inferential how but is mentioned by the he so long the Advisory all this is obvious this and see that slowly the point that it is the is the 1 above all that so that you can use a primal dual uh by the difference quotient what's the difference quotient of the world the difference in the of the of the of the the monastery all the plugging in the definition so this gives this is the the clearly although the try to and so this limit is this is the limit of the How much of that will allow you to detect all of these essentially you to the fact that it is bounded if you Saul when using these the the of on the bounded about a sequence of times this is this is the most so as I have said that if the data is that legible it's still which is good for all of us all in their life rule and now I'm well if it's if it's complex defensible is ill at any point in different possibilities of a lot of the entire thing so if it's you lost and then use by the field and found last time later quoted about uh which was theorem which number 20 something 26 and you would use that as an elected and the policy and so that they could have a policy this so what is that old age all say in has a policy of and policies for the field of filling in the sense that includes itself which is the 1 and so all this is some the reason for had that again this is a lot of it is given by its final state of the art is often councils who and solid effective policy resulted in order to allow in a in a in a lost in the field and also looking for OK so if you have that now going good shape as he sees uh we can make the lightest pose usually easy of the product of that which means they have a policy use of were you so if you flip the to the the nature of the divided by the square which means a tree has a fever and so forth the but is the an infinite which is a nice life power series and also this convergence of the board like this on slowly was conditions in quantity of

13:36

video so fit in the use of the the OK so this is that it's solid solutions this was not all of you who so this also is that it falls it and you know she the OK and so that we use and we found that the definition of a definition of that land and since this is a conversion policy is on the of the ability of the definition of a agents and this must coincide with just by definition of a model additional you see what I can say that and then you would like this by the conditions at all that's left and so this is 1st of all for because the knowledge the In the ball and a continuous analytically with the policies to the point he would feel is can we will look at along on the the entire in the in the show him for losing the policy is all this of functions that the that of the world and OK so this shows that and we'll singularities in this expansion is bounded from which would also it has also a wide this is proved not give these functions and differently as it all of them using radical so what what is the argument would have been occasionally and pays the the and that for at exactly that that is the logical that we have a lot of advantages analytic but what we have that up to here is 1 of the things she we all know what time the time and the and the way the thai need this His name for the thank you but the it was found on the a lot of the the the who think for if you have a chance do the the thank you the in OK so for who gets the removable singularities in the next statement on the author this is the I think you I think this if you think that we and the policy reversal of policy the show you FIL this year 1 who's the I think it would be the 1 to say in this class about policy laughter so I the so you want to consider what defines it the on and how how long have said monthly in order to determine the pose of the of the function and the crucial time is a constraint sets both of which are not big enough in order to the policy so this stuff and let me recall would look at the nation's point what quality is so all of them in the of the is listed on the the lecture the theme of the both of them in the what can I just follow my point by point this is from the from the a in it so what is this thing here FIL about the the of the the being that such that in the form of what this is called the conditional and slowly so think of the pieces were that with the fact that the sequences consists of points distance to the point the point that the accumulation point of the set of the the lively this OK this is the sociology for the defined discrete that the flow of let from he find it here so in the same way in the space in the complex plane but I want this and all we'll see around there and said which but in the end contained in this call at this community on the other this cold is in so if it has no inflation . 1 and so the the on the right to it think examples the no isolation I should be careful you and that's it has come to see that I I want you to look at the set 6 year and a half and so and with the stuff and the this community what so that these have lived in the that's good only that that what is in love the increase in the each client point in the field of the with there it is only finitely many of the quality of cannot let me cannot be the hydrogen abutments infinitely many things that will be such a sequence and in fact I can make this small enough have a neighborhood of a given point this need anything point of and this but the things the In this chapter so I'd like to make simulators which contains a only the which contains the what is the None of the difference in both so that means for the 1st for all the points you move through some In all of

