So you've also got a sky bit league two season begins against Exocity on Saturday. How are you feeling going into it? Yeah I'm feeling really good to be honest. I mean we've had a good pre-season everyone's got minutes and they've got these formations that you want wants to you know go out with and you know we're all looking forward and ready to go. So you've played six of the seven pre-season games how do you think your fitness is feeling going into the game? Yeah I'm feeling good I mean the first few games were like you know trying to get that game fitness back into me but you know I think game in game out with more time and stuff I'm feeling a lot fitter and just ready for the first game of the season. Oh is he joining the club in the summer how are you finding it with your new teammates so far? To be fair they've brought me in really really kindly and got good great lads and you know I feel like a part of their family who was here last season as well so you know we've got a good tight knit squad you know there's no there's no bad egg in here and you know I'm just like again I think everyone's looking forward for the for the first game of the season. So how do you assess pre-season overall then some really positive results like what's your overall take on it? Yeah I mean it's great results I mean playing playing against Blackburn playing against Zackington you know Leagues Above Us and we've done really well against them we show nice stuff our capabilities and you know sometimes we need to take out our negatives out of that game and turn it into a positive the next game so I think we can do that and hopefully on Saturday will be a great game. Of course a new manager at the club as well Derek Adams how are you finding working under him? Well I wouldn't say new but yeah he's you know I've had him a couple of years and I know he's working his philosophy and stuff so you know it's the same as as old times you know he wants you to work hard and put a good grafting but also you can have a laugh so it's a combination of the two and I think he's going to get the best out of the squad. Obviously if he were to be extra on Saturday how much confidence did you think that would give the team going forward? Yeah I mean it'll be a great confidence going forward you know it's always positive to get to get you first win in the first game and getting that momentum but you know it's going to be tough no doubt about it extra great sides you know we mean the Gaffers got a bit of a well Naz as well got a bit of not hatred but you know Derby's sort of coming from Plymouth right now so you know it's going to be again it's going to be a tough game but we're looking forward to it.


Hello and welcome to my latest book review. I am joining you this evening, literally just off the plane here in Brussels and I would like to take you through my experience of reading 50 things that made the modern economy and it's by Tim Harford. This is a very well known book and it is one that took me a while to read but it wasn't a difficult read and just a very reflective read. So before I start telling you about what's in the book I just want to first of all point out who might be interested in picking up this book and having read it. Two people in particular. The first one is the person who's tired. The person who is tired of constantly thinking, constantly problem solving in their own business and what they would like to be able to do is just to stop and be transported into another world. But they don't read fiction because that they might think isn't exactly the best use of time. They still want to do something very productive and this I have to say is perfect for that. So I'll tell you where I bought this. For my brother-in-law's 50th, we went to Le Hinge and specifically we stayed in the cliffs of our hotel at Lourdes and I went down the day before on my own and I said I'm going to take a day out and I have to say the scanner is just one of the most beautiful parts of the world. So I said I'm going to take a day out and I'm going to do exactly what I just said to you and I'm just going to stop thinking and I'm just going to reflect. So I was in the Le Hinge bookshop and that's where I picked this up. So I started reading and I'm not joking you. This draws you right in. It draws you right in. I'm telling you it does because what it does is it gives you 50 small case studies of inventions that have completely changed the way we live. So what it does is it draws you into this place of where you start thinking through, oh yeah so since that happened then that led on to something else which led on to different business models and so on and so on from there. The second type of person who I think would really enjoy this is the person who wants to challenge assumptions. Now when you think about some sort of business models like Airbnb, Airbnb challenged the idea that you would take somebody who you don't know at all into your house, walk away, leave them there and expect them to treat your house as you do yourself and that was an assumption that people assumed could never be the case and Airbnb thought about that and said hold on now, hold on now. What if we could do something about that so that we could in fact create an environment where people could make money out of their homes when they're abroad themselves or when they're away or if they have a spare room or they have a second house or whatever. So if you are somebody where you're trying to be innovative and you're trying to think and you're trying to look at things differently. I had a phone call yesterday morning, a Skype call with somebody who said there's really no point in doing things over and over again that don't work but how can I come up with new ideas that may or may not and this is where this book comes in is because it just makes you think and stand in the shoes of other people who thought like that and led to completely unintended consequences. So I'm going to give you three examples right of the 50. So I got in a flight today, left Dublin airport, flew over here to Brussels and then got to my accommodation where I am now and now I'm sitting here talking to you. Okay, something that I do regularly, something that you've done I'm sure many times. So what is it about my journey this evening? A rose because of somebody challenging assumptions. Well, one of them, right, one of them and the one that I'm going to tell you about is of the 50, it is number 22. Number 22 here, sorry, now that's very hard to see. Number 22 is the elevator and if you think about the elevator, think about what the invention of the elevator may happen. You might say, okay, well it made sense that if somebody is going up steps, now they can go up faster and also if somebody can't climb steps because they may have a physical issue, well then it enables them to rise. Yes, and that would be what I would have thought too. Okay, but the thing is what the elevator enabled was a safe way to move people literally vertically up into skyscrapers, which therefore enabled far more people to live in cities. And it also meant that people could build buildings that were going to be able for a lot more people to work in, to live in and to be safe in. So if I think about it this evening, as I was in Dublin Airport, because they have various different levels enabled by the elevator, it enables something like Dublin Airport to be a lot more concise in the geographical area. It's able to be in a much smaller area as a result simply of the elevator. So when you think of skyscrapers in, let's say New York is usually the place that we might think of or San Fran or Hong Kong or Sydney or wherever, think about how much wider that urban sprawl would be without the elevator. So that's the first one. The second one is where I am right here at the moment. The place that I decided to stay in, I had researched. I had researched so I knew where it was. I knew what it was going to be like to be here. I knew what it was going to be close to and so on like that. And the only reason that I did, you could say is because it has a website and because the internet facilitated this. True. However, to take this point further, what I would also say is actually that's because of seller feedback. It is going back to the Airbnb idea. It's because people leave their own comments on things like TripAdvisor.com. And that seller feedback has created so, so, so, so, so much. It has led to a lot of unintended consequences.  