Smoothness and analyticity
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Title 
Smoothness and analyticity

Title of Series  
Part Number 
8

Number of Parts 
15

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CC Attribution  ShareAlike 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
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Release Date 
2014

Language 
English

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00:00
Point (geometry)
Functional (mathematics)
Divisor
INTEGRAL
State of matter
Real number
Lemma (mathematics)
Kontraktion <Mathematik>
Theory
Mathematics
Theorem
Circle
Nichtlineares Gleichungssystem
Antiderivative
Set theory
Social class
Rotation
Gamma function
Neighbourhood (graph theory)
Curve
Time domain
Integraltafel
Equivalence relation
Proof theory
Loop (music)
Vector field
05:36
Point (geometry)
Functional (mathematics)
INTEGRAL
State of matter
Multiplication sign
Lemma (mathematics)
Translation (relic)
Inference
Mathematics
MKS system of units
Strategy game
Term (mathematics)
Different (Kate Ryan album)
Boundary value problem
Physical law
Gamma function
Set theory
Standard deviation
Time domain
Numerical analysis
Degree (graph theory)
Proof theory
Radius
Algebra
Vector space
Calculation
Order (biology)
Mathematical singularity
Right angle
Thetafunktion
Resultant
13:03
Group action
Length
State of matter
INTEGRAL
Multiplication sign
Equaliser (mathematics)
8 (number)
Mereology
Explosion
Estimator
Different (Kate Ryan album)
Circle
Partial derivative
Social class
Physical system
Computability
Complete metric space
Integraltafel
Flow separation
Proof theory
Arithmetic mean
Order (biology)
Right angle
Functional (mathematics)
Supremum
Link (knot theory)
Harmonic function
Event horizon
Rule of inference
Power (physics)
Partial differential equation
Inclusion map
Term (mathematics)
Wellformed formula
Complex number
Modulform
Boundary value problem
Theorem
Nichtlineares Gleichungssystem
Analytic continuation
Initial value problem
Set theory
Module (mathematics)
Complex analysis
Model theory
Limit (category theory)
Loop (music)
Radius
Diffusionsgleichung
Physicist
5 (number)
24:24
Point (geometry)
12 (number)
Group action
Functional (mathematics)
Dynamical system
Observational study
Differential (mechanical device)
INTEGRAL
Multiplication sign
Real number
Parameter (computer programming)
Mereology
Rule of inference
Theory
Dimensional analysis
Fraction (mathematics)
Derivation (linguistics)
Group representation
Manysorted logic
Different (Kate Ryan album)
Wellformed formula
Analogy
Stochastic kernel estimation
Gamma function
Partial derivative
Set theory
Social class
Complex analysis
Neighbourhood (graph theory)
Mathematical analysis
Physicalism
Integraltafel
Convolution
Holomorphic function
Category of being
Order (biology)
Resultant
30:47
Difference quotient
Logical constant
Density functional theory
INTEGRAL
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Differential (mechanical device)
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Sheaf (mathematics)
Parameter (computer programming)
Solid geometry
Mereology
Derivation (linguistics)
Sign (mathematics)
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Different (Kate Ryan album)
Social class
Concentric
Infinity
Integraltafel
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Arithmetic mean
Isomorphieklasse
Order (biology)
Chain
Linearization
Right angle
Resultant
Functional (mathematics)
Integrallogarithmus
Real number
Distance
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Power (physics)
Term (mathematics)
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Wellformed formula
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Stochastic kernel estimation
Chain rule
Supremum
Standard deviation
Lemma (mathematics)
Content (media)
Limit (category theory)
Sphere
Holomorphic function
Calculation
Schmelze <Betrieb>
Inductive reasoning
44:48
Difference quotient
Presentation of a group
Group action
INTEGRAL
Differential (mechanical device)
State of matter
Weight
Multiplication sign
Grothendieck topology
LagrangeMethode
Group representation
Derivation (linguistics)
Sign (mathematics)
Manysorted logic
Different (Kate Ryan album)
Series (mathematics)
Position operator
Physical system
Rotation
Counterexample
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Infinity
Category of being
Arithmetic mean
Restglied
Symmetry (physics)
Order (biology)
IRIST
Mathematical singularity
Theorem
Right angle
Mathematician
Identical particles
Spacetime
Point (geometry)
Polynomial
Slide rule
Functional (mathematics)
Civil engineering
Real number
Mathematical analysis
Rule of inference
Graph coloring
Power (physics)
Partial differential equation
Finite element method
Goodness of fit
Wellformed formula
Term (mathematics)
Operator (mathematics)
Boundary value problem
Modulform
Theorem
Nichtlineares Gleichungssystem
Stochastic kernel estimation
Gamma function
Antiderivative
Analytic continuation
Maß <Mathematik>
Condition number
Distribution (mathematics)
Shift operator
Neighbourhood (graph theory)
Analytic set
Limit (category theory)
Equivalence relation
Holomorphic function
Numerical analysis
Loop (music)
Diffusionsgleichung
Radius
Function (mathematics)
Physicist
Glattheit <Mathematik>
Gravitation
Coefficient
Inductive reasoning
1:00:05
Point (geometry)
Statistical hypothesis testing
Functional (mathematics)
Direction (geometry)
Uniform convergence
Mathematical analysis
Theory
Heegaard splitting
Group representation
Estimator
Wellformed formula
Term (mathematics)
Different (Kate Ryan album)
Geometric series
Divergence
Series (mathematics)
Area
Weight
Power series
Rate of convergence
Neighbourhood (graph theory)
Analytic set
Limit (category theory)
Numerical analysis
Arithmetic mean
Radius
Uniformer Raum
Normal (geometry)
Local ring
1:03:57
Difference quotient
Logical constant
Presentation of a group
Beta function
Differential (mechanical device)
INTEGRAL
State of matter
Multiplication sign
Equaliser (mathematics)
1 (number)
Uniform convergence
Parameter (computer programming)
Food energy
Dreiecksungleichung
Group representation
Mathematics
Integrability conditions for differential systems
Roundness (object)
Manysorted logic
Different (Kate Ryan album)
Geometric series
Matrix (mathematics)
Convex set
Extension (kinesiology)
Partial derivative
Rotation
Structural load
Power series
Infinity
Price index
Variable (mathematics)
Sequence
Flow separation
Integraltafel
Category of being
Order (biology)
Linearization
Mathematical singularity
Summierbarkeit
Right angle
Resultant
Sinc function
Point (geometry)
Standard error
Functional (mathematics)
Sine
Cone penetration test
Divisor
Real number
Continuous function
Theory
Power (physics)
Force
Term (mathematics)
Wellformed formula
Modulform
Theorem
Stochastic kernel estimation
Antiderivative
Set theory
Polyeder
Condition number
Multiplication
