ato products I like but just like a raw tomato in any form I'm just not really a big fan of except in like a salsa but then is that a raw form because it's been mixed with other stuff so is it really a raw form because other flavors have infused it right so they have a pre-order poster no I would say with tomatoes don't like them solid paste sauce only fruits and veggies in general see fruits and veggies don't gross me out but yeah tomatoes I'm just not a just not big into raw tomatoes in any form besides again salsa is not really raw tomato though because it's mixed with other stuff you know adds to the flavor profile I would imagine a future patch would give you the option to enable Bluetooth support with limited joycon connectivity right now you can connect up to 8 joycons I can see like oh but if you're using bluetooth headphones then you can only connect 4 joycons or something right but I have no idea the Skyward Sword HG codes from the E3 giveaway go out tomorrow or are they going out at an uh there's only one Skyward Sword HG code we only gave away one copy right I gotta check that maybe we give away two I gotta check that document they're not going the I feel a bit bad the E3 ones are going out tomorrow the E3 ones are going out in like a week it's weird it's just how the budget worked out the E3 ones go out in a week the ones won for this month go out tomorrow it's I know it's weird uh I do have a lot of I think my I gotta let the Amoeba winners know my the Amoeba have been delayed for me that was an announcement today Nintendo announced it themselves they promised retailers all these Amoebos and we have all these pre-orders of the Skyward Sword Loftwing Amoeba sold and Nintendo informed everyone today by the way most of you aren't getting your Loftwing Amoeba come back in August maybe you'll get it then that was announced by Nintendo oh Nintendo so my pre-orders were affected so it's kind of like oh I have people at E3 who won those I have pre-ordered them I was hoping the Joycon show up that we gave away we gave away two sets of Skyward Sword Joycon I'm really hoping they show up if they don't what am I going to tell the people that won the Skyward Sword Joycon if my pre-orders get cancelled like sorry sorry have a good day ugh I have to do something for them it's dumb the pre-orders for the Joycon and the Loftwing Amoeba I'm I'm not happy right now I'm scared that I'm going to find out tomorrow because they're not going to arrive tomorrow but I'm going to find out my Joycon ones have been cancelled ugh I'm scared because we did giveaways for them I'm not keeping any of them I don't have a Loftwing Amoeba coming for myself I don't have a Skyward Sword Joycon coming for myself we gave them away during E3 I had pre-orders in and I'm so worried that I'm just not even going to get the products even though I had pre-orders in it's so dumb I'm going to feel so bad for the winners when I tell them sorry they cancelled all my pre-orders and I don't know when they're ever going to do another run of these ever they might never do another run I don't know it might be a year I have no idea how long it's going to take ugh dude I'm on since E3 isn't that crazy a month ago today was Nintendo's E3 I wonder where my members are I remember we gained a ton of members during E3 so we're around renewal time we're around renewal time let me look at this quick I wonder how many members we actually have still ugh as we peaked at 131 yeah we dropped a lot yeah we're down to 95 members we dropped a lot so many members just left us after E3 ended yeah we were at 131 now we're at 95 but oh well what are you going to do right what are you going to do nothing you can really do about people leaving it is what it is still more members than we have before E3 so we'll see a lot of them are due for renewal here coming up in the next few days so we'll see what happens ugh yeah yeah dead man walkie 89 you're right I'm scared because we gave away I think two loft wing amiibo and one loft wing amiibo I don't know I had to check the document I had two pre-orders in for each I had two pre-orders for loft wing amiibo and I had two pre-orders for joycon and I'm scared that all four are going to get cancelled and I got to email those people and be like I'm so sorry it wasn't a fake giveaway I swear but they cancelled my orders and I don't know when we're going to get new ones because I don't want to pay scalpers right I don't want to go buy it from a scalper so it's like I really hope they don't cancel on me as cool as it is to get my money refunded oh great your money's refunded I don't give a shit I gave these away I guess that's the risk you take in trying to give away pre-order items though I suppose you know you pre-order something and like I have to switch all that pre-order from oh I know that's going to cancel on me so who the hell knows it's so hard to rely on pre-orders for shit that are oversold how you doing man looks like we're giving the real switch pro you talking about the the steam device the steam deck 80 copies at your local Walmart nice do you really think Nintendo will release another pro um do I think Nintendo could release another model of switch absolutely will they be more powerful I hope so but I don't know when um you're welcome Tierra Hughes yeah I talked about it on Twitter and on the YouTube channel and I saw a number of people surprising I think I helped inadvertently I think I helped like 25 people pre-order a switch OLED today crazy right crazy I wasn't I don't it's like people didn't know it's like the all the people they follow on Twitter weren't wasn't updating for them um it's weird but we we helped a number of people get switch OLED pre-orders today it was crazy there's a chance you can find them in store my game stuff said they got a bunch of boxes this morning in the back that they were going to unpack tonight in the morning yeah I don't know the amiibo are sold out and Nintendo Nintendo themselves are the one that we're like yeah we oversold that what we produced sorry check back in August it's like Nintendo really so we'll see what happens maybe maybe my pre-orders will arrive in August maybe they won't arrive at all maybe I'll get refunded I just feel bad because they're collectors items and I had the pre-orders locked and loaded that's what makes me feel bad not for me for the people I gave them wait it's like I had this shit locked and loaded this was taken care of it's frustrating it's more like RGT today got frustrated when he had the switch OLED in his card on Best Buy and couldn't get checked out he got frustrated imagine having pre-orders locked in for over a month and then like the day it's supposed to come you get told oh by the way you might not get your product have a good day just like the Zelda Game & Watch next thing you know they're going to tell me cause we gave away a Zelda Game & Watch next thing you know they're going to tell me Target I got it at Target I think they're going to tell me that I'm not going to get the Zelda Game & Watch I pre-order it too that'll be the next thing they tell me winners are announced tomorrow don't worry about it now I fall but I still worry I do worry about it because we have all these giveaways from E3 which I'm still taking care of but these are the ones that you're waiting you're waiting for them to come out it's frustrating maybe I just don't give away pre-order items anymore I want to give away a switch OLED and there's a chance I might get a second pre-order in but it's like do I give it away on hand then give it away even though I could change the shipping address and have it shipped directly to them like if I get an Amazon pre-order in I know I can change the shipping address and just have it shipped directly to them like it would make a lot of sense oh I got an Amazon pre-order in give that pre-order away directly ship it to them so it's just I get frustrated I think everyone's frustrated by that situation anyone who's been told maybe August really? E3 stress is going to weigh on me until every single digital physical item is taken care of and that's not going to be done until the Zelda Game of Watch comes out so I'm going to have this E3 stress of the giveaways isn't even close to over for me isn't even close to over so I'm just anyways um can't even pass 10K views for his video since E3 the Alexander 707 says OMG Nintendo brain can't even pass 10K views for his videos since E3 look how this sensationalist over happy Friday's Stalin he deserves it so a little fun fact three months ago my videos were basically getting like a thousand views every now and then you get two or three thousand right now right now my videos the actual videos not live stream live stream views all depend on how long you're streaming for what the topics are all that jazz my standard videos my last one earlier today I got three thousand will probably top out around four by the time I wake up 7.3 for the one before that 6.8 we had 2.6 then we had 5.7 then we had 9000 4.4 3.9 6.3 6.8 6.9 7.9 so like yeah I'm not getting 10,000 but I'm getting like 600% more views than I was getting three months ago so yeah we're not getting the same amount of views just like there will be for the next Nintendo Direct and there hasn't been as big of news of course but it's like my viewership is way higher than it used to be what the fuck am I complaining about people always look at it like oh man you used to get 10,000 I'm like yeah that was a special period of time where do we settle when E3 is done we've settled where every video I upload is probably going to get at least three thousand views I was getting 1,000 three months ago so every video I upload is basically going to get 3,000 several of them are going to double or triple that I'm good just because I don't get 10,000 views of video okay cool cares who cares I'm happy man we'll troll somebody else but they already marked it up 50 dollars but they really want to hire to a model that's another 50 sure, Nintendo doesn't care um should I buy a Skyward Sword HE digitally and play tonight or just wait and pick up my pre-order in the morning I mean I'm picking up one tomorrow I don't know it's up to you can you wait? I don't know I mean I know it's like available to play like right now um can't pace everyone on a spike yeah everything on a spike yeah like that's the thing like yeah every year by the way the lead up to E3 plus E3 itself is always my highest viewership period you always are doing numbers that are bigger than normal the question is when the dust settles and E3 is a month behind you are you getting more views now than you were before the E3 rush factually yeah my channel is doing way the hell better now than it was three months ago cool that's a win that's a win now the growth is kind of stagnated and that's going to happen but a lot of that's on me I haven't been uploading this one you guys know I'm not back to my normal stream schedule right I haven't had a week where it's been Monday Tuesday Wednesday Thursday Friday and forever so I haven't had my five streams a week yet in a while and I haven't been doing two three videos a day like I used to so yeah as I get back into a schedule here after my vacation sure things will probably pick back up because I'm not making more content more the