It's enabling me to have that information through the eyes of somebody that's been here performing. But also what that has also led to is completely different marketing. Is that now, rather than let's say buying expensive marketing space or advertising space, now what influence or marketing is totally enabled as a result of seller feedback. And the other thing as well is that seller feedback has led to buyer feedback also. If you think about it, if I was to stay here and I, let's say that this was Airbnb instead, well then what the provider, the accommodation provider could also rate me. So I got a My Taxi today, I used the app My Taxi to get today because it was a fantastic event this afternoon and I had to get to the airport really quickly. Now, I used My Taxi to pay in the taxi and to get there and that's all great. But if there was negative feedback left by the taxi driver about me as the buyer, well then I may not be able to use that app anymore. So what seller feedback has actually led to is a lot more unintended consequences around accountability of the sale. And the third one that I'm going to pick up on this evening, right, is that I, as I was landing down into into Brussels, well, I wasn't landing anything. It was the pilot obviously was doing that. But you might think, okay, so you land in Brussels and I got off on my merry way and scrabbled in here into the city centre. But if you think about it, what enabled me to land? Yes, an airplane. Yes, a pilot. Yes, an airport. Yes, airspace. Yes, EU regulation. See what this book does to you. This is what it does. It makes you think and think and think and think of all the other elements that happen that lead to unintended consequences. But the key thing, the key thing that enabled me to land in Brussels today was radar, was actually the invention of radar, was to enable yes, that airspace and that airport and that pilot all to communicate together and to coordinate all of the planes that land in Brussels airport simultaneously so that they don't crash into each other. So in fact, actually, it is the radar that underpins the entire air travel. And that is what I mean is that you just get transported. You get transported away into a place where you start thinking through, hmm, this leads to this leads to the other leads to that leads to the other again, and so on and so on from there. And I'm telling you, you cannot think about your own business as you read this book, because what it does is, as I say, it just encapsulates you. Now, I finished the book. I told you where you started it. So that particular occasion, my brother-in-law's birthday, that particular occasion happened at the start of October. I finished it. I finished this book on a nine and a half hour train journey from Nha Trang to Da Nang in Vietnam when I was on holidays. And the reason that I mentioned that is because it's a super holiday read. It's a super holiday read for business people, or for people who think a lot about their business or their career or whatever else. It's a super, super, super read. And I tell you, what will happen is that you put down that book several times, you put down the book and you'll just go off and you start thinking, and then it really will spur you on to challenge assumptions and to think, what if this happened or what if this didn't happen? Or what if I could turn that on its head? That is what this book does to you. So there's three particular things that I wanted to mention about how I have used this as that type of read in my business. The first of all is I've learned to appreciate, really, really appreciate consequences. And this book has taught me how to think five steps ahead. So if X happens, how does that lead to Y, which affects Z, which influences A, which directly impacts number two, three and four in an equation that leads to an output of O. That's what it does. And it does take you through that. There's a very global perspective in this. Lots and lots of stories, you know, they start off with such and such a person was in Kenya and then. Or there was one story actually about radar in particular that speaks about it starts off with the, I was only talking about it the other day to somebody, the volcano in Iceland that erupted, for example. So it brings you on a very global journey and it brings you back in history. It brings you back to the world war. It brings you back to conversations that were had pre-accounting. That's one of the 50 things actually is the double entry system and accounting and what that has led to and accountability and auditing and financial statements and all that sort of thing. And it also very interesting one about market research is how market research started. And the thing is you often think, but you didn't everybody research the market before they got into business. Well, no, why, why would you think that? Why would you think that something was always there because it wasn't somebody had to start it and it probably didn't start with the outcome like he talks with the iPhone, for example. The iPhone didn't just create out of the iPhone. Apple didn't just come up with the iPhone. It came from an iteration of several other things before that. So that is what this book has done for me is that it's given me that type of a focus on looking at things and thinking things through five steps ahead. I watched a documentary. I always watch documentaries when I travel and I watched a documentary on Angela Merkel one night. Very interesting actually about her whole history. And again, she is a woman that has honed that skill of thinking about consequences because she's a chemist. She's a PhD in chemistry. So she's always thinking about chemical reactions of A and B or to be combined together. What would she look like and how would that affect things? So that's how she does it. But this is my way. I'm not going to be doing PhD in chemistry. I did it with a leaving search. Great subject, but that'll be where it stops. Anyway, so that is one of the key things. The key business outcomes are practical outputs that I got from this book. The second thing is, and I've mentioned this now a couple of times. It has taught me how a kind of a behavioral mindset around challenging assumptions. And I do all the time now I think about, okay, so what if I didn't have internet at all in a certain place? What would that lead me to do? Or what if we, right, here's one that I've really thought about. What about if we were to eliminate air travel, right? So forget about the radar. And what about if we were to think about, if we were to eliminate air travel and simply think about using under the ground to travel? Like Hyperloop, as many of you might have heard about what Elon Musk is trying to do now about, say, joining cities right around the world using hyperfast, like hyper, hyper, hyperfast trains under the ground. Removing gravity so that you can travel in a vacuum the same way that particles arrive at a TV screen. And if we were to do that, what would that do to the world? So that would clear completely clear airspace, for example. What would that do to the hospitality industry around airports? What would that do to connectivity? What would that do to hubs? What would that do for all of the people employed in that particular area? What would that do for structural engineers? What would that do for, and so on and so on? So that's what this book has done. And when you do that, you start revealing things about your own business where you start to challenge ideas or challenge your own thinking. That can be very refreshing. It can be really, really, really refreshing. And now, it may be refreshing for you, not for everyone around you, but that's their problem. But no, I really do mean it, though. It can lead you to thinking a constantly of new things. Now, some of my own staff say, oh, here she comes now with another idea. And a lot of it can come from this. Now, some of my ideas work. Some of them don't. But it's a great way of creating ideation in your mind. And therefore, also, you can bring that. And when you can codify how to do that, like Tim does. Tim, my friend now who's on first name terms with me, it's a really helpful skill. And the third thing is your brain needs a break. It does need a break a lot of the time.  