22:12

the system of a little bit about this point the the city and it's that closely the affect the little man talking and solve that move the density of this little ball means of we set S of couples of the point is that in the the OK right so that's a discrete set of examples the and some examples of policy of the only thing you can do it he now things see the 1st is the use of all the what a more sophisticated in fact what is the the little over a year from on 1 of the it will pay for this this set is a subset of the because we actually want to however is a except that of the the the law a lot of money in the the thank you and I think that this this is the on only economic point in C so if I remove it from i said you that the value I can say that with this we have all the without so about on but snow on the in the in the so here I think it will made him on this you know what I would depend on you if there is a lot of care and that in a kind of said which is always the data in our life 1 of his most of you is a of so all of the simplest financing in and then just the 1st thing you have to have the final thing I used to use some of the this the is classified uh winemaking where we're interested in discrete states and fact vector the dual 0 set of what month extension is a discrete set at all the functions identically 0 that's next then statement so walmart 10 solid move on now you VAT me it's my extending this assumption that it goes cases in the time that is allow me can quote it nevertheless is the so is the thing and in the most exciting thing and then but the following alternative I've can't come is identical to say well some of the there it is 0 for all that annual all what can be said of the of this is the sort of the system and the need to use the traditional means you face a serious need MIT cool at the end of this notice that triggered 28 scale on 1st z in new so 0 2 . said the gaze that this is the on the cell so what do we know already we know for instance if he had this example in C here and given function which manages at all these points and the sign of the entire plane the the entire complex plane C. then what it has a non-discrete 0 set so then actually you will love of French nearly all the zeros defined on all of the of the mass energy density on unit on what you do so and so why is it and 1 would the domain well it's it disconnected is at the mall and I said well in the sun and then we could define NASA identical to 1 of the 1 component and 0 and the other thing the sitting will be false of the car but I expect you to be trapped I mean you can become a ball in a the other celebrity many proofs this whole or exist along it's the 1 cell maybe start with something that I could claim is that instead of showing the share of

29:02

accumulation point of his it was there in groups and their uh their use people and also when an OK so here looking in the claim is that this center is still high but it's a little indirectly defined I know that you know that's the 1 thing instead of relation points is open so that if the has what accumulation points that it's 0 would be as the case for the examples which I mentioned then I have little neighborhood of 0 rare as the bear some of which I also can approach the the others in the sky and the example so that means that actually has to be an identity 0 look at in a neighborhood of such a point in an OK let me prove this sloWNet so take a point making I mean his may also be and didn't care right now 1 I want to show is a drop in local so I take a point when was in and who so that's an accumulation point can be significant uh and also and it was the of the CEOs wasn't the very so that means that make this kind increasing sequence of the oath of their silver extinct saying he changed sequence the eating of points in and after that that they should not it me such that so and also meanings that x of beating benches and also this season's converges to be OK so this is 1 of the start working and in the in in this setting him and now have this strategy is I want to give you a policies a policy areas and development of my country and and be this because the and so suddenly is concave suddenly we can write it the as a mobile a lot of attention in the neighborhood of the can when you write it as opposed to tasks such as that seen you blokes worms in order to us in the world said that skip use excluded and so forth uh at the watching is the and to come this is also not yes and this holds for snore ing In ball the eye of meat which is contained in the OK well and is in the middle of the proof in this job now let's look but what do we want to show we want to show that this power areas xt vanishes what if if I have such a point then this and power series vanishes what can I use well I can use that in the case I get 0 so let's let's do this back to the uh working uh working someone AEK to the next so what we do is we 1st look at in auto repair here that's something that a knowledge well by definition this is just F of F of the point now develop F of the well a now I want to invoke the B case while cross this is f of limit p all and now I want to use the continuity in their and Services Ltd that's been carried In order to get to my zeros so this is f is continuous so then I have limit of f of p k each of them is 0 so this tells me that a naught is 0 bulking cell uh now I know this is not there what can I do in order to show that a 1 a 2 and so forth that vanish while I divide by Z minus the and to use the same argument became so now it and then I keep on going I have not I consider f of centered on the part of a Z minds being which is an people using the power series a 1 plus a 2 so that when speed as a fleet statements Ms. credence influence and this is a valid for standards for B also the minus the point Karen Moser point these however the right hand side is defined also in the and is a nice continuous function there so what that means that this function on the left hand side has a removable singularity at be at that because the and I can actually remove it just by the value here at z equals B which is anyone who are called can so I can say that this has been will removable to the moves of the singularity and uh z equals the and and this comes from the fact that this is a nice continuous function and so you could say in particular its continuous and put z equals deep nets so or therefore I get is is a removable singularity well now what is OK it's a can be removed the value a 1 of