content drives the views drives the subscribers drives the discovery if I'm not making enough content then yeah I'm not I factually have barely made content since E3 like in comparison to what I was doing before E3 so I'm not worried about it we'll be fine everything I do make is performing outside of my expectation even my video today I didn't think that video was going to get 3,000 views this is old news from yesterday I'll be lucky if this breaks like 1,500 here we are 3,000 so it's like dude every video I make is like outperforming what I expected to do so even this live stream we've been consistently 130, 140 plus all night we've been struggling to get over 100 live in some of my recent streams and yet here we are we're back up we're back on the up and up and as long as I keep the consistency of my streams going the viewership is going to continue to climb I just haven't been consistent you know I know all the YouTube game works I've been doing it a long time well aware of how the YouTube game works but people like to credit things to be insensational or whatever I don't even know clickbaiting I guess so excited for next year's E3 I hope Verizon announced a 6G alright TfSouls I've been here a long time your live streams are so fun and you're funny too you're absolutely amazing thanks I try alright we're going to call that a night hey we have 171 likes tonight cool thank you guys so much for tuning in I had a lot of fun tonight I hope you had some fun too we talked a lot about switch OLED and the valve thing and we went until a fun little story on the first night that I ever got smashed you guys are awesome have a good night you're playing Skyward Sword HD tonight have fun enjoy I'll be joining you guys tomorrow night I will start my save file tomorrow night live on stream I know the intro can be a little long to the game hopefully we get past the intro during the live stream we'll see what happens have a good night guys


We don't try to achieve stillness, we value stillness. That's how it's achieved. We see it as, that's very, very useful. When I'm not thinking, something is happening. See, we've been conditioned to think that when I'm not thinking my mind is blank, nothing is happening. Here's the crazy thing. When our mind is rehashing and reanalyzing, when that's happening, the truth is, the mind is blank. That's actually when the mind is blank.


Okay, so welcome to this course on advanced complex analysis, so what we intend to do is to give us given this course a selection of topics from advanced complex analysis. So of course we assume that you have already done a first course in complex analysis basically you know covering the notion of an analytic function and then Cauchy's theorem and then the idea of Taylor series, Laurent series, the idea of singularities in the residue theorem okay. So of course we have chosen for the topics to be presented certain important theorems certain landmark theorems which are usually not stressed upon in a first course in complex analysis and whose proofs are also not all that easy okay but they are very interesting theorems and they are of very geometric nature and that is what we will try to cover. So of course so let me start with what we will be doing in the first few lectures and that is about trying to look at zeros of analytic functions okay, so basically you know so we are interested in zeros of analytic functions. So this is a broad topic for the first few lectures alright and of course what I am going to do is state some important theorems connected with this theme okay. So of course let me first of all remind you and when I say analytic function I think of a function which is defined on a domain in the complex plane okay, so a domain is an open connected set okay. So the fact that it is open means that given a point in the set there is a small disc surrounding that point which is contained in that set, so the fact that set is open is being is the same as saying that the set is a union of discs okay and of course you know we always work with open sets because if you want to study the properties of function at a point especially if you want to take a limit at a point then you should be able to approach that point from all directions and so you must have a nice disc surrounding that point where the function is defined so you can actually take the limit okay. So we always study only functions at points where in a neighbourhood of which the function is defined okay and that is the reason why we always study functions defined on open sets and of course we also study functions defined on connected sets because I mean if a set is not connected then it falls into two pieces and essentially a function on such a set is a different function on each piece okay, so you can reduce the study of functions to just studying functions on a single piece okay and that is why we always study functions defined on open connected sets which are domains alright. Now so we take a function defined on a domain in the complex plane and we of course assume the function takes complex values so again the core domain of the function is complex numbers and if you remember from the first course in complex analysis there are several ways of trying to define when the function is analytic at a point in the domain. So of course the simplest definition is that of and it is the most common definition it is that the function should be differentiable not only at that point but in a small disc surrounding that point okay and we also use the word holomorphic function instead of the word analytic function that is common in the literature and we say a function is holomorphic or analytic on the whole domain if it is analytic at every point of course the function being analytic at a point can also be described in several other ways one way is the way that I told you that it is differentiable in neighbourhood of the point that is the first derivative exists in the neighbourhood of the point. The other way of defining function to be analytic at a point is saying that the function can be expressed as a convergent power series centered at that point in a small disc surrounding that point and this should and if this happens for every point then we could call that function an analytic function so basically there is one definition of analytic function which says that the function is differentiable once in the neighbourhood in a small neighbourhood of the point there is another definition which says that it is represented by a convergent power series centered at that point and the relationship between these two definitions is that they are equivalent and that is the great thing about complex value functions because power series if you would have learnt in the first course in complex analysis is infinitely differentiable within it is region of convergence the region of convergence of a power series is always a is a disc centered at the point and then probably some points of boundary may or may not be included but within the disc the power series always represents an analytic function and not only it is once differentiable it is differentiable infinitely many times so this is one of the striking features that differentiate real valued differentiable functions and complex valued differentiable functions ok. So if you assume a real valued function on a subset of the real line see on an interval open interval is differentiable throughout the interval there is no reason that other higher derivatives exist in fact there is no even there is no reason even that the derivatives continues whereas if you assume a complex analytic function of a complex variable on at a point is differentiable in neighbourhood of the point the amazing thing is that it becomes infinitely differentiable which means all the derivatives of all orders exist in they are all continuous this is the greatness this is the amount of power that one time differentiability gives you infinite differentiability in the neighbourhood ok and that is the characteristic feature of studying analytic functions and of course the power series if you take the coefficients of the power series they are going to be related to the they are going to be related to the Taylor coefficients ok and these Taylor coefficients can be gotten by using Cauchy integral formula ok. So you have this notion of an analytic function either you define it as something that is locally given by a convergent power series or something that is a function that is differentiable everywhere ok differentiable once everywhere and of course the usual way of checking that a function is analytic is the is by checking the so called Cauchy Riemann equations so what you do is that you check you take the real and imaginary parts of the function this is how you try to check a function is analytic usually you take the real and imaginary parts of the function and then you write down the Cauchy Riemann equations and then you check the Cauchy Riemann equations are satisfied and then you also probably check that the first partial differentials are continuous and then you conclude that the function is analytic. Now so there is a way of checking that a function is analytic using Cauchy Riemann equations as well but nevertheless the point is that we are interested in zeros of analytic functions and the first important fact that all of you should have studied in a first course in complex analysis is it the zeros of an analytic function or isolated ok so that is the first important fact that means given a zero of an analytic function there is always a small disc surrounding that zero where there are no other zeros ok so this is called so if you have different zeros they can be separated from each other by small open discs centred at those zeros and this is what we say this is what we say this is what we mean and we say that the zeros are isolated so zeros of an analytic function are isolated ok. Now you see so then of course that comes the question when you what is the problem with looking at the zero of an analytic function well you take a well you take a function which is having a zero at a certain point which is analytic at that point if you take a small neighbourhood you know there is a small neighbourhood where there is no other zero because zeros are isolated now if you invert the function in that disc then you know the reciprocal of the function is defined except for the zero there ok and then this gives rise to a pole at that point and this is one example of what is called a singularity ok. So analytic functions there are points on the boundary of the region where the function is analytic where the analytic functions are supposed to have singularities so there is so called singular points and again you would have studied about singular points in the first course in complex analysis so basically one is always one always worries about so called isolated singularities because one does not want to the case of non-isolated singularities is far more complicated to analyse. So for example if you take the function log z then you know it has several branches you have to define various branches of the logarithm but to define a branch of the logarithm you will have to make a slit in the on the plane for example you have to slit the plane along the negative real axis and then you can