I often find that I love flying, right? I love the literal blue sky thinking that's around me. When I'm up at the clouds, some people might say I spend a lot of time there. Anyway, sorry, I'm only laughing to that myself. But if I spend a lot of the time, when you're literally in the sky and the clouds are surrounding you, you have a very bird's eye vision. And none of this is a myth for us. I find it's a great space to think. But if you think about it, I don't have Wi-Fi. The laptop can only do whatever is on it. I can't talk to anybody, person, person to the left or to the right of me. But I can't just be distracted by my phone. And that's what this book does is it transports me into that type of a mindset. So on that note, I'm going to thank you very much for joining me. I can see the people who've been here. So I'm going to wave. I can see the people from Ireland and Australia have been on this live stream. So I'm very appreciative of that. This, of course, is going out in our monthly newsletter, which will be arriving. For those of you who signed up for it on the positiveeconomist.com, it will be arriving into your inbox very soon. And between now and my next book review, which I will be happily telling you about next month, I wish you an absolutely fantastic February, fantastic Valentine's and fantastic mind space thinking. Thank you and good night. Thank you.


Okay, here's what I want you to know. I am trying to zoom out. I want to show you the visual of an epiphyseal plate so that you can imagine the process of interstitial bone growth that increases the length of a long bone. And it happens by the time you come out of your mama's belly, all you have left are epiphyseal plates, where interstitial bone growth can happen. Otherwise, all bone growth is oppositional.   There are layers of, like there are places where we can identify the things that are happening in the process of interstitial bone growth at the epiphyseal plate, okay? Do you remember what did we have in order for bone to grow? We had to start out with a model. We had to start out with preexisting connective tissue. In the case, in this case, it's hyaline cartilage. So look, zoom in. Do you agree? Hyaline cartilage, look at the lacuna, the little space. Look at the cell inside. That's a little cartilage cell. That totally works. That's a little chondrocyte. You look at this tissue, you're like, it's pretty clear, it's nice, it's probably hyaline cartilage and you are correct. Now, we have zones. This area is the zone of resting cartilage. Here is my epiphyseal plate. Do you see how, like, what is going on here? There's something funky. Well, first of all, look. Look at how the cells, here the cells, the lacuni are big with a single cell inside and all of a sudden you end up with, like, these kind of stretched out long lacuni with lots of cells going on there. This is the zone of proliferation. This is where the cells are like, dude, something bad is going to happen, we better start multiplying. Then they start to hypertrophy. So here's proliferation, we're getting big. Hypertrophy, the lacuni get really big and the cells inside are really small. At the bottom of hypertrophy, you actually have the cells producing calcified cartilage matrix. And so they're pooping out cartilage at this point. That's the zone of calcification. Can't really tell the difference between hypertrophy and calcification but then we actually have the zone of ossification which is where the bone, the new bone is actually growing. All right, we're calling it good. We will see you in class. Bye-bye.


Okay, we're back here live. This is theCUBE. This is SiliconANGLE's program. We go out to the events. Extract the signal from the noise. I'm John Furrier, the founder of SiliconANGLE. I'm here with Courtney Nash, editor at O'Reilly Media. O'Reilly Media is putting on this conference, the Fluent Conference, which is really targeting the developer community, the JavaScript world, jQuery, .NET, Ruby. All the different aspects of how JavaScript and the developer community on the front end, user experience is expanding their horizons, increasing their capabilities, new protocols, new standards. It's really where all the action is and there's a lot of exciting things here happening at Fluent. And again, this is where all the actions for developers we're covering two days. And my next guest here, we're going to talk a little about how that intersects with kind of what powers the developers. And that is ultimately the cloud infrastructure, tools, technologies. What's under the covers, what's under the hood is really what's driving. Courtney Nash is editor of O'Reilly Media, covering the velocity market segment. Not necessarily the conference, they're putting the conference together, but you have to have the 20 mile stair, the wide, deep, kind of the beach head of the marketplace. Welcome to the queue. Thanks, it's nice to be here. So one of the things that the queue we like to do is just kind of lean back and let some ideas flow, but also talk specifics around what's happening in the marketplace. So folks watching can learn a little bit about kind of how things are connecting. The Fluent conference is a developer conference and we had Brady Forrest on yesterday, he's an amazing guy doing some great stuff. And he made a quote, we haven't had an Amazon web services moment yet in his world, which is essentially the Internet of Things, the Maker Faire, but the Amazon impact of the developer community really with the public cloud has been really amazing. And the guys here at the conference and Gals here are really exploiting that capability and doing new things. So that whole generation shift to data centers, to cloud, to new software techniques, virtualization, all this stuff is the meat and the potatoes, the technology under the hood that's powering here the development of Fluent. So you got to look at that. So what's your take on the intersection of Fluent? Connect Fluent and the Velocity Conference, which is coming up on, by the way, to plug with you guys on, and Santa Clara June 18th, 20th. That's right. So the Velocity Conference is about speed and power on the web. And this came out of a need, Steve Sowers is one of the co-chairs for the Velocity Conference, took a look at the landscape of the web at some point and said, how do we make this thing faster? And the influx of mobile devices and all kinds of devices, people's expectations for how quickly things should happen online or on your device or on your tablet are continually going up. And so the speed of what people are delivering online needs to match those expectations. And there's real revenue involved in this. And I think that's where the interesting piece of Fluent and Velocity coming together, you can talk about tools, you can talk about libraries.  