35:53

these he and what is a 1 uh so value at singularity hope and singularity said it was the is anyone in our ideas I apply the same kind of argument is here sigh right anyone is the limited work OK this time I must however when you consider the F of said well that's right 1 more step up its efforts that over Z minus the where uh be or BK approaches uh or have it z approaches b and and it is not equal to the but so those parts of points decay I know that they uh they give me such a sequence so I can simply right this is the limit of the f or the kano over whether it is the Carolinas speed but and now this these uh 0 up here and so this is is a word sequence and so this is 0 I'm working so showing me that anyone who is in the and now I divide my function F set of a set of of uh is it must be once again by factors that modesty and repeat the argument and so on cell induction an indirect she who are following exactly the same argument and gives me that in a n equals 0 for all and then they should not be and so that means an event to s has a power series representation which vanishes so that means f vanishes on an open of the open disk uh since this is really the quality of the entire this an S vanishes on a b of it is the hour so s is identical to 0 all on be all of these all came in particular if I have such a point B and of a neighborhood and that so it's so this means claim the more on OK so this the difficult that's enough across let me now come to the is bit and who which is a connectedness of more than argument I will show that not only and is open that also M is closed and the different ways to show at 1 would be 1 would be a diagonal sequence sequence argument so if you have and I know net b yeah if you have a sequence of accumulation on a sequence in and so each point in the sequence is an accumulation point so each point is approached by sequence well then you take a diagonal sequence and show that the limit who is also be where intersect has an approaching sequence who never way to show this An is the following who was on 1 to show this and I know you so m is closed or you without em is also open is opened up more care how do I do this well if I take a point and you are in the complement of the and that means in the neighborhood of this point I have only finitely many uh much cited in many zeros of sex news so if it's there than in the center of but he said it is it is not an accumulation point out now often then but enough itself and 0 set of this no and then then I'll according to the property of a amazed there exists a neighborhood there exists a neighborhood of the the existing neighborhood of this point or z containing can only encourage sharing and containing only finitely many zeros of so manifest renewed sign at many points at all uh moments but also in a sense yeah so I choose a point it's not a a an accumulation point of ineffective that means that there is a neighborhood where have only finitely many zeros well and that means that that in fact I can take his entire neighborhood and it must to belong to the set a is you there was not In the case of the seller this neighborhood this neighborhood and I should have given it a name neighborhood belongs to I is also in what you said minus and you minus and right so all these points in this neighborhood can also not be approached by points in the 0 set since they have a finite distance to the boundary OK so that means and that means you if you if you without and the complement of the complement of M is open and that means

42:42

and itself so in itself is open and close open and close to regain and now you know that in a domain M. sets in you domain well I the only connected the only open and closed sets are either into sets of entitlement so that means I have I have and N. equals you will be in this case the 0 set that is all of you and the function is identically 0 all M is the empty set well known in which case they air the and uh 0 set of f has no accumulation points so as discrete and its discrete and this is precisely this alternative here is precisely the claim so we've proved we wanted to and perhaps is a chair of edges strides and the consequence which has nothing to do about it so let me just write it on the board and then is a break so now I want to apply the lemma of we apply the lemma to the difference of functions of 2 or more of functions what do I get well I've ever they agreed and they agree at most on a discrete set of all they are identical and this is the uh this is the identity theory here is that in the identity theory of fear Amphipolis serious that says if S G where our to holomorphic functions so what was the functions and then uh 1 of the main monodomain is again important on a domain you so the OK then what this the tentative of a lemma for f minus gt means the then I this preserves the order it's not there OK so then I have if it is identical to the all all the coincidence sets they coincidence said discrete Cohen's the consists search is so what's it is coincidence that the set of that's where f of sick people Jia said that is discrete in you so that man it may still have a regulation points of the boundary of S O discrete set is actually the notion of a finite sets but adapted to open it to open sets uh in closed sets are closed and convex sets and discrete sets and this is the final OK so this species and we know very well the policies are and I will draw consequences uh after the break OK I would like to continue so and so on there's 1 more formal statement I would like to make and then I will discuss examples so performance statement it's just a little 1 that perhaps worth pointing out how will pose serious which is that and if I have a power series I say P of z equals some a k a or Z minus B to the end of the world that could if I have a power series and has radius of convergence where was work and suppose radius of convergence is positive now then M I claim that this power is no larger value is um but it is no at at sign and save is not positive EPS island such that I can continuous function PE effort to a ball slightly were matter radius opposite sign on with a we're logic functions so then band of hunters and this and the search for no positives and sign on on theories varies a lot extension defined on say a slightly larger disk the out as epsilon speed but we can to see all of his decided on here continuing will maulstick and continuing will say and such that S is restricted to the body coincides with the capital of Kabul idea can learn coincides with peak OK so that means if you have if you have a radius of convergence of a policy areas it's not possible to define a function of any larger to define a warping function on any larger disk induces helpful and in discussing them discussing how policies and singularities and this is a very straightforward to prove so I assume you have such an who gay men and women OK by analyticity there's a power series of that class have so far this so Ganjin 26 analyticity uh if has power series converging policy areas on this entire ball was the well I the I did theorem then and the 2 the 2 months since these to agree on the ball B I by the identity theory and this is uh is actually also the policy is given up there so what entity during the that tells me tells me that if he quotes PE on be up and so that means that means and P is also the power series of but will be defined was larger radius well then the radius of convergence would also be R + at silence and this and so on so therefore P's P has radius of convergence how the abiding by the theorem among policy reason presentation and is city we know that the radius of convergence is optimal step size contradicting the assumption of of a statement of the proposition OK so if I have a radius there are of convergence I I cannot extend extend the function analytically tonight entire neighborhood of today is also cause it may be possible to an extended to certain points on certain larger domains and has so in order to discuss examples let me know OK let me make 1 more definition there and hold on to see is saying and the gauging higher the OK then and function F is say how whom C 2 E is called analytic continuation it has a meaning undertaking current creation sheet back little left if and only if