define a branch of the logarithm and then the whole negative real axis becomes points of singularity for this function. So this tells you that the singularities are not isolated because they are they continuously lie on the negative real axis but of course these are not the kind of singularities we are interested in one always studies isolated singularities and isolated singularities basically are of three types if you recall the first one is called the removable singularity and the removable singularity is essentially the is a non-singularity okay for example a function like sin z over z there is 1 by z times sin z if you look at z equal to 0 if you try to directly substitute in the function you will get 0 by 0 which is not a defined value but of course you know limit as z tends to 0 sin z by z is 1. So if you define the function to be to take the value 1 at z equal to 0 this gives rise to an analytic function and therefore the point z equal to 0 is what is called a removable singularity for the function f of z is equal to sin z by 1 by z into sin z and how this is reflected is of course it is reflected by looking at the power series expansion if you take the power series expansion for sin z and divide that by out by z you see that you essentially do not get any negative powers of z and that tells you that essentially this is a Taylor series not a Laurent series and therefore this is not really a singularity and therefore and you will also see that if you take the power series for sin z at the origin and divide by z and put z equal to 0 you will get 1 and that will tell you that 1 should be the value that you should define for the function to become analytic at the point at the origin okay. So this is what is called an isolated removable singularity then of course come the so called poles of a function and the poles are well they are supposed to be thought of as zeros of the denominator okay. So I mean the simplest example is you take if you are given a point z0 then you look at the function 1 by z minus z0 to the power of n where n is a positive integer and then you know z0 is a 0 of z of this function of the denominator of this function which is z minus z0 to the power of n it is a 0 of order n. So 1 by z minus z0 to the n has a pole of order n at z0 okay so the pole is basically a 0 of the denominator that is how you should think of it and well so a pole is really a singularity it is something that you cannot tinker with to make the function analytic at that point okay and in a way a versatile kind of singularity is called an essential singularity and that is a singularity for example you take e power you take exponential of 1 by z at z equal to 0 that is an essential singularity and both poles and essential singularities are really the bona fide singularities the removal singularities are actually non-singularities because you can always get rid of them you can get rid of removal singularities by redefining the function at the point but you cannot get rid of a pole or an essential singularity at a given point okay and of course you would have also learnt how do you distinguish between a pole and an essential singularity and there is a so called Laurent theorem which is an analog of the or you may even call it an extension of the Taylor theorem. So the Taylor theorem is a theorem that if a function is analytic at a point then you can express it as a convergent power series around that point that is you can find a convergent power series centered at that point which point wise converges to your given function in a good neighbourhood of the point this is Taylor's theorem and this is the theorem that actually tells you that once differentiability implies infinite differentiability this is what gives you the equivalence of the seemingly weaker definition of analyticity being once differentiable and the stronger definition of and the stronger you know implication that once differentiability implies infinitely many times differentiable so that is the Taylor theorem. But the Laurent theorem is kind of extension of the Taylor theorem it tells you that if you also include negative powers then you can get a series involving also negative powers in a deleted neighbourhood of the point and that is called the Laurent series and for all you know the Laurent series may have negative powers of arbitrary order ok.  So this is called a Laurent series centered at z not and the function converges to this equality means that this Laurent series converges if you plug in a value of z in a small disc surrounding z not then this series converges to a value which is equal to the function value at the at that point and of course the point should not be z not because you cannot substitute z not here because you will be dividing by 0 all these negative terms and the fact that if z equal to z not is a removable singularity then all these negative coefficients will be 0. So your Laurent expansion will actually be a Taylor expansion that is exactly what happens when you look at 1 by z times sine z for example z equal to 0 ok and ok so this is so called Laurent expansion and the point is that among the important singularities namely the poles and the essential singularities you can also distinguish the type of singularity by looking at the Laurent expansion if you get infinitely many of these negative powers ok then the singularities and isolated essential singularities for example exponential of 1 by z at z equal to 0 and if you get only finitely many of these negative terms then it is a pole ok and the order of the pole will be equal to the minus of the largest negative subscript you get here ok. So that is another that is one way of you know trying to distinguish between a pole and an essential singularity there is one more way of distinguishing between a pole and essential singularity and that is by taking limits ok if you take the limit of the function as the point tends to the singularity and if the limit exists and is equal to infinity from all directions so of course this means that you have to make sense of what limit equal to infinity means I mean the limit of a complex quantity you say it is equal to infinity if the modulus of the quantity becomes arbitrarily large ok no matter how you approach the limiting point. So you know if the singularity z0 is such that as z tends to z0 in no matter in whatever direction the mod of f of z goes to infinity then you say the limit of f of z as z tends to z0 is infinity and this is a situation exactly when z0 is a pole ok and if there could be at what happens in the case of an essential singularity the limit will not exist in the sense that you might get limits differently limits as you approach in different directions for example you can take exponential of 1 by z and try to calculate the limit from by approaching the point z equal to 0 from the positive axis from the negative axis real axis positive real axis you will see you will get different values the fact that you get different values from different directions tells you the limit does not exist and that is precisely that is precisely the condition that tells you that it is an essential singularity ok. So the most important thing about singularities is what is called the residue of the function at an isolated singularity and that is supposed to be the value of this coefficient A sub-1 which is a coefficient of 1 by z-z0 ok. So you know residue of f of z at z equal to z0 is A-1 ok and this is a very very important value for the function because it is connected with the residue theorem ok which tells you that this is what you will get if you try to integrate if you integrate the function over a curve surrounding over a simple over a close curve that surrounds this point simply for example a circle a circular or a circle that surrounds this point if you integrate the function what you will get is 2 pi i times this this quantity. So for example you know you know you can take a very very simple example take g of z to be 1 by z-z0 I mean this is the simplest thing you can think of ok and you well here is z0 on the complex plane and then if you want draw a circle look at the circle mod z-z0 is equal to rho is the circle and obviously this is an analytic function except for the point z equal to z0 everywhere else the denominator never vanishes z0 is the 0 of order 1 so it is a simple pole ok a pole of order 1 is called a simple pole and if it is order is greater than 1 it is called a multiple pole and well if you try to integrate if you calculate 1 by 2 pi i integral over if I call this circle as gamma f of gz I will end up with 1 ok. So in fact you know if I if you want I can even put a lambda here where lambda is any complex number ok and if I integrate it what I will end up with is well if you would have done this several times. So if you want to integrate over a contour the method is that you first parameterize that contour and then you make a change of variable mind you whenever you integrate something when you integrate a function over a contour you must understand that the variable lies on the contour ok. So that means that you should write an equation for the contour and that is called a parameterization of the contour. So the parameterization of this contour is z equal to z0 plus rho e to the i theta where theta varies from 0 to 2 pi so this integral becomes well if you write it down perhaps you have already done it you certainly have done it but let me just recall quickly you will just get lambda. So you will get 1 by 2 pi i integral theta from 0 to 2 pi I am going to get g of z is z0 plus rho e to the i theta of course I have forgotten to write dz there which is the variable of integration so I will get dt z0 plus rho e to the i theta and this will turn out to be 1 by 2 pi i integral 0 to 2 pi what you will get here is 1 by z minus z0 if I substitute this I will get rho e to the i theta and if I differentiate this I will get i rho e to the i theta d theta ok and what will I end up with I will end up with my rho to the i theta cancels my i cancels I get 1 by 2 pi i integral 0 to 2 pi I think I forgot a lambda there so there is a lambda on top so I will get lambda t theta and that is just lambda ok. So the moral of the story is that you see if I look at this function and integrate it over a small over the small circle surrounding this point which is a simple pole I pick up this question and you know actually about this point if you try to write the laura expansion this is the laura expansion the laura expansion for lambda by z minus z0 is lambda by z minus z0 ok it is already the laura expansion and a minus 1 is the coefficient of 1 by z minus z0 and that is lambda and that is what shows up when you calculate 1 by 2 pi i and that is the residue for the function. So this is the simplest illustration of the philosophy behind the residue theorem the residue theorem says that if you integrate over you know the point which is isolated singularity and assume that there are no other singularities then what you will get is the residue and you will get I mean 1 by 2 pi so let me write that so to be more precise you have the residue theorem which is the residue theorem is the starting point at least for our discussion. So you know so basically you have let us assume that you have a nice contour like this and