But Bing and Google actually got together and did a study where they intentionally slowed the web down. I don't know if anybody noticed this when it happened. And they found that a two second delay or a two second slowdown in search results had something like a 4.3% revenue per user impact. And the Obama campaign is a great example of this. They geared for speed. They designed out of the gate for speed. And they had a 60% faster site and that was something like 14% increase in donations for the campaign. So speed is money. And that's really where I think that Fluent and the developer world connects with some serious fiscal reality. And what I like about the program and your job is that it's not just cloud. It's not just mobile. It's not just social. It's a system. It's a holistic the future world we live in. It's operating environment. You called users and software developers. So talk about the speed thing because what does that mean? I'll say one trend that we've all been covering is DevOps. DevOps has evolved very fast. Now that's actually being kicked around in mainstream marketplace and the enterprises and kind of no one really knows what it means but you have a developer and operations which is classic. You have the cloud guys. You have private cloud. How does a person out there understand the velocity marketplace that you were mentioning? Is it one thing? Is it holistic? What's your take on that? I think of the velocity marketplace as one thing but I think the reality is it's still a fairly fragmented marketplace. If you go to the velocity conference you talk to people who are talking about DevOps which is really about breaking down silos that people work in today. That all makes sense to me. That's all one world there and people are all trying to deliver an experience that works well for people but the reality is there's still a lot of silos out there and I've always thought of the DevOps side as almost being more about the stronger aspect of velocity so we talk about faster and stronger with velocity too. So it's not just speed but you have to have a base on which you can go fast. You can't put your hoopty on the freeway and go 90 miles an hour. The thing's gonna fall apart.  You're gonna have problems. So the strength is the foundation of the speed and the DevOps piece is really about people. It's a cultural phenomenon. It's people looking at saying why isn't our organization working well? Why is it taking us six weeks or two months or years to roll out features and looking at the people that are actually the part of that technology organization and when you can smooth that stuff out you can make your operational environment faster. You can make your website faster, your mobile web faster. So that's what I see the DevOps piece being but it's kind of a buzzword now I'd say almost. People don't all know really what it means. And I saw your eyes rolled when I said this. I don't mean to roll, I love. No, no, it's just one aspect of a bigger picture. It's one pillar. I'm a DevOps enthusiast and people will say no doubt there but it is a piece of that velocity culture and that velocity world that people are just starting to get their arms in. You know what's interesting, what I love about the O'Reilly conference is that there's a lot of intersections between multiple what I call pure play market segments. So strata straddles a bunch of different markets. Velocity, you're talking DevOps, which is strength. You have this virtualization, you have cloud, you have mobile, all these things strength and agility and speed kind of make a big deal. So I got to ask you about a couple of key trends that are emerging, I'm not necessarily sure they're pure play segments but obviously I wrote a post yesterday on Forbes that we post to and talk about NetApp and Silicon Valley and they're really transforming around software-defined storage which is a buzzword that's popped out because when Nasir got bought by VMware for a billion dollars they only had 50 million dollars of venture from Andreessen Horowitz. Everyone took notice, this software-defined networking trend has taken hold. So now it's software-defined networking, software-defined storage, software-defined data center. Everything is code. Everything is code. Infrastructure is code is a term that we've been kicking around on orchestration. There's a lot of serious stuff going on around this area but software's the key to it. And so how do you look at that? I mean, because you have to kind of vet, I mean is that infrastructure, it's a hardware box from Cisco, Juniper, whoever but where's the software? Do you look at the software piece and how would you share your perspective on the software-defined movement or? I don't know if I look at it in that way that you're thinking about and what I actually think is more interesting about the velocity world is that velocity is starting to look at and has organizations that are looking at the boundaries of humans and software. So when you said everything is software, I nodded and then I thought, well actually I don't agree with that entirely because the singularity is not approaching as John Allswell likes to say. Everything will not all be software and there are these intersections of people and software and you can't ignore those at your peril I guess. You can't automate everything, software is not going to solve all of your problems. There's process issues, right? I mean, we were on the cube at Strada, Dave Vellante, my co-host is not here, said people focus on the container not inside the process. So again, there's people dynamics. It's not just human, it's also organizational. Yeah, absolutely. Are you finding that? That's a very strong theme that comes out of velocity. I think we've heard some of it here at Fluent, we've heard some of the key notes people have been talking about those things, you can't, you don't get work done without other people and what I really like about the conference in particular as a view into this world and this market is, unlike other conferences, even other O'Reilly conferences, velocity has a history of sharing our mistakes, you know, of sort of showing the warts off of using failure as a learning tool and so I like to joke that failure actually is an option, even though that's sort of our tagline for velocity is that failure is not an option. Yeah, yeah. So people are building failure into their software now, right? So Netflix has the Simeon army or like Chaos Monkey, they intentionally take their systems down in production and as a developer, you have to try to develop towards that, right? Yeah, absolutely. When I look at software, I can't not look at the people side of it. So web velocity is a conference, it's going to be in June, their web performance and operations conference, that's what the tagline is and are the focuses and the tagline is building a fast, stronger web. So we checked the box, we promote the velocity conference. Let's step back and talk about kind of the marketplace. You're out there as an editor, you're talking a lot of stakeholders, partners, customers, end users, developers, programmers, et cetera. What are the biggest challenges and opportunities that are facing folks as they look at the landscape of what velocity looks into? What do you see as the biggest challenges and yet the opportunities? Well, I think what's interesting is that the web is only marginally getting faster, right? So I think everyone would agree it should get faster, we need to respond to what customers and users want, but the reality is the people who've been driving the increases in speed on the web in a lot of ways have been the browser and the ISP folks, right? So browsers are doing a lot of, browser manufacturers are doing a lot to make browsers faster and