52:12

1 of the best of F tilde is analytic and saying it continues net is this will what it says so the is an undertaking out on C and uh s till the restricted to allow is at OK this is jason win where we're given in many cases namely uh with many examples for this name you think of the exponential function sine function so consider 1st as a real function well Bayes a and then we have analytic continuation is by just replacing x really by complex Z and this gives us an analytic analytic functions and this is also a unique way of well this energy continuation is unique I should write this out so I say if I mean analytic continuation exists up to 20 years she exists how and then you need I ran into is unique I well why is this well I B M by the and uh lemma and which is gone now uh at 80 find a function is defined uh and analytic function is on French is already defined on any non-discrete set hours non-discrete yeah so it determines the power series and of the the polis serious of of the of f is already completely determined on our and so on and that means as the result of what the power series is unique and so and the analytic continuation is unique if it exists that so all I uh by the lemma what was inspired the lemma telling me that that s is wasn't by amount that S is already determined on any non-discrete set that and to OK it's not necessary it's not necessary that we have such a continuation by parents and so so many examples it's uh and this is um is is um worth considering and also remember our initial discussion of the exponential function in an analysis was what did we do are where we derive basically heuristically derived from the differential equation as prime equals F we derive the exponential serious on on it was and then I said something like well this is only 1 way to make this a and a a policy is also on the complex domain just by taking replacing X by their but it's by this very same principle I mentioned now there is a unique analytic continuation from our to so this would have no choice yeah so the and if you want to uh extend the exponential function for real numbers to the complex numbers by this principle is a unique way of doing this so if you want the functionally Trajan all the differential equation also work for the exponential function then there you end up with with the same exponential function on what C will get there would be an example of cell it this possible explains this to you right and node explains x from c yeah with from artists these will add to our OK yeah thing is n this is a special case of analytic continuation actually I could uh replaced uh I could replace year the real numbers by any non-discrete said you yeah and say of architectural and another that i guess i want and from discrete when the to me and interests and and that the this colleges had a and or am on was discrete sensor it's coded yeah and it's this so you have a child that's use yellow and here to replace our lives OK and let's we place on the continuation shed on you which is a subset of and as we'll see a subset of up the superset model and sorry yeah and then I call it an analytical intonation there's the relations is analytic and restricted to the set M it's s and all I need to do is I require M. in a non misleading doesn't have to be over just non-discrete a sufficient in order to uh the stories in that Member policy is that is modified at any hour on um could be from mistakenly where you see an OK it's actually detective a consequence is that yet hack should be feature that yeah I chose since during pretty tells me tools such condemnations must agree they do agree on a non discrete set it used to be I and now it's my learn discrete set M and then but you know all sustained by the fear of death they must be thank you OK so so that means and I can talk about um I can talk about and is a continuation in general now there is a little problem about it but let me 1st and it which I want to show you an example but let me 1st show if you a nice example where everything works as of nicely so 1st example of how an considered essentially on the real x maps to 1 over X squared plus 1 or more OK so this is a simple function defined on all of our which uh has uh is 1 and 0 and decays to 0 at plus minus infinity OK what about the policy is at x equals 0 what in terms of the geometric series it's easy to write it out but I don't want and 1 to do this it all I'm saying is there is a power series has power and uh developed etc. so it's and what is the radius of convergence when you use a geometric serious and it's obvious that you can compute or convince yourself that that the radius of convergence is 1 call legends are you goes along how OK so what is particularly about his value 1 year whole minus 1 in the area of a real so nothing right there why should what what's reason for will policies of this function converges exactly on this interval year it's hard to tell however if you look at the the complex version then it's obvious now H makes allow we extended to z maps to 1 over z squared plus 1 in the complex plane and well what do we discover while by the standard polynomial trick this is 1 of a Z plus time times Z minus y and so that means if 2 singularities and plus on and that minus sign the rain man and some of this function is actually defined on see how well removes the points plus minus sign well by the previous here is now clear that if I every development point b equals 0 the maximum radius of a disk for which this function here which this function here is holomorphic has radius 1 right well OK pretty pretty poor picture of the disk of radius 1 but I hope you need that so here's 1 and here is minus 1 then it's OK I so you see right away that this what this radius of convergence is cell maximal