you have a function f defined on this on a domain which contains this contour and the interior of the contour and assume that you know there are well several isolated singularities z1, z2 and so on zn if you integrate the function if you write 1 by 2 pi i integrate over this contour of fz dz what you will get is some summation i equal to 1 to n residue of f of z at zi this is the residue theorem. So what I have done here is I have taken I have simply taken the function to have only 1 singularity isolated singularity and I do this integration and what I end up is the residue at that point but if you have several and of course you should assume that there are no singularities on the contour over which you are integrating okay. So you know the assumption for this is that the function is analytic in the interior and also on the boundary which means that to say the function is analytic on the boundary means actually it means that is analytic in a small disc surrounding every point on the boundary which means it is actually analytic in a bigger open set a bigger domain which actually contains this boundary and the interior okay. So then this is the so called residue theorem and in the simplest case it reduces to this and you can also see that you know if you take this function and you instead of taking lambda by z minus z0 suppose I took this power series I mean if I am not power series if I take this normal series and I integrate it okay around a contour like this then of course the first thing is that the integral of this whole series is the same as integrating term by term that is you can integrate term by term and then take the resulting series and this is correct because you can interchange integration and summation provided the series of functions converges uniformly okay and it is a theorem that if you take a Laurent series then within in the region where the Laurent series is defined if you take a closed disc in that region then the Laurent series will converge uniformly okay and of course whenever I say Laurent series should be it should be a deleted neighborhood you should not include the point of course because you cannot substitute the point because you will be dividing by 0 for the negative terms but of course there is a similar theorem for power series which says that whenever you have power series which is converging in a disc then you take any closed disc inside that disc the power series will converge there in fact you absolutely and uniformly okay. So because of this uniform convergence the integral if I calculate this integral for this function I can actually integrate term by term and you know if I integrate term by term from here onwards each term will give me 0 because it is Cauchy's theorem, Cauchy's theorem says that if you integrate an analytic function over a simply closed curve there is the I mean the integral is going to vanish you are not going to get anything okay and so the integral of all these terms will go away okay the integral of this term will give you a-1 of course okay and the integrals of all these terms will also go away because 1 by z-z0 to the power of for example 1 by z-z0 the whole square has an anti derivative which is just 1 by minus 1 by z-z0 to the power of 1 okay. So all these negative terms from power 2 onwards they all have anti derivatives and it is a version of the fundamental theorem of calculus that whenever a function has an anti derivative then the integral is just the anti derivative evaluated at the final point minus the anti derivative evaluated at the initial point but then this is a closed curve the final point is same as the initial point therefore you get 0. So if you integrate this term by term the only thing you will pick up is a-1 and that is the proof of the residue theorem if you had a single if you had a single singularity okay and then if you have several singularities this follows because of Cauchy's theorem because what Cauchy's theorem will tell you is that Cauchy's theorem tells you basically that if you take an analytic function and integrate it over a simple closed curve the integral is 0 okay but this is the so-called simply connected version of Cauchy's theorem which is over a simple closed curve where your domain which means the region inside the curve has no holes okay but then there is a different version of Cauchy's theorem which says that you know if your domain has an outer curve like this and you have inner curves you have inner curves and of course you know in all these issues the orientation of the curve is very important we always take the curves to be oriented in the anticlockwise sense and that is called the positive orientation okay and if you change the orientation the sign of the integral will change that is how it goes and if you apply Cauchy's theorem to the region which is this the interior of this curve and the exterior of all these little curves that the function is of course analytic and therefore you will get that the integral is 0 but that will amount to say that the integral over the outer region the outer curve is the sum of the integral over the inner curves okay and but then sum of each but each integral will give you the residue at that point as I explained here and therefore you get the residue theorem basically you get the residue theorem from an application of Cauchy's theorem and literally this kind of argument okay fine so you have the residue theorem now the you see the so let me having told you so far let me also tell you what kind of theorems we are going to prove okay I think I do not know how many lectures it may take but probably a few lectures so you see the kind of theorems we want to prove are actually theorems about zeros of analytic functions and well so yeah so glimpse of the you would like to prove so probably some of you who have done a little bit of further reading beyond the first course might have seen two of these theorems but this is where I would like to start the course so the first theorem is so called argument principle and the this is so called argument principle and the and what is the argument principle so let me state the theorem that I want to prove one is argument principle probably it is just some kind of corollary to the residue theorem if you look at the logarithmic integral if you remember it from the first course but anyway I will recall it the second thing is using the argument principle or even otherwise you can prove the so called Cauchy's theorem okay then we would like to study we would like to prove Hurwitz theorem and of course and thereafter one would like to study you know the open mapping theorem prove the open mapping theorem and of course also one would like to prove the inverse function theorem so these are the I mean the first set of theorems you would like to prove probably you would have seen the first and the second maybe but anyway it is they are the starting point so I will make it a point to recall them okay so let me explain let me explain what these what these theorems are so the first one is the argument principle so what it so what does it say so I will briefly describe what the statements of these theorems are and you will see that they are actually connected with I mean they are the right theorems that will come out of this topic trying to study zeros of analytic functions okay so the argument principle so the argument principle is well you know 1 by 2 pi I integral over a simple closed curve so when I say simple closed curve let me explain what that means first of all a curve is said to be closed if its initial point is the same as a terminal point okay of course by a curve generally we mean the image of an interval a closed interval if you want the closed interval 0, 1 on the real line a continuous image of that on the complex plane it is called a curve for example the circle of the curve because it is the image of the interval 0, 2 pi under this function theta going to if you want z0 to the i theta okay so it is a continuous image of the interval and it is a closed curve if the initial point is the same as the final point the fact that there is an initial point and there is a final point tells you that the curve is already oriented okay that means there is a direction for the curve and that is that direction is given by the direction of the that is given by the direction of increase of the parameter the variable that is used to write the equation of the curve okay and when I say simple curve it means that the curve is does not cross itself does not intersect itself so it is not something like a figure 8 or more complicated curves it cross themselves one segment of the curve twists and turns and comes back and hits itself at some point again crosses itself okay there are no such self crossings so such a curve is called simple curve and since you are going to do integration okay the curves that we will always deal with will be piece by smooth that means the curves if you write down the parameters parameterization for the curve okay then the parameterization will always come will be defined over some interval and the fact that you can divide this interval into sub intervals in each of which the function you write down is actually differentiable it is differentiable and continuous okay so this is what is called a piece by smooth curve. So for example here the function theta going to z0 plus rho e to the i theta where theta lies from 0 to 2 pi is of course a continuously differentiable function of theta which is the parameter okay but more generally a curve need not be given by a single parameterization it could just break down into several pieces and each piece may have a different parameterization okay one piece may be say part of a circle another piece may be part of a parabola the third piece may be part of a line but it does not matter the point is piece wise it has to be it has to be smooth so whenever I say simple closed contour whenever I say contour it is always something that is piece wise smooth okay. So the argument principle basically tells you that if you are looking at a function which is a function defined on a domain like this okay with the property that this the function is analytic on the on this on a domain which contains this whole region okay except at finitely many points which line the interior which are only poles okay you assume that they are only poles okay and the function should not have the function should not have any zeros on the on the boundary of the contour okay then the 1 by 2 pi i integral over gamma of d log fz it is called the logarithmic integral okay which is will give you the number of zeros minus the number of poles inside the inside region okay. So this is the this is basically the argument principle and of course d log fz of course means d log fz means f dash of z by fz what you must understand is that because f is analytic wherever f is analytic f dash is also analytic because as I told you a function that is analytic is infinitely different so and this is a quotient of analytic function it will be analytic there the only problem is the denominator might vanish. So wherever you have zeros of f f dash by f the logarithmic the so called logarithmic derivative of f will have a pole wherever f has a zero and of course if f has a pole then f dash have a pole okay so the