Validity (statistics)
Weight
Model theory
Equals sign
Bound state
Content (media)
Calculus
Limit (category theory)
Numerical analysis
Formal power series
Radius
Loop (music)
Calculation
Thetafunktion
Coefficient
Local ring
Maß <Mathematik>
00:06
hello we will come to you too complex and knows this and we are in the the middle of proof or before giving approval the uh but let me still give a little review of the last class and of the main theorem the the a the main theorem that we had the the considered the implication that as well if as is all the more of the man the uh loop integral of this vanishes for contractible loops and in fact this is the core she integral theorem and it is also an equivalence and the way this was proven was noting that this uh is equivalent to the you Riemann equations the and because the many equations mean from a real point of view that this is a vector fields uh with no rotation that the integrality uh the for uh this these intervals is satisfied and so and the other so in particular we can use this well finest due to um define local primitives have had has local primitives now given hopes for a minute 2 is given by integrating out so F of so that this interval from some fixed point to z if I OK enough of data the OK this this it's uh it's 1 of the I think we covered and uh we also had a little lemma which perhaps I don't want to recall which tells you the following which I don't want to recall literally which and uh tells you the following in the case that the new it will does not manage but for general functions say let's consider this loop integral um and think of a case that it's nonzero along loops see while what the Member concerned is in you can extend this circle to some other circles gamma in a way that this function is holomorphic here Honolulu neighborhood of this cell essence holomorphic here then these 2 integrals uh coincide and in fact the same theory applies to any change that you can make with a a domain by editing joining something uh to a domain where the function is holomorphic so you may also think of situations like I know this year here there you know that s is holomorphic all on the set I did status but it's the same it's the same idea Philips C or even you might even consider cases there you have 2 intersecting curves but still you know that on a neighborhood of uh what a sheet here uh the function is holomorphic then you can say that the 2 loop integrals uh I the same by 1st considering this little contour say and then joining these all those 2 to it and the Indigo will always remain unchanged under the assumption that f is holomorphic in on the different sets the OK so they call she integral formula uh gives you quite a strong tool also for integrals the loop integrals by the integrand viviendas interval does not vanish OK so we had I finished last class with a co she integral formula so let's recall that that was the 23 co integral co she co she's integral formula the and that tells you that and you can be expressed as as the loop integral you forgot factor 1 over 2 pi as an loop to well over db of bees say later integrate the function F of state state of minus so the
05:37
desired results for all those that good be uh in within this this could be all speak under the assumption that let s be holomorphic led air from you to see the whole the and this disk should be compactly contained In you'll be are of it's compactly income then follows and so the situation as I explained last time is that you have a domain you say you have this ball centered at B of radius R and you have at any point z with In this ball but in order to calculate the value of the function and that's all you need to do is integrate over this boundary contour here OK so this is as we will see later on today is a very strong tool whole question this should be digitized thank you for anything else OK thanks so this is a very strong tool as we will see later but let me and 1st proven it explain to you last time proves not hard so we consider we consider this function here and F of state over Z have minus Z and perhaps give it a name so the clergy uh . 4 per name B G of carried G be a function on you would be this function defined on you were thought z where g of Zadar equals f of theta over C Z minus theta said minus that sorry wrong way the displayed call and now I use this lemma here which my proof last time I want to say that OK to say that interval of OK now the boundary it'll which I'm considering sonnets copy Beth of Satan the timing of that this is I want to uh OK I want to use now a smaller ball like this so this will be the last fall and I will use a smaller belong db of I what can we do all case or from a point set of to make it visible so uh when I move to say to here and consider a smaller with respect to the center which of radius little are such that it's contained in this uh larger disk uh br OK so I should achieve to start out some um the OK for B on a of z compactly contained in my larger ball the the I have 5 Lemma I want to express this as a boundary into go over a general boundaries so all while since the function and the difference said there is no singularity of this function of the only singularities it said so in here there's no singularities so I can this automorphic functions so I can apply this lemma and say that nobody was coincide look at this so this is the yellow boundary db of standards same time to uh state said DCA can and now I really I want to slightly changes by writing to terms and 1 term what I introduce artificially as of said so 1 term it's like this the same integration indeed out that and the other term now while is to correct this so what do I need f of data miners inferences of the to of of OK and let me write this essay 1 and this is true and consider both terms separately
11:15
it we look at it and some and the room carries so 1st of all the strategy is actually the term 1 that's a good that's a good 1 that would give the left hand side it'll say it up to this vector 1 over 2 pi this I will show that to you when the manage OK so let me 1st of rewrite 1 the the 1 OK well note that the numerator F of Z 2 in 1 so it's again the wrong OK uh to be it and the but I'm sorry the I have a hard time writing intervals depending on the type but now should be correct so the Duma now it's obvious that the numerator doesn't depend on the integration variable the so I can then pull it out of the integral er and right this is f of z in front of the integral over the remaining 1 of the zeta minus the zt these data consists of again all of the yellow contour line degree of certain the OK what is this well this is just a translation of 3 as 1 over function to this point uh uh Z so this gives me a 2 pi the the but by the calculation we did before the
13:03