you see all the commercials now about faster is better, right? On the ISP side too, right? So all of the cellular providers are doing a lot. But at the developer level, things aren't really getting faster. And I found that was really fascinating. Steve Souters just put out a big summit about this or a big summary about this. And there's a lot still that developers can be doing to make their sites faster. They have a lot of control that they're not taking advantage of yet. And so, and I think video is going to be a big piece of that that we're seeing, right? So you're looking at page weight over, year over year, like our website's getting bigger. Are they getting smaller? They're getting bigger. We're at developers to add if so much you can do a JavaScript round, right? So much flexibility, so much customization. But our page weights are getting bigger. And it's not monolithic anymore, it's decentralized, right? So you got social media, social network. You got the party stuff that's coming in, right? Yeah. I mean, I think the social media thing's interesting with Hadoop we saw at Strata is that, yeah, you collect data. How do you put that stuff back in? You're only taxing the infrastructure. And we were just talking to some folks at EMC at EMC where all of them are like, the guy who runs the big drives, the ones that is quote being killed by the open source, their business is up, people are still buying more storage, right? So the roads are getting clogged, more and more clogs, you know? So that's a big issue. So what do people do about it? I mean, the enterprises in the area we also cover. And they're kind of still in the old days. I mean, they're modernizing and it's clearly a mandate. You guys are just seeing that, we're seeing that modernizing this modern era of infrastructure. I mean, where do you tweak it? You got like an airplane at 3,000, 3,000 to 5,000 feet. How do you change the engine out? It is, it is. And I think enterprise is an area for velocity that has a lot of challenges, but a lot of opportunity. Bill Scott's keynote this morning at Fluent, I thought was really spoke to this phenomenon really well. And I'm going to come back, I'm going to pound this drum again, but you can't change the technology in an organization without changing the culture, without changing the people. But I think if a larger enterprise can figure out what those business metrics are, what their users need, and they can start tying that back to what they need out of the technology, eventually we can start switching up. And there is hope for change in enterprise with this kind of stuff. I really believe there is. We're here at Courtney Nash's editor at O'Reilly Media talking about the velocity landscape with the conference coming up. But it's the landscape of the world. And it's the flat world, as we say, and certainly the tech business is going to be more software driven, whether it's a thermostat from Nest, or something cool that Brady Forrest invest in. We had a guest on earlier today talking about here at Fluent, about JavaScript, blah, blah, blah, blah, blah, the stuff we're talking about. I want to tie it to a question for you in Velocity's world. And the comment was two comments. One was the theme of Fluent is the web, the web, the web. Most people think, oh, web and mobile, different methodologies maybe. And Agile, you have different app stores, different distribution, different business models. At the end of the day, it's still the web. In fact, Jeff Frick and I were discussing and debating Agile methodology in mobile versus web. People on Twitter are talking about responsive design and so on and so forth. So that was kind of a cool check. I think we all agree it's the web. Then the other comment he made was the biggest concern of uncharted territory is performance on mobile. So I got to ask you, with Velocity, how do you see that agenda coming into that landscape? Mobile is becoming a much bigger deal for Velocity than it was a year or two. There is an actual mobile track at the conference now. And just from a larger marketplace landscape, people are really starting to pay attention to us. There's fundamental things. Like you said, mobile is the web, web is mobile. There are fundamental things that you should be doing across both of those, but there are obviously significant challenges in mobile. It's pretty inefficient right now. I'm wondering, it's exploding in growth. So you have growing pains, right? Right, yeah.  Latency, ultimately, Elia Grigoric gave a really great talk, a workshop earlier at Fluent here. You can't control the speed of light. You run up against certain barriers. Physics. Physics are sort of difficult to define that front. But there are things that people can be doing to optimize for mobile. I mean, you mentioned responsive design. And that's definitely going to be a big focus for the conference and also just for us in terms of publishing. And then on the big data side, there's some big data angle where you can look at it and saying with Flash, Fusion IO or violin, these companies are building kind of great caching layers for open source software. I think that's cool. Virtualization is kind of changing, morphing. It provides us being commoditized, seeing a lot more different software things. So a lot of cool things happening. But I want to ask you kind of as the parting question is just your thoughts on what's uncharted in the velocity area. What are you seeing as an editor? You have to kind of keep an eye on things. The normal areas. I gotta watch that community. I gotta watch that community. So outside of the known areas on the radar, what are you watching personally that you've got your eye on that's not yet baked out or revealed itself that you think is going to be a trigger? Well, I mentioned the topic of failure. And I think that's tied to a certain degree to this idea of risk. And a lot of interesting people in academia are starting to look at risk and risk models and how do you model risk? And I think there's some really interesting stuff coming down the pike if you take more academic and theoretical models of risk and start applying them to your systems, to your front end systems, to your people systems, to your operational systems. And to a certain degree embracing risk and having a different model of resilience of what your site should do. So instead of having these bulletproof, bombproof kind of sites, how do you have sites that can flex with sort of chaos and things that you simply cannot plan for? There's a lot of interesting things happening there. And I would say that that's a couple of years out for people to start figuring out what's going on there. The adaptive learning machine, learning software. Yeah, well, it's sort of, you know, yeah, websites and entire systems that are not so fragile and that are more resilient. There's, that's a model of thinking that I think is only starting, it's on the fringe still for people who are just right now trying to figure out how do they optimize the JavaScript? Very, very relevant focus. Love it, love the, just in summary, fast and speed is the velocity, being strong. But what you just talked about was systems, people, organizations and technology and doing new things, holistically. Check out the velocity conference with the Riley Media, a lot of great stuff. Again, it's the intersections, the confluence of many things coming together. This is what our world is like now. I mean, it's not one, it's not a one-trick pony for off-the-shelf software, general purpose computing. All this is kind of converging in. It's exciting time and we'll keep on covering it here to Riley Media and theCUBE. We'll be right back. The next guest, Courtney, thank you for coming on theCUBE. Stay tuned, keep watching. We'll hear all for the rest of the day live on theCUBE, we'll be right back.