only poles for this function we assume the only poles for this function are some zeros of f inside the zeros should not lie on the boundary and some poles of f inside and they also should not lie on the boundary. So the boundary should be free from both zeros and poles and there are only finitely many zeros and poles inside your domain okay and so this is the argument principle so computing the log if you integrate the logarithmic derivative the argument principle tells you that you get the difference between the number of zeros and number of poles so that is the so called argument principle okay and then let me quickly tell you about what these other theorems have to say so I am just now giving an overview of results but then we will go into them more deeply so what is Roush's theorem so Roush's theorem is so as I told you this whole exercise is to somehow study zeros of analytic functions okay and basically for example you want to count the number of zeros inside a region which is bounded by a let us say a curve a simple closed curve so the point is that you do not get in general a zero of f will of course be a pole for f dash by f so you cannot avoid considering poles also okay so that is the reason the argument principle gives you zeros and poles also so in particular if there are no poles then you will be counting the number of zeros and of course when you count number of zeros mind you every zero has to be counted with multiplicity so for example you know if you take the function lambda by z minus z not this has zero of order one at z not so the number of zeros will be one if you take a simple closed curve enclosing z not if it is not going to enclose z not the number of zeros will be zero okay but if I replace this by lambda by z minus z not to the power of m then the number of zeros will be m though physically there is only one zero at z not but it is order is m so it is also the order of vanishing okay it is a number of times the factor z minus z not z minus zero appears so whenever you say zeros or poles you have to count them with multiplicity zeros also have to be counted with multiplicity for example if I take the function z minus z not to the power of m where m is positive then z not is a zero of that function but it is it should be counted as m zeros so zeros and poles have to be counted with multiplicities and then this formula holds so this is the kind of counting principle okay that is one thing then Roush's theorem is something more the philosophy of Roush's theorem is that you know you take a you take an analytic function in a domain and suppose you are interested in the zeros inside the region the domain that is you know that you get the interior of a simple closed curve okay then Roush's theorem says that you know if you perturb the analytic function little bit okay even after perturbations the number of zeros will not change okay that is if you add to the analytic function another analytic function which is small enough that means you add to the analytic function as smaller analytic function of course you know there is nothing called smaller or bigger in complex numbers because complex numbers are not ordered but then whenever we say smaller or bigger we always refer to the modulus so you know what Roush's theorem says is that you take a function f of z which is analytic in a in say a in say a bounded region surrounded by a simple closed curve then the number of zeros will be the same for f of z and f of z plus gz where gz is a smaller function smaller on the boundary okay so that is Roush's theorem and you think of adding g to f as a small perturbation okay so I will just write it in words the number of zeros of zeros is not affected inside a simple closed contour is invariant is invariant means it does not change is invariant under small perturbations. So you take the analytic function and add to it a smaller analytic function function that is smaller than this function on the boundary on the boundary contour then even after adding it the number of zeros is not going to change okay so the addition of another analytic function which is dominated by the given analytic function on the boundary is called perturbation if you want okay and it is a small perturbation because what you are adding in modulus is strictly less than the modulus given function on the boundary okay. So this is Roush's theorem right so one version of Roush's theorem will tell you that suppose you want you have two analytic functions f and g okay how can you conclude that they have the same number of zeros okay so the answer to that is you calculate you know if you calculate mod of f plus g triangle inequality will always give you mod of f plus g is less than or equal to mod f plus mod g okay now the question is on the boundary if you get strict inequality if you get mod of f plus g is strictly less than mod f plus mod g on the boundary then both f and g will have the same number of zeros inside that is another avatar of this Roush's theorem so it helps you to also to compare the it tells you that two analytic functions will have same number of zeros if the sum of their moduli is strictly greater than dominates the modulus of their sum on the boundary okay that is another avatar of this okay so it helps you to compare number of zeros. Then the third one is Hurwitz's theorem so this Hurwitz's theorem is again you know it is again a very beautiful theorem what it says is that you know if you have sequence of analytic functions which is converging to a given function okay in a domain and assume that the convergence is going to be uniform on every close disc in that domain so this is called uniform convergence on compact subsets okay the other word that is used in the literature is called normal convergence okay so if you have normal convergence okay which means uniform convergence on compact subsets for example the convergence is uniform on every close disc in your domain. So if this is a normal convergence and if f has a zero of order n at z0 then what happens I will draw a diagram so z0 is a point where the limit function f of z has zero okay now some fundamental complex analysis will tell you that because the convergence is uniform and since each of these functions is already analytic f will also turn out to be analytic okay that is again an exercise that you can easily try to do and in fact the derivatives of all these will also converge the derivatives of f okay and all this is just because of normal convergence uniform convergence over complex subsets and of course in that case the integrals will also converge because the moment you have uniform convergence integrals derivatives everything will behave well with respect to limits okay. So suppose the limit function will be analytic and if it has a zero of order n at the point z0 what will happen is you can find a small enough disc surrounding z0 small enough radius such that beyond a certain stage all the fn's they will all if you take each fn it will have a n zeros inside this each fn will have n zeros with multiplicity which means that some of the zeros may be multiple zeros and the beautiful thing is as you if you plot those n zeros okay so I was so you know if you plot those n's those capital n zeros and if you make this small n become larger and larger then these various zeros will slowly come they will coalesce they will all tend to the point z0 okay. So what it tells you is that when you take a nice limit of analytic functions then the zero of the limit comes from zeros is a limit of zeros of the same number of zeros of the functions in the sequence okay so if f has a zero of order n it is z0 then you know then let me write it out somewhere here so then there exists a row greater than zero such that for large n fn has n zeros in mod z-z0 fn row which converge to z0 as n tends to infinity. So what Hurwitz theorem says is that the zero of the limit comes by you know you take zero beyond a certain stage it is the zeros of the functions that are giving the limit zeros of the function that you are taking limit of it is those zeros that slowly you know coalesce together and give you the zero of your function. So this is Hurwitz theorem and then I will quickly tell you what the open mapping in the inverse function theorems are the open mapping theorem is a very beautiful theorem it tells you that if you take an analytic function and you take a point where the function is non-zero I mean the derivative of the function is non-zero then there is a neighbourhood of that point where the function is an open mapping which means that you will map open sets to open sets and this is a very deep result okay because it is very rarely interpolish that you get open maps okay and for example a bijective continuous map need not be a topological isomorphism but if it is bijective continuous and open then it is an homomorphism because the openness of the map along with the injectivity will tell you that the inverse map is contiguous so you know so what it tells you is that an open map is as good as an isomorphism except that you need to know that it is injective if you know it is injective then it is an isomorphism okay so you can invert the map and all this is true analytically complex analytically also so the only condition is that you know the derivative should not vanish at a point and then in neighbourhood of the point everything is beautiful it is a local isomorphism okay that is essentially the open mapping theorem and let me quickly tell you about the inverse function theorem the inverse function theorem is that again in a sense you can think of it as another variant of the open mapping theorem what it says is that whenever the derivative is non-zero okay at a point then there is a small neighbourhood where you can invert the map okay and the inverse function can be written again using a Cauchy integral okay there is an integral formula to write out the inverse so you can invert the function there is an explicit formula okay so this is the so I am not writing more details we will go into them in the succeeding lectures so the point about these two theorems is that you must be essentially looking at well there is a more general version of open mapping theorem will tell you that all you do not even need the derivative to be non vanishing essentially you need a non constant analytic function okay so you need a non constant analytic function and it will always map open sets to open sets okay and in particular if the derivative is non-zero then in a neighbourhood of that point where the derivative is non-zero the function is actually a holomorphic isomorphism it is an analytic isomorphism which means it is an isomorphism it is injective onto its image which is open and if you take the inverse function that is also holomorphism that is also an analytic and that is given by an Ix formula integral formula that is what the inverse function theorem is okay. Now all these are somehow connected with 0s of analytic functions and they can all be derived starting from the argument principle which essentially is I should say the residue theorem applied to not to f but its residue theorem applied to the logarithmic derivative of f so the root for everything is the residue theorem okay so we will do this in the forthcoming lecture okay so let me stop here.