last class we calculated by 1 over Z integrated although loop of a circle gives to buy and 1 over set to any other power gives 0 OK so and so this is just 2 pi i being the value of the integral times as 5 Example of by this theorem actually last time the answer OK so this is really what we want to obtain you then writing the uh 1 of the 2 pi i uh on the other side I get that the is that the desired integral is f of set up to the time to but now I would show you the 2 vanishes How do I want to do this well as note that this term is not a function of little r the OK I mean In order to show that something is 0 yeah it better not depend on this well I left out the little also here sorry the so this is again over be part of state OK hold out it's complete cell this term here doesn't depend on on the left hand side obviously doesn't depend on our right so that means what if 2 out of 3 terms do not depend on hour well then this term also won't depend on it although it's uh it although it's an ingredient there but in fact this term must be independent of little or and so all I need to do is calculate the limit for are tending to 0 and in order to convince uh to convince you that this term vanishes limit vanishes then the quantity and luminous OK so 1 and there's not deep 1 little on and also uh and so while this side of this with the left hand side up there does not depend on little they the on the door and soul the soldiers to OK I will use this in a minute so what this tool while I'm In order to show something is 0 it's always a good idea to show its modulus is small the and what is the estimate we can use to estimate the model as well he's the length estimate link system and and manage I I mentioned uh so what's the link for estimate while it's the length of all the on a circle of radius R times the supremum times the of function of a function of modulus which is f of theta minus as of the date Over zeta minus the and the this all is extends over the be wanted so this is very Wang for estimates at I showed you before OK but now we note that is zeta has distance off from z while this is just a constant to us what this means I can raise well I can you raise this or replace it by 1 yes so all I need all I have to write it is through time the throughput of of zeta minus 50 of them again over stated in dB they no this is the difference and the function f is continuous well then for small and this must be small effect this can be made uh can be made less minutes I by choosing a are sufficiently small delta by by continuity rule so that means this is arbitrarily small well this must be 0 the whole thing must be 0 event and so on that's like sound and they the f continues the when it means that the modular stuff tool can be made less than a silent for small arms and this means uh while if for any of silence in the at dissonance sorry this means that um and factual must finish um end OK together with the fact that this does not depend this quantity here are just not depend on our at all it's constant so if it's constant then it's equal to the limit
18:44
the limit must be 0 so it's in welcome of the tool is independent of r on we have the true the modulus is equal to 0 or to itself is 0 and this too is several men method there with the proof of the the core she integral formula this the OK so we have we have this formula at hand and in a minute you see why um why I find this really this is really the key this formula's the key to everything uh in complex analysis um so this is step back and best and as see what we've done so far most of it is actually on the upper blackboard small lined and I lower along from case so a look at what we've done and with the coach here even equations partial differential equations which correspond to the fact that F the holomorphic as is see linear has seen a derivative so we have partial differential equations course you buy kosher relent we show what their um the local integrals vanish and by the fact of the loop integrals manage and that we have affect we used is that this integral is uh 2 pi but an explicit computation which is very important these 2 things are the only major ingredients for uh for the proof of the Gaussian integral formula I've shown to you so I I would say that that course Raymond um cautionary man partial differential equations that gives they co she integral ferent the and together with the fact that and the integral the integral over 1 over z over the roof I of 0 is 2 pi i viscous physical she integral formula the so the message here is and we started with a with a partial differential occasion of and end up and in an interval formula which represents the value by interval you would see uh pretty soon why this is such a great thing so let me let me only say that there are other cases where you can derive integral formula such as the coach formula formulas I are also village all of them for many pieces for many partial differential equations it must be linear however um want to write out the partial differential inclusions for instance and perhaps perhaps some physicists know what I talk about it please the plus equation the plus you equals 0 uh the form of a similar formula which is report some formula that this is not such a surprise since actually if you have a harmonic function you can find take it as the real part of morphic function and find it uh an imaginary parts to that uh such that you just IV is a little more think and when we are in our setting all but nevertheless this can be done completely in the real setting and also for safe over heat equation you cheese equal slope fashion of fuel where it's just the heat equation so model so here you start with but as you start with some heat to you in the models how it dissipates and then there's a similar form relied on Wall went on the right about that then there is a formula like this I don't tell you really into it all in a list out which is important you have been various such a formula uh and you're integral well it depends on initial conditions of I won't bite everything out here but this to show you that there are some such formulas bear the initial value in this case initial various services view of you of 0 of Y commas 0 there's some values for special for some special values here is a world in our case on the on the boundary are integrated are integrated and they give you back the solution and this
24:24