I was I was like literally sitting down getting ready to watch the game. Oh, yes, Saturday Night Football Coast Patriots should be a good one and tomorrow It's like the hits just keep on coming They literally just keep on coming and it's non-stop because now Sammy Watkins is officially out of the game tomorrow and He has been being put on the the COVID-19 list. So another one Another one. It's like again all around it and that failed. There's this this thing just been cray it's been non-stop literally non-stop and He's just another one on the long list So, I mean, I don't even really know what to say about It was a Rashad Bateman should have already been starting But it should have been because of play not because of injury or COVID So this is unfortunate for for Sammy Watkins Just hope everything gets good with him and I man this thing just wild man Don't even know how to explain it don't even know how to even talk about anymore. It's just It is what it is and it just keeps happening over and over and over and over and over again Now again, even though Ravens they seem to be getting hit with something right now It's not nearly as bad as it is for other teams, but With what the Ravens have already been through this year with injuries They can't afford to lose nobody else, but they keep losing other people Every week is something new whether it be the injury and now COVID Keep losing people every single week. I Would love to remember a week when there was when Ravens didn't lose anybody when they they lost like nobody If anybody can remind me or show me when that would that happen. I would love to see it, but I Don't think there was one So anyway Here's what it is Sammy Watkins officially out and This will give the Ravens. I mean if you want to look for a a sort of Silver lining with this whole thing. Obviously, it's not good at all for him to be placed on the COVID-19 list. That's all bad But now It will force the Ravens hand with the incorporation of other wide receivers now with Greg Roman it It is short notice and we know that there's been a problem when it's come to adjustments in game and adjustments on the fly There's been a problem sometimes with that, but He does have A night to prepare not even a full 24 hours, but all right, Sammy Watkins is on the list Y'all are professionals You're a professional so this is your opportunity to show just how professional you can be and Find other ways to involve different guys and now you have no choice. So Hollywood and Rashad Bateman, those are your starters or your starters because Ravens do a lot of rotation But those are your starters as they should be But now with Tylen Wallace, I don't know if he recovered from his concussion yet or not. I'm not sure but Duvenay Pro Shae, I will I don't know what Miles Boykin status is But yeah other guys they they have no choice but to involve other guys now So we'll see exactly how they do it if they do it and Tomorrow's gonna be a lot of fun. It's gonna be a lot of fun So team keep it clean appreciate y'all love y'all and just like Sammy Watkins Uh, just like Chuck Clark just like Tristan Cologne Castillo just like Chris Westry Um Just like I mean half the roster is I'm out


Let us continue with whatever we were doing. So you know let me recall few things. So you see we started by asking what the image of an analytic mapping is okay and of course the if you we have already seen that you know because of the open mapping theorem the image of a non-constant holomorphic map is an open set and in fact it is an open mapping so it takes open sets to open sets so the image of a domain is again a domain okay and then I told you that we wanted to know what we wanted to know more about the image set okay namely the set of values that the analytic function takes and you know somehow the open mapping theorem tells you that for example you cannot expect the image of a non-constant holomorphic map to be to lie inside a curve for example okay because it is it does not have the property that it can accommodate as sufficiently a small disk okay. So then of course we where we stated the so called little Picard theorem or the small Picard theorem which says which deals with the case of an analytic function which is analytic on the whole complex plane so called entire function and the theorem says that the image will be either the whole plane or it will be the plane minus a single point okay and interestingly the proof of this theorem which is usually stated only stated in a first course in complex analysis involves interesting amount of analysis and that is what we will try to do as part of our this series of lectures and I told you the key to this proving this little Picard theorem is a so called big Picard theorem or great Picard theorem which has got to do with the image of a deleted neighborhood of an isolated essential singularity under an analytic mapping okay and the great Picard theorem what does it say it says the conclusion the great Picard theorem amazingly is the same as that of the little Picard theorem which says that you take a you take a deleted neighborhood of an isolated essential singularity of an analytic function then the image of that under the analytic function will be again the whole plane or it may be the plane minus at the worst one point or one value that is missed okay. So it is either the plane or it is a punctured plane and I told you that the proof of the little Picard theorem that we will give is going to be as a corollary of the big Picard theorem and to study the proof of the big Picard theorem or the great Picard theorem we need to study what are about Meromorphic functions okay and these are functions essentially these are functions which are these are analytic functions which have the only singularities as poles okay and so this led us to understand the and recall rather recall the notion of singularity so I told you that singularities of analytic functions are of two types namely the isolated and the non-isolated ones okay and of course the classic example of non-isolated singularities are the points on the negative real axis which have to be cut out if you want for example define an analytic branch of the logarithm of log z logarithm of z okay z being a complex variable. And I told you that these non-isolated singularities require much deeper techniques for example the study of Riemann surfaces to deal with them but then we are going to be worried only about isolated singularities and I told you the isolated singularities come in 3 categories of groups and these are mutually exclusive the first kind of isolated singularity is called removable because the idea is that the singularity can be removed in the sense that you can redefine the function so that it becomes analytic at that point and so it is not really a singularity then there are the poles which are thought of as 0s of the denominator okay and of course if the function does not have a denominator which is not writable as a numerator by a denominator then poles are just 0s of the reciprocal of the function okay and I told you that a function has a pole of a certain order if and only if the reciprocal function has a 0 of the same order at that given point and then what we are left with is are the singularities which are neither removable