1:01:42

all maximal disks or an error of about 0 had about 0 on which S is designed on which the 1 over so that was once defined is turned is it is the of radius 1 and sell and actually there exists no who cannot continue at this function velocity anything beyond I plus minus i since and this is a poll where function uh tends to infinity so there's no continuous function including these 2 points which extends it so that means that by hand so formerly was essentially the last statement propositions 31 yet we don't have we don't have a holomorphic extension to anything larger due to the fact that these 4 2 . suppose so that means I know right away but the radius of convergence is 1 of them so taking taking propositions so this is 1 thing it's at least so this shows shows me are as a least 1 and all the other on the other hand propositions 31 tells me or is at most 1 since there's no uh analytic continuation so that means uh is equal to what walked without any computation right but also the way how's it going to to be in the 1 1 over limited superior for an intruder also want any an formula this is the crazy Adam out from in there right so that the net OK so no need to appeal to the head no means the to kill to push it had a man is not yet so that means if I tell you that they have a lot of attention and he points that's not defined and and he has a point being than know right away radius of convergence of the policies is a maximal radius of the disk is contained in the domain that franchisal more often so it really is the radius of a cluster singularity if if it's a function of this kind yeah so this is a very nice to meet future questions they're distinguish on a sorry yes or else it would be contradiction thank you short OK so this is this is very neat now there's a little complication about um uh analytic continuation which I do not want to hide and in fact it leads to some modern mathematics you to services and we will look at and not to discuss it with a look at another example namely we look at the numbers there are a number of moons in the Polish the fall in the and it turns so the next example will introduce the is the load of every man so and so remember that the complex logarithm is given in the form of a logarithm of the modulus of Z + are at times argument of said and we usually define it honestly domain so what you find on the complex plane without the air without the negative area lights is and without yeah on on the state it's OK now have now here's a nice problem about analytical enduring durations and look at the complex plane look here say the 1 1 this function here is for instance defined on a disk of radius 1 more year a on a disk of radius 1 know about the . 1 yeah it it doesn't have the 0 which which is dangerous hunting so here we have look see defined OK I will never get is defined at 0 show that I can now extended state a said in or I can be extended to set like this no why not move OK so it analytic so analytic extension continuation I going mutation he signed and and she by this formula we precisely know what happens is the logarithm of the of the modulus of said and the argument would go from 0 to pi over 2 I don't know I persons OK or I can look at another analytic continuation and which would say go from the desk but this time let me just go like this right OK and never analytic initially OK so that means that it may depend on where a I continue by a function what the value at a given point is right so here what's the difference well you know what the difference is it's just at 2 pi i in this case the argument here would be here the argument is 0 pi over 2 so if I come from yeah it's pi so here I have in the yellow version I have pipeless something for the argument on the other hand the orange versions as I get something like almost minus pi and will for for the imaginary part here so the difference so in here I have a difference difference of to pi I right and so it turns out that in fact analytic continuation to the complex plane or to a maximal subset of the complex plane uh is uh not really the best idea so what you need to do is sort of keep on going by analytic continuation last term I showed you I showed you and the helicoid services near which is a graph so well it's not a global graph but it it's given by uh by taking straight line and and applies to motion to it and such that it goes up and up like a helical into a set of stairs so that would be the actual actually the nice domain for this logarithm function to be defined you want it to the final infinite coverings and that's the soccer agreement surface of a function in the case of its parents you which is the true copies going round tries to have in an analytic function well defined I will not develop this any further that but it needs to the notion that leads to the notion of a manifold and since Rhemann surfaces as was already discovered to uh and uh in the 18 fifties also 50 years before manifolds voraciously invented so and complex analysis was a forum members mistake who who BNC women also talked of an annotation embedded was a long series of before uh before it became conceptually we what to do that but it's the same thing as the angle with the angle the function really defined well if you work on the unit circle the angle is not continuous there's a troop I get where is the the angle function