is very I think we have uh with the co she into the formula it given a partial differential equation if you're lucky you can find representation formulas the time derivative is it's a it's partial derivative of reality the this is the should be this have more so this is the time derivative and you have sort of a large increase in heat at a given point if the sort of and in in the surrounding you find the neighborhood uh you find lots of derivatives of you lots of heat coming on OK perhaps you will see these things later uh in your studies so i for what it's worth mentioning in this case and it's also common is also connotation which is particular common physics if you have such a formula you call this well this thing here is an interval kernel and you look at is this is the convolution integral value folder function with the kernel same thing up there but I won't explain this to you in detail about in the course notes I have included uh references he or you need to do is look for some the entries on the Wikipedia and you find it but certainly in physics you will I need to to use this formula and in the end they come what may also come up with differences in the country at and click through dynamics and so forth the the OK so now that I have to coach into a formula uh I can sort of present all wonders of complex analysis the yeah but ended that's only the dimension it's really I mean it depends on the but else it only depends on the dimension whatever it is uh don't be afraid I will not ask for this in the written exam here it's pretty obvious that only wanted to give you an alligator analogies for reason with them I don't know when I'll have 1st learn complex analysis I was quite surprised above results in and couldn't sort of couldn't see where it came from I think I now understand it and want to communicate it's not that surprising OK so now we can harvest and say properties of the Morphic fractions properties also will freak functions the OK so all we do here is our is drawing consequences of the of course she's interval from you OK so this is something I announced thing in the 1st uh class when we started with complex analysis the 1st result is um that holomorphic functions are infinitely often differential so it's movements all of holomorphic functions so something which is not at all true in real and the real theory of other functions which you can differentiate once but not twice a year for instance you know such a function of the French from are 2 of which you can differentiate once but not twice I for instance x times more than 6 something here there are many such functions that 1 derivative exists to find 2 derivatives well impossible in the complex setting you've always infinitely many derivatives In order to what will we do uh in order to show this so methods will be that differentiate differentiate the course integral formula now in order to differentiate an integral formula uh I have to say a word about uh differentiating integrals uh this is something we covered in analysis tool in the real setting so let me uh not at let me use this results in the end just say apply this to real and imaginary parts so this is now a little long to ride out but it's not hard so this is likeness rule let me interest groups that which tells you is use autumn and you have a function depending on and on you and on a parameter in an interval of the real interval real interval yes the interval of me and US to see if f is continuous 10 and the differential above with respect to the variable in him so suppose suppose that um S that if I may have to have the team then this is motherfucker it is
30:48
more on you for all t in a b the and also a visit David Sewell mosaicking can be differentiated with respect to z wildly wanted the reside is still continues you can and and now is the question how to write the difference in the uh derivative of this function so my notation is partial derivative but if you like you can also write to standard t for this side and I really don't know what's best so if the if this which is also f prime for this function here depending on a parameter T uh if this is continuous and it's continues then I can differentiate under the interval then the integral yeah the so do I still have the null the race to Koji integral formula but while in a minute you'll see it so then I know over the integral with respect to the variable t so this would be a line on log integral in the minute so this integral is holomorphic as well so who will freak and uh fact I'm allowed to differentiate under the integral sign wall Street on new with the I'm allowed to so this integral differentiated the pieces of if T and the integrated from a to B OK hope this time everything is correct so is the same as integrating out under the integral sign OK I'm allowed to place it's inside OK and this is actually known to you have the real case uh section together with 1 an approved for being a sphere and we differs same thing well let's see this is readable at all yeah the OK so uh at the end of last term we prove the same thing for it in the real valued setting so unfortunately the letters uh do not correspond nicely that and but again we have a parameter in which case in this case the parameters the 1st variable X and the integrated with respect to x and for each value of X this is partially due continuously partially to partially differential both with respect to the other variable here called wife so why it will play the role of the it over there OK and the proof was just OK make sure it take the difference quotient distance Gaussian by the linearity of the integral can be written under the into will but they're all in order to show that this uh actually converges you need to swap limit and integration and this is ok in case this function here converges solves the written there converges uniformly and there was to be proved so that was the real theorem and the complex field follows from taking from applying there's real theorem to the real part and the imaginary part separately that's all I want to see here so f efforts holomorphic men we have the 2 D F uh what let's use standard these DFT is equal to uh D S T real to uh the d x differential of real part class again the DD X of the imaginary part multiplied with I want case of this this formula uh we had early on and actually use before and now 4 here you can apply this theorem Bush separately to real and imaginary parts ended up and then you get the complex motion of the we share believers assumption of that the whole final reveal function f is continuous is missing here for real life so this is this is correct me this should be continues to be assumption OK so this is all I want to say um say about the proof OK and so this allows us to differentiate the cost interval formula under the integral sign yeah and perhaps I rewrite it because integral formula is that efforts that can be represented as the integral over db our a capital I used to have lost the also associated state of mind at the same time the so what I want to do now is differentiate with respect to that variable which is here in the interval so it's a parameter dependent integral depending on the into on the parameters that and I want to read differentiated and with respect to
37:10