nor poles and they are cleverly called as essential singularities and they are very very important as the for example the great Picard theorem tells you. Now what I am going to do is first prove a certain weaker form of the great Picard theorem so called Casorati-Weierstrass theorem which says that you know if you take an isolated essential singularity then you take a small neighborhood of that deleted neighborhood of that no matter how small the image of that will be dense in the whole complex plane namely it will you can always find a sequence of complex numbers in any neighborhood such that the values approach any given complex value okay and that is a very deep that is already a very deep result but it can be you know reduced from the Riemann theorem on removable singularities so what I am going to do is now I am going to tell you something about the Riemann theorem on removable singularities because it involves a lot of nice concepts okay so here is Riemann's theorem theorem on removable singularities and it is a very very deep theorem because you know it already uses some of the most basic theorems in complex analysis in its proof okay so we will see that. So let me state that let Z not be an isolated singularity of the analytic function f of z then the following conditions are equivalent so number 1 Z not is a removable singularity that is a first condition and so here you can give several definitions of removable singularity but we will give the most natural one the most natural definition for removable singularity is that it can be removed namely that you can the analytic function can extend to an analytic function even at the point of singularity okay what it means is that you can find an analytic function which is also defined at the singular point and which equals the given analytic function outside that point okay that is what it that is what it means to say that the analytic function extends to an analytic function also at the singular point so that means you have by extending it you have actually been able to remove the singularity alright so of course you know the standard example you should think of is sine z by z at z equal to 0 you can you know the limited at z equal to 0 is 1 so you know you can take the function that takes the value sine z by z at z not equal to 0 and at z equal to 0 you can define it to be 1 and then this turns out to be an analytic extension okay so let me write that down that is there exists an analytic function function so g of z analytic at z not such that g of z is the same as f of z for z not equal to z not in a small neighbourhood of z not. So this is what it means to say that the singularity is removable I am able to extend the analytic function to the point to the singular point okay and to be able to extend the analytic function to the singular point means that I am able to find another I am able to find an analytic function which is also analytic at the singular point and restricts to the given function outside that point okay so let me rewrite it is very important in mathematics to be able to say things in different ways verbally without using symbols or notations as far as possible because that will help you to get a good understanding of the ideas okay so you should be able to state theorems at least as accurately as possible without using much notation and just using concepts okay this is something that you should strive to do you may be able to write technical mathematics namely you can write a theorem with all the technical symbols and so on but then it is very very important for the purpose of communication and understanding that you should also be able to say things in a way that does not involve any notation okay and so in that sense to when you say something is a removable singularity a point is a removable singularity what it means is that the analytic function extends to analytic function at the singular point okay so let me write that down in other words f extends to the to an analytic function g at z dot okay so this is the I mean so this is actually the definition of water removable singularities okay so this is the first condition okay so you can see that what this theorem does is that it gives you various equivalent conditions for a singularity to be removable so what is the second condition so you see in all these in all theorems connected with characterization of singularities there are usually at least three statements one is about the one is essentially the definition of that singularity the second one is the behaviour of the limit of the function as you approach the singularity okay and the third one is the behaviour of the Laurent series around that singularity okay so for example if you take the case of poles which I stated in the last lecture you see I stated that theorem in the last lecture and I wanted you to try to prove it and I do not know if you have done this exercise or not but essentially you see if you try to do that exercise at some point you might have to use Riemann's removable singularity theorem which I am going to actually prove now okay so but nevertheless the purpose of the exercise was to make you realize that you need to you might need to use this okay so you know in the case of a pole the condition for the first condition for a pole was that it is a pole namely which is the basic definition of a pole which is just that it is a 0 of the reciprocal okay the point is a 0 of the function which is the reciprocal of the given function okay that is the first condition the second condition is the behaviour of the limit of the function as you approach the pole and that condition turned out to be that the limit of the function turned out to be infinity okay and that is to be interpreted as the limit of the modulus of the function goes to plus infinity okay as you approach the singularity alright that is another characterization of a pole and what is the third condition the third condition involves the Laurent expansion and what is the condition based on the Laurent expansion the condition based on the Laurent expansion was that the Laurent expansion contained only finitely many negative powers of Z minus Z naught okay so you know a Taylor expansion is something that contains only positive powers and zero powers and this of course zero power corresponds to the constant term okay and the Laurent expansion is something that will also contain negative powers of Z minus Z naught okay where Z naught is the singular point and if you have a Laurent expansion which has only finitely many negative terms I mean terms with negative powers of Z minus Z naught that is an indication that Z naught is a pole okay now that these three conditions are equivalent was a theorem I stated last time so in the same way for removable singularities the first condition I have stated is what removable singularity is essentially and I will give you two more the two other conditions so the second condition is going to be a condition that is got to do with the limit of the function so the second condition says that the limit of the function exists as you approach the singularity okay just the existence of the limit is the second condition okay and why it is significant is because if the limit exists what you are saying is that the function