1:11:11

nicely defined well it's on the covering as on the real line which I imagine as they're going round infinitely often only units of and same thing here for the logarithm with respect to seize the rest of the group and I so this much about and the need theory of power series and now let me use the last 15 minutes or so famous theorem you are in the name of a theorem by we'll deal all of which is I can again and again and then the consequences are or what's going on up there any this is a class you know it OK so the units here so what we did what we look at our function as functions defined on the entire complex plane some of the something called the entire functions an entire function the kinds of food so so as now entire functions and it is just a file is any will wanting to function as automorphic functions defined on all of us smooth function if finally entire planar on that will high seas OK and enter many examples the famous examples of on polynomials in polinomial so what else do we have the exponential signed calls OK but not 1 over z squared plus more so that many of who are have all K if you look at these functions and but with control point of view of boundedness the as well then you realize they you know it's on a bounded x is not bounded well sine and cosine look in the real version bounded right well they're bounded functions been restricted to realize however if you go to the purely imaginary values then you know that the a uh sign i t who is not bounded since it's a need to be know it's the to the ones that I T 1 is e to the minus side t over tool or censure something like what is it seems that I'm missing into I here sorry to correct now before as men and I think this 0 of the 2 video no here the eyes sorry yeah you're right here's the ice I don't need it and now but that means I must flip the order from minuses her OK I'm sorry hotel work done the potassium salt whatever visitors it's a cinch function over to no over pressure of minus and they are all it takes to clusters and Schlangen and whatever and In any case this don't have limits at plus minus infinity if you go to which T 2 plus minus infinity so the analytic continuation of signed the sign of real so cosine defined reals is unbounded in so all of these functions are unbounded on a on on the complex plane and the theorem is that this is always so with the entire functions from parents or a fair amount he had the ability and and assess each of the of OK each well if it's not if it's not unbounded when it's bounded what is it's bounded when it's only a constant each bounded entire function entire function is constant it yet so essentially each interesting entire function is of is an unbounded that's in the way it is stated OK and you so once again statement certainly not true and for real analysis and her but in fact true for some of the partial differential equations locate miss something to say but it's much more complicated so I don't want to mention that uh in detail OK so here's a simple proof the proof is just inspect the crochet um precaution formula for the 1st derivative with the course she integral formula for that and we derive the error in the course of showing that F is small and the right formulas for as prime and the instrument here we need a formula for f prime locations so I start with said to pick a point in seeing and then I have this formula by co she which was ferent find and we have said as prime wanted is 1 over 2 pi i and it will be in and sectorial but this is the 1st derivative so as 1 and it's the boundary The B art of certain Lauder quiet shell and capital losses are will be large in minute and if has of Satan and now unlike the original crochet integral formula this comes from differentiating so it's zeta minus Z. score and decent who came and this has since we have an entire so this originally of this works for any ball in the disk contained in the domain of definition you will here it's an entire function defined and C so this works actually for all our the come emphasis coming f and time so that we can use any discussion we like what then all we need to do now is that this user links estimate and the boundedness so they are enumerated is bounded all this is a circle of length 2 pi so you see and this is approximately 1 over R squared so there this will altogether given 1 of our bad links estimates a let's do this in detail and found it clean so saying if it's less than some constant C. OK and now we use see and length estimates in order to instance the modulus of f prime do which is 1 of the 2 times a times the length of the circle who know what the that's just let it out to be ourselves z at times 1 won't want universe love soup of f of Zadar minus seats grants and now this suit

1:19:43

taking over DVDs are and now this is how to plow it's a circle and this here is estimated by while the numerator by C and the denominator is a constant has just ask them so what do we get our together we get a 1 over r squared L 1 of our so why actually equal but he must be a this way but obviously the smoking OK and this is 1 now 1 over on is remained the form grapes has C over this C over what openly correct now so what is the mean well this is valid for all for all of our are larger than 0 well that means that as prime of of z is is 0 net primary so that it must be 0 well that this is for arbitrary z and see well this means F aside and if Prime is identical to the Earth and of s is identical to 0 when f is low and defined on the entire plane the method is constant background intensity so it's really and that if you like it's a simple consequence of the co she integral formula differentiated once know this is this comes from differentiated coach the formula once and then you plug in the length estimate and then see what you get in there you are it's an amazing feeling on over here and there was no reason all bounded entire function and this except for the constant questions that ask Mr. Flanagan and users where do and what of OK right show since it sort of its these things here against this here right thanks yeah OK so and by many interesting consequences from which you can draw from this sharing an 1 1 I will show next time is the sum of theorem of idea but I giraffes loaded follows easily and so next time wrong better than I did not hear her which she which some exciting but the famous theorem and I'll let me also make an announcement which I forgot to making break and Justice asked me you know what if a camera it is whether they can register for the English calls complex analysis uh they seem to have trouble to register for the English version but can only you could only the register fall from time to according to missus move was of this is not true so try hard to if you want uh if you want to register for if you're physicists and want to register for complex analysis it should be possible and if it's not possible for us to see Mrs. some from physicists there was who should know but it should be possible it OK so let me draw let me draw a comic Francis well 1 thing is and it's a problem for our which is the show that an entire I then is I have a constant and he or if you look at the entire range it in the the see that's that's while this just a little smaller trick which leads you from the unit sphere into this statement and so also known as well this means that in such a nasty holomorphic F. could miss many values in fact there is a a theorem which we will not through so called peak little cerium what is also great fear that this is not a theorem as telling telling you that a f n time this is at most are ministers who are at most 1 point and most warm point always constant or is constant so yeah same intended it is it's not constant in time time function which is not constant so which is or who implied by which is not bounded can must take on values except for 1 of them what's a value misfiring every exponential this 0 right that's an example that it may be possible to miss 1 value yeah I mean polynomial signal values as we will see that uh but 1 point can be missed OK and another consequences and whose occurrence which and would like to mention the that if you consider s on the entire complex plane say to move the unit disk be what I love in session as cannot exist by hearings is is a bounded set him in time and not entire budget logic and by objectives who was the bora objective it does not exist now this is an immediate consequence of humanitarian why is this interesting well this genetically this means there is no angle preserving map of 2 spaces which are topologically from point of holes cannot be distinguished in have complex plane and be the unit is fire some uh what these are homeomorphic sets so I wouldn't like there's homeomorphic but if you restrict yourself to angle preserving maps or mosig maps and you can distinguish these 2 sets its or what's also interesting since the Riemann mapping theorem Lehman mapping sharing he mapping theorem says that if you have a said uh you know which is a proper subset of the complex plane and which is simply connected and simply connected and I then contrary that is a holomorphic map then there exists f will LoFreq from you to be sold by analyzing holomorphic by jective and 1 or more of the it's and homomorphic so that means that any any nice any nice simply connected set for instance this 1 here can be mapped in an end of preserving way to the unit disk now also this little domain narrow whatever whatever you like all these things always nice interconnected