Your coca-cola bottler presents Claudia Claudia based on the original stories by Rose Franken brought to you transcribed Monday through Friday by your friendly neighbor who bottles coca-cola Relax and while you're listening repress yourself. Have a coke and now Claudia David you haven't forgotten have you what today we fixed breakfast for mama She's not going to like having breakfast on the tree. She'll hate it. No, fine. That's Harry. I'm ready.  I feel like Netschenski who Netschenski I don't know him Hey, watch where you go. Who put that chair there? Who do you think? The kitchen's hot open the windows. I'll get the cream. We don't have We don't have to whisper anymore now What I do next you set the tray and I boil the water for the eggs fry the bacon make the coffee toast the toast and squeeze the juice All at the same time. Oh, it'll only take me a second.  No, I got everything ready last night. Just turn the stove on will you yeah Juice is all squeezed must be gremlins in the refrigerator.  There's money in it these days and no Responsibilities can you bottle? I've got a white coat. That's for dentists tails are for button. Oh tails are unfashionable I think even butlers are Say David, are we gonna be rich someday? I doubt it. That's good. I don't think I'd like to be rich I married a wonderful woman. Why not? Oh, it's so complicated. That's just as good a reason as any sides I feel rich now. You do a very Pick the eggs are ready. I do not good. Then you hold the tray. Okay. I've got it now toast cereal bacon Eggs are two minutes the coffee's two minutes the napkin No room to this room where here under your arm. Claudia. I'll drop everything. No, you won't all right.   That's right You can grab it with your teeth Claudia. I can't take another thing. Don't talk so much. You'll drop it I'm going on here. No, I've got to wait. I forgot something. Oh, lift your other arm I'll get even with you for this.  Hey, Claudia. Candlesticks are very important every well-served breakfast tray. Okay. Now come on Breakfast is brown breakfast ready What's going on here? Morning, ma'am. Good morning, ma'am.  There you are. Just what you ordered, ma'am You forgot the napkin. Use the sheet. Lovely manners. Here are the napkins under my arm.  Why is it a big picture fool? But David and I intend to help you. We should have served it in Aunt Louise's soup terrain, huh? Soup terrain. Don't forget, ma'am. We're bringing it back this afternoon.  All right. Stop arguing Dead or alive, that soup terrain goes back.  Four eggs. I can't eat four eggs. That's funny, David. Now I can eat four eggs. Then eat them. Mrs. Brown, are you going to want all of that toast? Go on, take it. You don't like bacon either, do you, ma'am? Take it all. Lucky for us, isn't it, David? She's such a fussy eater. We'll just have to leave her a candlestick to nibble on. Take the tray, please, David. You haven't eaten anything. Hand me my kimono over there, will you, Claudia? I'm so modest all of a sudden.  It's right on that chair. You've got as much on as any bathing suit. I'm in bed, not in the water. Here it is, modest. Thanks. Hey, where do you think you're going? Since this is your breakfast, you get into bed and you eat it. Mama. I'm going into the kitchen and make myself a decent cup of coffee and I shall drink it standing up. Dear, I'll be awfully glad to put this soup terrain down. It feels as if it was full of soup. Don't look to me for sympathy. This was your idea. Oh, I don't like a store like Pacquiao's. Am I hat on straight, Mama? I can't tell. Is it the both of you? No, of course not. Is it? Yes. Oh, dear. Push it over my eye for me so I can make a good impression. Which eye? Left.  No, not yet. Just a moment, please. I'll be right with you. Poor thing. He thinks he's going to sell us something. You should have warned him. He's got to get his order book. Mama, you tell him. Me? I'm just here for moral support. Well now, what can I do for you, madam? It's my daughter. It's about this soup terrain. I received it about three weeks ago and... Yes, lovely, isn't it? One of our most choice pieces. Oh. One, Claudia. I like it very much, too. You've got another one like it. Is that it? Yes. Yes, that's it. So you wish to return it? I'd like to, if it's all right. Do you wish it credited or will you purchase something else? I'd rather purchase something another day. Or could I have the money back? Shameless, but sensible. Of course, madam. Your name, please. Mrs. David Naughton. But it was sent to me as Mrs. Claudia Brown. Oh, I see. A wedding gift. We get a number of exchanges on those. I'll just take it to check the original purchase. If you don't mind, and I'll be with you in a moment. Well, he's awfully nice about it, wasn't he? They're used to it. Mama, look over there. Don't look. How can I look if I don't look? Cheating shows, that's all I can say. Cheating shows. What are you talking about? What shows? Cheating. What do we do? What is it? It isn't an edit, Aunt Louisa. That's not funny. Now, don't fool about such things. I'm not. What? She's right over there in the flat silver. I don't believe it. Oh, yes, I do. It is Aunt Louisa. She's looking this way. What do we do? What do you do, you mean? Mama, you can't leave me. Why not? I told you I didn't want to be a party to this. Aunt Louisa lives at Pacquets. I bet she has a bed under the counter. Be quiet, and maybe she won't see us. She sees us. Here she comes. You talk first. I will not. You better, because I'm not going to open my mouth. This is all your fault. My fault? I said you should have some sentiment and keep the terrine. It's too late. I lost. Why, my dears, what a nice surprise. Aunt Louisa. Hello. And Claudia. Hello, Aunt Louisa. How nice to see you, child. So sorry I couldn't come over Tuesday evening. It's these allergies of mine. I think I discovered a new one. Silver polish.  It's too late. Dear, dear. When are you going to bring that husband of yours over to meet me? Oh, soon. Your generation has no family feeling. You go about your ways as if relatives didn't exist. Now, if your father were alive, we won't talk about that now. Claudia, what are you twisting about like that for? Nothing. I was just looking. The service here is slow, but good, very. You never have lovely things at that age. Yes, I always buy my gifts here. I got Claudia's soup terrine here. Yes, I know. Thought of sending the child a check. Oh, Aunt Louisa. But then I thought a check is so unsentimental, especially for my dearest brother's only child. Oh, here you are, Mrs. Norton. And now about this soup terrine. Yes, have you got another one just like it? What did you say? Claudia. Have you got another one just like it? Well, but I thought that you were... Isn't that the terrine I sent you? Yes, it is. Yes. Louisa, Claudia feels...  Didn't I, Mama? Yes, you didn't. What was that, Mrs. Norton? Yeah, that's really charming. But Claudia, can you use two soup terrines? Yes, David's crazy about soup. He likes one at each end of the table. That's rather an attractive idea. Never thought of it. But Mrs. Norton, a moment ago... I know, but have you got another just like it? Yes, we have dozens, but I... Well, could I have just one more? Just one will do fine, please. That's David. Why doesn't he use his key? The lock doesn't work for him yet. Well, go let him in. You go, Mama. I should say not. The old soup terrines. Mama, what did he say? How would I know? You married him, not I. Yes, but if it weren't for you, I wouldn't be me. Aren't you going to let him in at all tonight? See, that's an idea. I wouldn't try it. Well, go on. I'm going. The stand is still. Who am I? I'm coming. I hear you. Hello. Don't you say hello to Mama? Hello, Mama. Hello, David. Is this where I really live? What do you mean? This is a fine reception for a stranger, but for a husband of so many hellos and so little else. Don't sit there, David. Come into the bedroom. Wait a minute now. Let me sit here for a second. Well, now that you're home, David, I'm going to put this up around. Don't go, Mama. What's the hurry, Mrs. Brown? It's the chicken, David. I want to watch it. Well, watch it later. Claudia. Yes? Claudia, what is that I see over there? Where? On the table in front of the window. I don't see anything. There is two of them. You're seeing double. I've been drinking again. Men are all the same, aren't they, Mama? Probably thinks he's seeing two suites. Huh! Huh! It's right. Huh! What's going on here? I don't know. We're asking you. That chicken needs me. Bye-bye. Mama! Claudia, is that or is that not two suites on the table? Isn't it R? Isn't what R? Well, two suites are plural. Shouldn't it be R that... Don't get grammatical. Just answer me. It R. It is. Claudia, slowly, dear, in words of one syllable, how come when this morning we only had one and you went to return it, tonight we have two? Oh, darling, you know how suites are? No. You tell me. Oh, David, how could I return it? He was so nice and she had her allergies again. And she is my father's only sister and I'm her dearest brother's only child. Oh. She was there? What could I do? Aunt Louise, it must have been very, very touched. She was. It was almost worth it. Oh. Don't you think we can learn to love them, David? If I live long enough. They'll grow on. They've grown enough already. Don't worry. Darling, I'll think of something. They'll come in handy. How? For what? For why? For when? Oh. For bathtubs. For babies. Claudia. I was just thinking, David. Just thinking. All story material used in this broadcast of Claudia was under the supervision of Rose Franken and William Brown Maloney. When company comes, how grand it is to be able to say howdy neighbor, have a Coke. Just like old times. Now you can stock up, order Coke by the case if you like, and keep it handy when folks drop in to visit. Now you can step up to the familiar red cooler at your favorite refreshment stand, lunchroom or service station, and ask for Coca-Cola and get it. And the price is still five cents. Every day, Monday through Friday, Claudia comes to you transcribed with the best wishes of your friendly neighbor who bottles Coca-Cola. So listen again tomorrow at the same time. And now this is Joe King saying au revoir and remember, whoever you are, whatever you do, wherever you may be. When you think of refreshment, think of Coca-Cola. Or ice-cold Coca-Cola makes any pause to pause that refreshes.