z and you see I mean if I if I'm allowed to apply this formula if I'm allowed to differentiate under the integral sign what the z dependency is absolutely easier it's it's 1 over Z was constant over the over a constant minus that 1 you can differentiate infinitely many times so whatever the calculation is is which comes I find of this right now OK so let me draw the concentrations theorem 24 um so each holomorphic function each homomorphic function and as from say hold domain you choose to see has arbitrary arbitrarily the many derivatives many complex derivatives so each C 1 function is infects the infinity it's statement and actually the we can write out what this means precisely by writing out the result of n times differentiation of the course into a formula so this statement is sperm so so for the bone the art of so is not compare the content and you and and in the end we have infected from the following formula the nth derivative is inspectorial over to Part I times the integral over db our all of said non which is contained completely contained in use so f is defined on that set f of zeta the and all of the heat up over a data miner's said to be in class 1 DC attempt and resource for all said in with simple board said in the yeah of not so we have an explicit formula obtained by and obtained by differentiating the Gaussian integral formula the as you what's see in a minute this to this before the break uh world community I'm there and so thank you the or Korean solids do the differentiation proved the proved prove um for OK let's just do it and then verify verify that the assumptions our of the and like this through a satisfied so assuming there are I can I can put it applying the Laplace rule I can and to the co she integral formula I have that this is the desired of the left half of the turned over zeta minds the mean of that user in time and what is this small as we all know the B was with little on a couple of a couple of of of his OK and so different note that these are constants so um all I need to do here is right take the they take these constants and use OK this is this is to the minus 1 so I get minus is to remind us tools so this is the minus the schematic and with the minus sign however the derivative by the chain rule there's another derivative of said so the minus sign goes away and I didn't use it here which is the formula uh and I've which is a very formula for the case n equals 1 conclusion and now you can keep on going so take a 2nd derivative apply the same thing so differentiate this under the integral using the likeness will well then you get this to the power of 3 and you get a minus 2 out of the minus what kinds of of the chain will once again but you get a 2 out what this to that men becomes part of the and sectorial right for for that that's the 2nd derivative 3rd derivative you have the 2 already here and you differentiate and then comes the 3 and so forth and so let me just write uh continue by induction that today performance it's pretty obvious so why are the assumptions of the lab it's rule satisfied measure coming this achieving notes on the correct here so had to realize cell I or II I write that for each OK so what's a situation here um we differentiate in a in integral depending on Z and status of parameter right zeta plays the role of the and z plays the role of set up their local so of what I need to verify is that for each state has been DB are also that what I want that this function this function of this function here the kernel is a nice function and isomorphic function and in fact it is since I walk I work on the so this works for is that in the are what is is not cost and
44:51
so then I have is since the state of of a boundary and the symmetry of a ball there's no singularity of this function it's a nice holomorphic function here this is all I need to see solved for this we have that z maps tool f of zeta over Z minus 1 nite shifts of state of mind is the is a horrible thing function this is a whole wall function with continuous derivatives but this is what I need to verify you look up there uh is holomorphic on you the derivative with respect to the right operator with respect to z is continuous so it has a continuous derivative in this holomorphic and also when I use induction I say the same thing for the already initiated function of and I have the to the here and say the same and the rest of the statement um is exactly the same OK so that means that the light that's rule applies so that means website that here right it's magnets rule applies the OK and then you can just differentiate and the integral sign and that's so whenever we have such an integral representation of the solution of the partial differential equation and we know that the solutions are uh infinitely often differential and in fact this is also the case for and solving the sound as I mentioned for the Laplace equation for monarch functions is also true that even if you start with at time 0 with the heat distribution which is not differential it as any logical in any positive time but diffusion has acted that you get smooth solutions to the heat equation so the heat function of of uh as a function of space and time it's the infinitely often differential again just by the same device differentiate and of into the site OK so let me make let's make 10 minutes break and ran a continuous analyticity OK I would like to continue and so so we've shown now overhead automorphic functions are infinitely often differential and the it is a it's clear what we showed is that a C 1 and complex the principal function is the infinity since we have the assumption that the whole more uh I edit the assumption that a automorphic functions is the 1 in order uh uh 2 works so remember that we had complex the frangible just as a definition for the difference quotient limit exists and for holomorphic we added the C 1 assumption now um and it's a nonstandard assumption if you don't go for rotation business but you some of the stuff when you don't need it well it turns out that actually I also don't need the but it's a little more complicated looking at this fearing we proved last time it's visible yeah before equivalent statements idiom implies loop integrals vanish implies a local primitives exist uh it turns out that this actually works also with office he 1 assumption and it was just see it in the limit of the difference difference quotient to exist but that it needs a careful analysis and so I will do it also it seems to me like an academic uh function uh a academic question uh unnecessary questions uh since with methanol gap between C 1 and C infinity as at once defensible means infinitely often differential and now and I will question is so far how weak conditions you can still make this fear so I will not go into and the details of the question but uh let me just mention that it can be done what I believe is more important is the next the next subsection and it's even more surprising is that and