1:28:14

sets which are not see itself however the itself is the only exception where such a map of the leases so you can distinguish different domains and classes you can make image it's the equivalence relation from the biased saying that subsets you are equivalent if such maps Projective for warping exists and then all of the sets come into 1 class let's see comes into another class surprising although it's homeomorphic to conformal type is different OK so this is not looking next time we will prove the from the fundamental theorem of algebra

00:00

Resultante

Lineares Funktional

Subtraktion

Punkt

Auflösung <Mathematik>

Ereignishorizont

Ausdruck <Logik>

Eins

Singularität <Mathematik>

Prognoseverfahren

Flächeninhalt

Sortierte Logik

Theorem

Analogieschluss

03:58

Mittelwert

Lineares Funktional

Singularität <Mathematik>

Einfügungsdämpfung

Punkt

Gruppenoperation

Relativitätstheorie

Bilinearform

Analytische Fortsetzung

06:08

Ebene

Subtraktion

Folge <Mathematik>

Punkt

Natürliche Zahl

Zahlenbereich

Kombinatorische Gruppentheorie

Komplex <Algebra>

Topologie

Ausdruck <Logik>

Negative Zahl

Theorem

Inverser Limes

Inhalt <Mathematik>

Analytische Fortsetzung

Lineares Funktional

Differenzenquotient

Validität

Schlussregel

Biprodukt

Unendlichkeit

Quadratzahl

Menge

Beweistheorie

Konditionszahl

Basisvektor

Körper <Physik>

Potenzreihe

Ordnung <Mathematik>

Aggregatzustand

13:36

Nachbarschaft <Mathematik>

Ebene

Nebenbedingung

Subtraktion

Folge <Mathematik>

Punkt

Klasse <Mathematik>

Analytische Menge

Bilinearform

Komplex <Algebra>

Gesetz <Physik>

Raum-Zeit

Einheit <Mathematik>

Vorzeichen <Mathematik>

Äußere Algebra eines Moduls

Zusammenhängender Graph

Ganze Funktion

Gammafunktion

Zentrische Streckung

Lineares Funktional

Parametersystem

Erweiterung

Finite-Elemente-Methode

Zeitbereich

Ruhmasse

Vektorraum

Physikalisches System

Dichte <Physik>

Teilmenge

Arithmetisches Mittel

Energiedichte

Singularität <Mathematik>

Menge

Sortierte Logik

Rechter Winkel

Beweistheorie

Konditionszahl

Körper <Physik>

Billard <Mathematik>

Wärmeausdehnung

Sigma-Algebra

Ordnung <Mathematik>

Term

Numerisches Modell

Aggregatzustand

29:00

Nachbarschaft <Mathematik>

Folge <Mathematik>

Gewicht <Mathematik>

Prozess <Physik>

Punkt

Momentenproblem

Statistische Schlussweise

Gruppenkeim

Gruppendarstellung

Vorzeichen <Mathematik>

Nichtunterscheidbarkeit

Inverser Limes

Abstand

Analytische Fortsetzung

Leistung <Physik>

Einfach zusammenhängender Raum

Parametersystem

Lineares Funktional

Konvexe Hülle

Kategorie <Mathematik>

Relativitätstheorie

Stellenring

Teilbarkeit

Ereignishorizont

Stetige Abbildung

Randwert

Singularität <Mathematik>

Lemma <Logik>

Flächeninhalt

Menge

Rechter Winkel

Beweistheorie

Mereologie

Strategisches Spiel

Potenzreihe

Ordnung <Mathematik>

Diagonale <Geometrie>

Standardabweichung

42:41

Nachbarschaft <Mathematik>

Resultante

Punkt

Extrempunkt

Minimierung

Konvexer Körper

Gradient

Endliche Menge

Vorzeichen <Mathematik>

Theorem

Gruppe <Mathematik>

Lemma <Logik>

Offene Abbildung

Analytische Fortsetzung