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If you had it like you wanted it, TV show or podcast. Podcast. I don't believe that, but I'm gonna ride with you. Podcast, podcast, I could do what I want. Radis 76er. Alan Iverson, not even close. Bubba Chuck. The doc say, hey, anyway. I love doc. Love doc, I love Barclay, I love- Radis Boxer Ever. Yeah, yeah, yeah. I'm gonna give it to Floyd Mayweather. I'm gonna give it to Floyd Mayweather because of the money he made. I'm about the youth and Floyd is showing these youngins that is no ceiling in the money you can make out here. Who's the best non-professional basketball player between 35 and 50? You look at that. You looking at him right now. David Gillett, Sam Giselle, Gilbert Arenas, Chauncey Gillips. I mean, y'all know.


These are my kids. We were hiking in the Oakland Woods when my daughter noticed that someone had thrown a plastic tub of cat litter into a creek. And she looked at me and she said, daddy, that doesn't go there in this really innocent smile and comment. And when she said that, it reminded me of when I was a kid growing up, I used to go to summer camp. And on the morning of visiting day, right before they'd let our anxious parents come barreling through the gates, the camp director would say, quick, everybody go pick up five pieces of litter. You get a couple hundred kids, each picking up five pieces. And within a few minutes, you've got a much cleaner camp. And I thought, why not apply that crowdsourced cleanup model to the entire planet? And that was the inspiration for starting Litterati. The vision is to create a litter-free world. Well, let me show you how it started. I took a photograph of a cigarette using Instagram. And then I took another photo and another photo and another photo. And I noticed two things happening. One, litter became artistic and approachable. And that was simply because of the power of Instagram. It kind of brought me closer to the environment in a way that I hadn't before. The second thing was, at the end of a few days, I looked at my phone and I had 50 photographs. And I had picked up and either thrown out or recycled each and every piece. And I realized that I was effectively keeping a record of the positive impact I was having on the planet. That's 50 less things that you might see or you might step on or some bird is going to get tangled up in. And so I started telling people what I was doing. And they started participating, again, using Instagram of all things, to photograph individual pieces of litter, tag them Litterati, and then throw out or recycle what they're finding. And it started to spread. And one day, this photo showed up from China. And that was the point that I realized that Litterati was more than just a collection of pretty pictures. We were becoming a community. And the community was generating a ton of data. Each photograph tells us a story. The easy way to think about it is that it tells us who, picked up what. A geotag tells us where. And a timestamp indicates when. So the next thing I did was I put together a Google map. And I started plotting the individual points where I was picking up litter. This is downtown Oakland. Those are the points where I started. And as this community grew, well, the data grew. My two kids go to school right in the bullseye. And I showed this to their teacher. And this woman looked at me and she said, that's awful. And I said, actually, it's way worse than that. Because this just represents what a few of us have picked up in the last couple of weeks. The reality is that we are surrounded. We just don't see it anymore. Litter is so commonplace that it is blending into the background of our lives. And we as a society have become desensitized. But I started asking this question, what if it weren't like that? What if it wasn't in the background? What if we could bring it to the forefront? How might we use technology and data to really start making a difference? I'm going to show you three ways that Litterati have started doing just that. The first is with cities. The first city that ever caught wind of what we were doing with San Jose. Some of you may remember a campaign that took place, which was I Heart San Jose. They ran it around Valentine's Day. And we worked with them to create a campaign. And the campaign was simple. It was a photo contest. They used it as a way to mobilize their community, get the citizens involved. What better way to show your love for San Jose than by keeping it clean? And we gave away prizes and tickets to Cirque du Soleil and the San Jose Sharks and vouchers for local restaurants. And San Jose was just interested in, again, mobilizing their community through this very simple action on Instagram to keep the city clean. Then San Francisco caught wind and they had a very different idea. San Francisco wanted to understand what percentage of litter came from cigarettes? Why to create a tax on all cigarette sales? So long before Litterati existed, the city put a couple of people in the streets with pencils and clipboards. And those people walked around collecting that information, creating a percentage which then drove a 20 cent tax on all cigarette sales, which as you might imagine, generated a massive revenue stream for the city of San Francisco. And then they got sued by the tobacco industry who claimed that collecting information with pencils and clipboards wasn't precise and it sure wasn't provable. So I got this phone call from the city. Hey, we heard about Litterati. Can your technology tell us what the percentage of cigarette litter is? And what the city didn't realize was that at that time, our technology was literally my Instagram account. And I was like, yeah, we can. Four days and 5,000 pieces later, we did. And Litterati's data was used not only to defend but double the tax, which now generates an estimated $4 million a year in annual recurring revenue for the city to help clean itself up. During that process, I learned two things. Number one, as you might imagine, Instagram's not the right tool. So we built an iOS app. Last week, excuse me, two weeks ago on Earth Day, we launched Android to 5,000 people just on the waiting list. That was the first lesson learned. Second lesson, if you think about it, every single city in the world has what we call a unique litter fingerprint. What you see on the streets of San Francisco looks nothing like what you see on the streets in Seoul or Stockholm or even Oakland. And if you could generate a massive new revenue stream just by taking the time to understand the data of cigarettes, what about coffee cups, potato chip bags, candy wrappers, soda cans, and the list goes on. Again, how might we use the data to make a difference? And if you could fingerprint San Francisco and Oakland, well, what about Central Park or the entire northeast of the US? Every country has a litter fingerprint. What we're seeing in Amsterdam looks a little bit different than what they're seeing in New Zealand or even places in sub-Saharan Africa. Second group that we've started working with are brands. Whole Foods ran an Earth Day campaign. Anybody who comes in with one literati photograph gets a free baked good. Worked out really well for Whole Foods for two reasons. The first was it was a great PR story. The second one was they knew anybody who came in for their free baked good was going to shop. It was a great, simple campaign for customers to get involved and benefited Whole Foods, it benefited the planet and it certainly benefited the customer. Here's an example of something that we started doing and it'll give you an indication of how we look at working with brands. As you might imagine, the data set that Litterati is collecting is very unique in the sense that we have all of this brand data and many people think that the brands are to blame. Our perspective at Litterati is we're all in this together. The brands certainly have a shared responsibility to creating more sustainable packaging. You and I have a shared responsibility to making sure we don't throw things on the street. Schools have a responsibility to making sure that they educate our youth, cities have a responsibility to making sure that the waste cans are properly emptied on time. This will give you an idea of how we think the right way to work with brands is. Again, downtown Oakland, this block is so badly littered, it's hard to imagine. 19th Street Bard is a few blocks away from here if that adds some context. So the Litterati community got together and picked up 1500 pieces, which I know sounds like a lot. I'm telling you it's nothing. 1500 pieces, we learned three things from the data. The first was most of the litter came from Taco Bell. Not a big insight, because there's a Taco Bell on the corner. Here comes the insight, most of the Taco Bell litter were their own single use plastic hot sauce. Those little plastic packages, right? But the real kicker was that most of those plastic packages of hot sauce had never been opened. Now you've got the root of the problem. It's one thing to go clean up a beach or a city block, but it's another thing to understand the root cause of the problem. Why? Because once you understand the root cause of the problem, you can have a proper path to a solution based on the data. So now imagine you could take this content, imagine you could take this data and go back to Taco Bell and say, listen, we're not here to shame you, but not only are you causing an environmental impact, you're also losing money. Okay, so the skeptic says, well, maybe one of those packages is a penny in plastic and a penny in sauce. We did one walk and found a couple hundred of them. Let's scale that out to a year and then multiply that by the 5,000 Taco Bells there are. Now you're talking about real numbers.  