automorphic functions are analytic and analyticity and others to sit analytic entities this T I can write it and then take it ended ticity action seems correct for holomorphic functions and also holomorphic functions so the next subsection so um what of analytic mean what it means can be represented by a power series that's just like the out in the form of the definition f from you to use C is called analytic is and only if uh OK for all points be in you know for all of these points were exists radius depending on the point and positive to be very correct end the power series coefficients powerseries coefficients in CEE such that we have a policy is representation such that f of z equals some from 0 to infinity if all of a and C minus B to the end and this uh holds for all but said in his little ball B on a all speak the meaning that and system you then we can and a point uh z will be man we can find it on a this contained in you we have a policy reversal presentation what remember the real fury for this case which very ugly and what do we do the real case um the yeah we use the Taylor series and the Taylor series has a remainder term so well 1st the Taylor polynomial and then you can you find just just can define the Taylor series by going to by taking
52:28
the infinite sum and then there's some ID theorems 1 of the ideas fumes small there too often but i only copied Lagrangian form onto the slide so you have F of X plus h equals f of and now comes the Taylor polynomial plus the remainder term and you don't quite know if a remainder term converges to 0 as n tends to infinity or not so you don't know if it Taylor seriously infinite Taylor polynomial if that represents a function or not and that is you know there are counterexamples to this fact given by the famous function e to the minus minus 1 over X squared so this suffering I has to recall recall that the function F from I to all given by f effects is 0 0 4 x equals 0 and E to the minus 1 over minus 1 over X squared else uh is the famous example of a function where the Taylor series Spanish is all a derivatives here are 0 and if any power series repeated if the policy is represents a function then a must coincide with the Taylor series by an identity variant so the Taylor series is identically to 0 and so is no power series representation of this function uh it's 0 has no um while under as there is no policy is coded in representation which coincides uh with a function so as power not note here this is the way to phrase that no power serious represents represents f in the and neighborhood in any neighborhood neighbor on also 0 and in fact Taylor series in Taylor series is identically to 0 the OK this is a famous really example which tells you read and Taylor series nice and actually for physicists I believe the world is an analytic so it's no problem but for mathematicians that have a serious problem uh very such functions which the infinity but not analytic not representable in a policies you may wonder however being a physicist you may wonder if and and it's is it is a good assumption here what does analyticity fire do this thing here then right of Pluto uh the molecules realize that the gravitational field is different him instantaneously with no time while probably analyticity is bullshit us physics in OK but it's a nice property of the colors here and all the real stuff is much more complicated than what we can do now so we have theorem 26 also due to co she met anyone mosig function can be represented by a policy was locally so let s be holomorphic the the therefore mu 2 so in be equal more fake and again I want to consider a little ball B are also be the and you say the then home care there and there exist numbers in the meaning what no such the rules while such that f agrees with the policies and uh in terms of sentient this follows and it's called each of but a in the world it's also worth an equals 0 to infinity and actually varies a nice representation formula for the a ends Sloan it's a quoted tool this all let me also say serious forgot to say to mention that the series is an ICU serious so it converges converges an absolutely and locally uniformly kernel of the mean care must say a word about this in a minute and but this is the sort of all has a nice convergence properties we uh introduced in the real case also ends as floor uh for the case that this little ball is a she compactly contained in you so if not uh if not for a standard if this doesn't hold right here then decrease or a little and then this will hold and so if it's compactly contained I have a representation formula for the end then I can write them as a kid needs of other stuff again as the nth derivative of be the divided over and trial and enter representation formula is 1 over 2 pi i but also f of Z to said minus B to the N plus one the and this time it's really gives us OK and in the integrate over boundary of this ball dt a D P I was be what OK so it's as good as it can be a um let me to have this slide handy for if here so in the real case just to for uh in in the real case just statements this statement is that you have given up OK because the radius of convergence given in terms of any of the policy is coefficients so there's no way to um to say rude uh directly in terms of the function F you want to represent by power series is no way of telling you directly in terms of F what has radius of convergence are it's as a formula once you know the policies when the offense then you know where it converges and this is for instance given by the because the uh radius where while it's sufficient to know that uh uh to recall that in R to the end
1:00:06
must be bounded and then you can prove its satisfies this formula and then you have for their respective policy areas uh it converges absolutely in locally
1:00:16
um uniformly on the ball of this radius and it
1:00:21