Regulator <Mathematik>

Auswahlaxiom

Sinusfunktion

Lineares Funktional

Strömungsrichtung

E-Funktion

Arithmetisches Mittel

Teilmenge

Randwert

Angewandte Physik

Menge

Potenzreihe

Ordnung <Mathematik>

Ebene

Subtraktion

Gewicht <Mathematik>

Ortsoperator

Komplexe Darstellung

Klasse <Mathematik>

Abgeschlossene Menge

Analytische Menge

Kombinatorische Gruppentheorie

Mathematische Logik

Term

Physikalische Theorie

Reelle Zahl

Subtraktion

Äußere Algebra eines Moduls

Holomorphe Funktion

Ganze Funktion

Leistung <Physik>

Analysis

Radius

Erweiterung

E-Funktion

Zeitbereich

Relativitätstheorie

Aussage <Mathematik>

Stetige Abbildung

Unendlichkeit

Komplexe Ebene

Singularität <Mathematik>

Energiedichte

Geometrische Reihe

Flächeninhalt

Differentialgleichungssystem

Numerisches Modell

1:01:40

Länge

Einfügungsdämpfung

Hyperbolischer Differentialoperator

Punkt

Nabel <Mathematik>

Gruppenkeim

Pi <Zahl>

Komplex <Algebra>

Eins

Negative Zahl

Einheit <Mathematik>

Vorzeichen <Mathematik>

Theorem

Nichtunterscheidbarkeit

Einheitskreis

Analytische Fortsetzung

Gerade

Parametersystem

Lineares Funktional

Verschlingung

Winkel

E-Funktion

Gleitendes Mittel

Ebener Graph

Teilmenge

Randwert

Polynom

Menge

Rechter Winkel

Beweistheorie

Potenzreihe

Ordnung <Mathematik>

Aggregatzustand

Ebene

Geschwindigkeit

Subtraktion

Glatte Funktion

Klasse <Mathematik>

Zahlenbereich

Derivation <Algebra>

Unrundheit

Analytische Menge

Bilinearform

B-Spline

Term

Stichprobenfehler

Physikalische Theorie

Ausdruck <Logik>

Überlagerung <Mathematik>

Logarithmus

Reelle Zahl

Flächentheorie

Inverser Limes

Integraltafel

Grundraum

Ganze Funktion

Topologische Mannigfaltigkeit

Analysis

Schätzwert

Radius

Erweiterung

Kreisfläche

Mathematik

Graph

Zeitbereich

sinc-Funktion

Aussage <Mathematik>

Primideal

Stetige Abbildung

Unendlichkeit

Singularität <Mathematik>

Komplexe Ebene

Flächeninhalt

Last

Klumpenstichprobe

Funktionentheorie

1:19:41

Ebene

Länge

Gewichtete Summe

Punkt

Physiker

Gleichmäßige Beschränktheit

Kardinalzahl

Bilinearform

Mathematische Logik

Raum-Zeit

Ausdruck <Logik>

Spannweite <Stochastik>

Einheit <Mathematik>

Theorem

Integraltafel

Holomorphe Funktion

Ganze Funktion

Einfach zusammenhängender Raum

Schätzwert

Lineares Funktional

Bruchrechnung

Kreisfläche

Exponent

Zeitbereich

Winkel

Einheitskugel

Primideal

Objekt <Kategorie>

Teilmenge

Konstante

Polynom

Menge

Sortierte Logik

Plancksches Wirkungsquantum

Funktionentheorie

1:28:13

Teilmenge

Fundamentalsatz der Algebra

Subtraktion

Menge

Homöomorphismus

Zeitbereich

Nichtunterscheidbarkeit

Klasse <Mathematik>

Projektive Ebene

Äquivalenzklasse

Kreisbogen

### Metadaten

#### Formale Metadaten

Titel | Identity theorem for holomorphic functions, Liouville's theorem |

Serientitel | Complex Analysis |

Anzahl der Teile | 15 |

Autor | Große-Brauckmann, Karsten |

Lizenz |
CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. |

DOI | 10.5446/34037 |

Herausgeber | Technische Universität Darmstadt |

Erscheinungsjahr | 2014 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Mathematik |