We don't need to be the villain. We have an opportunity to become the hero and here's what I mean. Taco Bell says, you know what? We understand that part of our packaging was ending up on the ground. So here's what we're doing. You want hot sauce? You have to ask. Or we installed bulk dispensers. Or we worked with a firm like IDO to come up with much more sustainable packaging. Now you have a brand who takes an environmental hazard, turns it into an economic engine and they become an industry hero. That's how we believe you work with brands. The last group and the group that I am most proud of that we've been working with are schools. Why am I most proud for two reasons? One, that I'll start it because of my kids. And two, if you are really going to create change for the long term that sticks, there is no better place to start than with our kids. Let me show you what schools have been doing. One, they've been creating art exhibits just as a way to create awareness. Two, they've started adopting literati as a citizen science platform. Schools are doing creative writing assignments. Go home, take one literati photograph, come back the next day and write an essay on how that particular piece of litter ended up on the ground. What that's leading to is the students are then having discussions about what can be put in place to prevent that from happening. An example might be a young girl takes a photograph of a Wrigley's wrapper that's on the ground. The story that she comes up with is that it flew out of an overflowing trash can so the next day the kids get together and design trash cans that have lids. Here's an example of one school that blew me away. In Modesto, California, a group of fifth graders picked up 1,247 pieces of litter just on their campus. And what they learned from the data was that the most common type of litter on their campus were the plastic straw wrappers from their own cafeteria. So these kids went to their principal and said, why are we still buying straws? And they stopped. Simple. And they learned that individually they could each make a difference but together is where they created the impact. The Litterati community started right here in Northern California with two little kids. And today it has spread across the world to 114 countries. We were averaging 10 pieces a day. Today we did about 4,000. And you know how we've been doing it? One piece at a time. Thank you very much. I believe that I'm now given the opportunity to take a few questions and answers. And I am happy to answer any question that you have. Yes, ma'am. Baseball, stadiums. And I'm just wondering if you've done anything with that and maybe if you went to the San Francisco Giants and said, if we clean up the stadium before we leave could they lower the ticket price for us? Something like that. Before we talk about baseball I'm curious about movie theaters. You mean just outside movie theaters? No, people bring stuff in. They're Coke things and stuff and they just leave them there and then the crew has to come and clean them out. Why can't you just take that with you and dump it when you're leaving the movie theater? But nobody does. It's almost like an accepted sort of thing.  It's just accepted that somebody else is gonna clean up after you. You know, it's interesting. This is one of those problems that nobody thinks about. First of all, it's not fun. It's not sexy. And most people think it's not my problem. Not only that, but it affects all of us. And it impacts the economy. The environment degrades our communities, kills wildlife and now poisons the food system.  The Ellen MacArthur Foundation and the World Economic Forum just came out with a report not too long ago. By 2050, there will be more plastic in the ocean than fish. So it impacts all of these areas but you're actually hitting a very different point which I think makes it even more complex which is you're talking about behavioral change. And that's real hard. Really, really, really hard. You have to give people not only a reason to care but also a reason to take an action that makes a difference. One of the things that we've heard from our community is that traditionally, people have picked up a bottle cap in Seattle or San Francisco and those have been isolated, quiet acts. What difference would it make if I picked up this one bottle cap? I'm all alone in this massive problem. What Literati is starting to do is serve as a unifying umbrella. The technology allows those individual isolated acts to become social and shareable and that means it goes from overwhelm to empowerment because now I know I'm not alone. Back to your question about baseball. The San Francisco Giants actually have a program called Clean Sweep which I thought came because they, a couple of years ago, swept the World Series but actually it was because they wanted to give back and they've done an amazing job working with a lot of schools. To your point about could they then incentivize people to reduce ticket prices for contributing to a greater good? We actually think there's a huge opportunity with that. So with our application, we're not there yet but we're approaching a point where, let's say you picked up your 50th piece of litter in the streets of San Francisco, what if in the application you could be served a $25 voucher for BART just as a token of thanks from the city? There's lots of interesting creative ways that we can start to align interest between city and the citizens that live there. You're doing a great job, thanks. Thank you. Yes, sir. Thank you very much for the presentation. I thought it was very informative and also it's excellent what you're doing. My question is around kind of your partnerships with corporations and to what extent are they starting to go beyond mainly marketing and PR? For example, the Whole Foods example and two real collaborations where you mentioned the Taco Bell example. We have not done any proactive outreach yet. It's all been inbound and frankly that's just been because of a lack of resources on our end. Right now we're getting inundated with interest and we need to be able to bring in the resources, capital, team, infrastructure to allow those things to happen. With brands I think that this is one of the most slippery slopes there is for us. Again, because so many people feel that the brands are responsible. And you can imagine working with a Taco Bell might look very different than working with another brand that has similar values like a Patagonia. So Patagonia is not a company whose litter is on the ground but they're very much about sustainability. Or imagine we could work in collaboration with Clorox or Glad Trashbags or Purell or you work with any top-notch advertising agency in the city and they could start to come up with some really creative collaborations. That is a long-winded answer to your question that we haven't really started to explore what deeply integrated partnerships could look like. But we have some ideas internally that we think could be pretty cool. Ma'am. What can we do about the people who litter cigarettes? And people who don't take their doggie poop bags and they just drop them on the sidewalk or along the grass area.  And it just irritates the heck out of me because when it rains it drains into the drain which goes out to the ocean and kills things. It just ruins the environment. I wish I had an easy answer for you. The reality is this is really hard. Like really, really, really hard to solve. I've picked up more litter than you could ever imagine and it's like a Sisyphian task, it never ends. You ever see, anybody ever seen Fantasia? You know the Mickey Mouse segment with the Sorcerer's Apprentice where the brooms just keep splitting and splitting and splitting and splitting? That's what this is like. Literally because everything is breaking down, right? That stat that I quoted earlier, more plastic in the oceans than fish is not off. That is nuts. And it's because so much of the plastic is breaking down into microplastics. It's really hard to do anything about people who just culturally throw something on the ground. It's not in all societies.  Tokyo's an anomaly. Back to this woman's comment earlier, it's about behavioral change. I don't believe in punitive measures so hard. First of all, it is really hard. Unless you're hanging outside a bar morning, noon, and night to catch people throwing cigarettes in the street. And here's another interesting thing is that most people think that all the litter on the street comes from people throwing things on the ground. That's just not the case. If you walk outside these doors right here, the streets are covered. You would think that you would be seeing people throwing things down all day long, realistically, other than a cigarette, but when was the last time you saw somebody throw something on the street? Doesn't happen that often, right? So that tells you that there's a more systemic problem at work. Infrastructure needs to change. I don't know what you do about the smokers. Other than continuous, make it easier for them. There was a great campaign that took place in, I think, Brazil. Here's what they did. They took Brazil's most famous soccer player and Argentina's most famous soccer player. They put a transparent container on a wall and it just said, vote with your butt. And so people would drop in their discarded cigarette butts. That was brilliant. There's lots of ideas like that we can come up with. The dog poop, you're on your own. As a dog owner, I couldn't agree more. Anybody else? Thank you all very much. I really appreciate it.