diverges outside the ball and there's nothing to say in general about what it does of about and what is locally uh uniformly mean um it means that if you are a fuel um if you take this a sufficiently small neighborhood then of uniform convergence the the so all respects excluded you to write this on locally uniformly and the means means uniformly on a small neighborhood have at the norm of the world and name the what and OK so we call of all these examples of policy with it well there's only 1 example which you really need to know the geometric series since that is the key of all the policies Theory and this for example for this the rate of convergence is 1 and it's not uniform convergence uh for on the the ball of radius 1 since all these terms are um well all these terms have this supreme 1 when you're close to r equals 1 so that means you cannot estimate the limit function the way you the difference to the limit functionally uh a you can be you cannot estimate the terms from say N on weights and bias uniform small number this will always be 1 in modulus OK so that means for this series you do not have you do not have uniform convergence of the ball but you still have local local uniform convergence by splitting off a little sector I did this in the 1st term perhaps it's good to look at it and actually the trick we will use now is very similar to that the OK so let me give approved uh to the of the power series representation of analyticity so how we take there will take any point said this correct I mean this year to get taken is that in b r of speed maybe oral speed is this ball contained in you know and uh choose um all yeah OK work that we let me draw a picture of 1st of all this be you is larger what kind of a test His B Mauer I want to look at that so this formula works for all z and B are of the so far live achieve is given them yes actually given in the statement for so far there is a little z Ingersoll and now what I want to do is I choose the radius of this is the ball of radius direct capital are there I should harassed but it
1:03:58
didn't in that I want a slightly smaller radius and the the this OK um and will not really be explicit OK OK I need I need a ball of radius R with respect to uh and then me see how do I do this same role we are OK and this is ah OK now it should be consistent right I have this little ball team be and now I choose uh I choose road such that search said z also is in the row also uh the the OK as in the fury of real power series I will use a smaller radius in order in order to get convergence nearby uh majorization of a geometric series OK so now we have a coaching integral formula all pushing integral formula telling us that f of z equals 1 over 2 pi i and integral over dp our and now I think she I want a smaller radius DBE row uh the 1 on as sorry this is something is something I had done well in the worst case have to changes this maybe I wanna Steradian OK so I have 1 of zeta minus C D is a term and I want to do right a power series um a power series of presentation directly from this let me just tell you 1st how it goes and then verified the steps OK I 1 2 I want to use the geometric series to represent this here as a power series so various and as yet has to be proven is 1 over 2 pi i integral of of GDI I hashing gets a small a she never right everything with respect to a low and I just don't know right now um anyway we are also do you and of Satan and now comes the it's a serious argument which gives the representation said minus B to the N over Z minus there's a time he zeta minus B to the N plus 1 in and this is integrated from n equals 0 to infinity this is integrated with respect to the same time and no OK with the we show how this how visited by a geometric series and how this can be written in this form once you've done this and how do we get to a power series well uh we have to um flip the order between integration and summation if you take the summation mn this this year tells me this is the Z minus B to the N is desired and all the rest goes into the coefficients cell that's just write it out and this has to be proven so this is the sum and so so I've pulled out of this sum to the French and then I lied this is 1 of the 2 pi i integral of db db our of the and half of zeta OK not everything except for those that minus B comes in here cell F of f of Satan over zeta minus B to the N plus one the commercial will carries integrated with respect to the zeta and this be not dependent on zeta right about here this a z of minus B to the N Paul OK so you see that if I call this thing here and I have to decide formula when I have represented F of said there's some a and set the set minus B to be OK so they do things to verify these 2 equality science what do we have to verify for the 2nd equality sign well when are we allowed to flip the order of integration and summation to pull this out well it's a theory of term 1 but we need uniform convergence units on the convergence of the integral as as a so uh depending on the index n if this is uniformly convergence convergence convergence uh for n to infinity then we can actually do step 2 and this we need to show later step it's just a an arguement aware what's down to uh arithmetic and the convergence OK so let's check these 2 things there is too low to high we know caring itself 1st 1st we need to various I so we want to verify 1 end 2 ships OK for 1 on it so 1 the I will I will rewrite this in terms of the geometric series were it's it would be good how all while nets to this 1st and then I write something in between so how do I know how to do I can and be able to rewrite this in terms of Z minus the wall I introduces artificially so this is said to minus B minus p z minus beta no OK nothing nothing changed now I pull out be 1 of as they minds the which is used here so what is remaining a 1 and this over this sector in 1 minus and remaining is set minus B over the zeta minus B OK nothing nothing certain and now all by the geometric sums formula this now becomes a there is some in terms of these things here so this is 1 of the zeta minus B these times some over there well said minus p to the end of over Satan minus TFI I to be n n equals 0 to infinity it all and I have to check something wait for that that this is a lesson 1 a modulus round was has to be done OK so let's do this why is this less uh then 1 in 1 in those well sins the that's what I did here 1 and since then zeta if I choose Satan in who are of the and my integration by variable is in fact in DB are of the around so that means that it's a timeline Z minus B the zeta minus B which is this number here which should be less lesser models all then 1 well in fact it is since z down data minus B is is before radius and that minus B is less than Rose C'est uh so the whole thing OK I don't need energy to quote this radius all I need to say is this is a number less than 1 who all yet it would be in it would be row over our which is less than 1 and let me call to as usual for the geometric serious and then we see that this representation is valid in fact it's valid for all the numbers in modulus and so on so this correct so where it's representation so we have to some and over there by just pulling this inside gives the n plus 1 gene which is here OK so step 1 is verified it's not hard to and how is in order to uh to see that to see this there to see well the 2 non that is known for the 1st time the following that and the gist of cocaine 1 where who won about uniform convergence so what is uniform convergence mean what it means if it that the tail of the sequence starting on it's state capital and but this can be estimated uniformly no matter what data is only on a very uh um condition that say there is an abundance of all so I want to make this term here small for large and while this is the case in the geometric solids let's just write this out so Z minus B over Satan minus B to the end In model this is less than and who are the if it's huge her we the of can so this is less than long usual triangle inequality then the modulus of the sum of the model I and this is a what but because of what is the modulus what is this number it's this number Q to be on so this is some of Q to be an and this some of Q to the ends is 1 over 1 minus Q E cell i in order to get a smaller I have to start to is not at 0 for which this is valid but it kept land you had to do I want to show that this becomes small if S . Nadia with large index and so let's just do it started and an index card and say here we will and me again that any equals and infinity now s and the same thing here all now this is just the same as before just that I can pull out the q to the capital and and front so I get this and now you see that I can make this arbitrarily small by choosing an large enough this will since this number less than 1 this goes tends to 0 and so this is uh means that this is a smaller models and actually I should no Q everything's fine OK OK so this shows Red um n h o and actually this is independent note that the right hand side does not depend on my integration variable stator right this is for the zeta in dB while no problem is uniformly estimated by number which tends to 0 for n tending to infinity yeah so uniformly uniformly full zeta in DB error I also be quite OK and so on if this is so this means uniform convergence of this time yeah well in fact we have a little different unaware of and has 1 year and f of theta and phi and and that's the integrand but that's only multiplying by and under sanctions cell all case so so this year was sorry I should I should perhaps led uniform convergence of all but this is not quite the function of interest to me I 1 of this function here how this this uh converges when n tends to infinity well just might apply if I when I look at this data maps to f of zeta of interest some nowadays said miRNAs and and zeta minus speed but now I have an n plus 1 I OK well this is the previous function this here multiplied by F over zeta in 1 factor here zeta minus B no ISS previous function previous function times what i times f's of data over 1 factor zeta minus B but this here of state over Z minus B is on the given sets the B R is a bounded function since B is in the center and not nothing wrong with the singularity here and F of Satan has a certain so that need certain value and so you this is bounded on the context save on the compact it on the db are B which is contact with a continuous function on a content on a convex set so this is also bounded well if you have is uniformly convergent seek sequence of functions and not functions and multiply it by a bounded function result is again uniformly convergent and was cell again uniform convergence are all who again means uniform convergence of this function here now if I have uniform convergence then I can I can called the summation the limits In this case the summation air outside the interval and this is what's written here and so I have the power serious representation okay out this is good enough although this is scribbled into the cone of the Blackboard and uh so we have whereas uh approved often notices city so once again if you I mean if you on the um if you're really good at power series and know how to manipulate it in terms of a geometric sum then this is sort of uh very standard procedure so uh it means that that that the power series but VV and this city of Mosul off extensions is serves as a direct consequence of the Gaussian integral formula once again and again also for other PbPb these Killackey you can use the the same method but as you see from the calculation it's much more special the other when what I did before just to differentiated into the formulas easier then to get to a power series representation OK so this means uh we've reset of come to a really to them a point where why should summarize all the results leave or obtained summarize all results book on each song do twist on afternoon From this I want to do and for Mousterian so we have as a function of which I assume is continuously differential the will 10 and I said for it is the kernel and who was falling so 1st F is homeomorphic not own and you decide that for yourself what you prefer way you prefer to viewers I would say it has a ceiling linear partial derivatives in yeah it's and it's differential in real sense with a Jacobian which corresponds to a complex multiplication and that means you have to cross the remaining questions will carry some the 2nd property is um en if have it is and I write it out once again is real differential meal differential as so I thought I said a minute ago real differential in satisfies collusion Lehman caution Lehman among the so perhaps here I'm sorry for what I said what is said applies to here so here think of difference quotient exists different limit of the difference quotient exists the princess here think of real difference ability of the Jacobian 2 by 2 matrix and 8 presenting a complex multiplication which is equivalent to coach agreements of cell f has local primitives in Paris that you is simply connected it actually has a global as global primitives primitives all this only at you determined only up to constants so the plural is there adequate in any way no in any setting and to the 4th property would be if it is analytic needed at nite and all these mn you can add them you can add to these properties what we stated in hearing before and that the loop integrals over contractible loops vanish him that yeah I there's also also into loads uh the and then initial contractible loops for all contractible loops in and the and intervals of this form point and all and intervals from 1 point to another only depend on the initial an endpoint the formal topic loops the formal because uh In coincide full model because considered but perhaps this is not as important 2 right cell depending on if you count these things here as full properties with likable and properties of oral morphic functions and none of them None of them and is a OK there are is satisfied in the real case at least not in the real case of several variables I mean for a function for my to our we have local primitives if primitives once they are the ones that continuous him images write out the sentiment theorem of calculus and get primitives but full sanctions from true tool are as I told you he need integrability conditions lie on the rotation and so this is they're not clear OK so I think we have uh right now and in the the idiom and a point which is nice point to stop even tho they're still 5 minutes to go but I hope that some time making taken over time perhaps you want disagree with me OK so let's stop here thank you for your attention to