extends to a continuous function at that point okay so mind you it is certainly weaker than the first condition because the first condition namely the definition of removable singularity is that it extends to an analytic function at that point whereas the second condition is the condition on the limit that the limit exists at that point only tells you that it extends to only a continuous function at that point okay so it is now so what you are saying is that the fact that you can continuously extend the function to the point already makes it analytic at that point this is a characteristic of removable singularity okay so let me write the second condition so here is the second condition the limit of f of z as z tends to z0 exists this is the condition okay so in other words in other words f of z extends to a continuous function at z0 namely we simply define f of z0 to be the limit at as z tends to z0 of f of z okay you define you redefine f at z0 and the resulting function becomes also continuous at z0 but what you do not have immediately is that it is also analytic at z0 okay and that is the serious that is the serious consequence that is the that is the Riemann removable singularity theorem okay just continuity at that point is good enough for analyticity okay that is the crux of the Riemann removable singularity theorem so this is the second condition and here is the third condition so I told you the third condition is something that has got to do with the Laurent expansion and you know that so I want to give you some background on this you know if you have a function which is analytic at a point then you have a so called Taylor expansion at that point okay if the point is z0 then you have power series in z minus z0 which involves 0 and positive powers of z minus z0 with some coefficients and of course these coefficients are you know they are just the related to the nth derivatives the derivatives of the function at z0 okay and so this is the Taylor's theorem that you can have a Taylor expansion for the function at a point z0 where it is analytic and an extension of the Taylor's theorem is the Laurent theorem it is an amazing extension because it deals with the case when z0 is not a point of analyticity but it is a point it is an isolated singularity what Laurent's theorem says is that you can still get a series but now this time you will also have to allow negative powers of z minus z0 that is the Laurent series okay and Laurent's theorem says that you can get that series and the coefficients are again given by integrals okay the see in the case of Taylor series the coefficients are given by integrals and these integrals are essentially connected to the derivatives by the general Cauchy integral formulas okay and in the Laurent expansion the coefficients of the negative terms the coefficients are all anyway given by the integrals okay there is no question of derivatives at the point z0 because at z0 is not a point where the function is analytic so the coefficients of Laurent series are given in terms of integrals okay. Now so you see the Taylor series is a very special case it is a special case of Laurent series okay and the fact that so this is the point if you have an analytic function at a point okay if you try to write the Taylor series at the point you will get the same result as if you try to write a Laurent series at that point okay see the fact is that if you have an analytic function we know that it is given by a Taylor expansion at that point but if you throw that point away okay then you get a deleted neighborhood of the point and in a deleted neighborhood of a point you always have a Laurent expansion for any function regardless of whether the function is analytic at that point or not okay and the point is that if you write out the Laurent expansion for an analytic function at a point okay you will get only the Taylor expansion you will not get the negative terms in the Laurent expansion the negative terms in the Laurent expansion constitute what is called the singular part or the principal part of the expansion okay and the principal part will not exist that is a sign of the fact that the function that you are actually expanding into a Laurent series is actually analytic at that point okay so that is the third condition there is a third condition in terms of Laurent series for a removable singularity is that when you write the Laurent series for the function at the removable singularity centred at the removable singularity you will see that the negative terms the principal part does not exist it is 0 okay so that is the third condition which is based on the Laurent expansion so here is the third condition the Laurent expansion of f of z at z not has no negative powers of z minus z that is it has 0 principal part principal or singular part okay this is the condition in terms of the Laurent expansion and then so these are the three conditions that you will always have as equivalent in any theorem on singularities characterization of singularities now there is one more condition is pretty interesting and it is a rather remarkable condition the fourth condition is the following if you have a removable singularity okay at a point you see the let us believe that is a removable singularity just to understand this condition then it becomes analytic at that point okay and if it is analytic at that point it is also continuous at that point okay and you know a continuous function on a bounded set okay if you take a continuous function on a close and bounded set it is going to be bounded in modulus okay because if you take a close and bounded set it is compact okay and if you take a real valued continuous function on a compact set the image is going to be also a compact set topologically the continuous image of a compact set is a compact set and a compact subset of the real line is going to be bounded okay so the point is that if you really believe that z naught is an essential singular is a removable singularity for the function it should the function should extend to an analytic function at that point in any case it is going to extend to a continuous function at that point for example that is what condition 2 says then in a neighbourhood of that point the function should be bounded in modulus of course see whenever we say bounded for a complex valued function we mean bounded in modulus okay that goes without saying okay so that is the next condition so it is an amazing condition you just assume that there is a small deleted neighbourhood of the point z naught where your function in modulus is bounded by a positive constant okay that is also as strong as saying that it can analytically extend to that point that is the really amazing hypothesis because it is a very weak hypothesis you see the first hypothesis the first condition is that the function is the it has a removable singularity namely which by our definition is that the function extends to an analytic function at that point okay. The second condition is slightly weaker it says that it does not extend to an analytic function at that point but it extends to a continuous function at that point okay that is a slightly weaker condition and of course the third condition has got to do with the Laurent series which says that essentially the Laurent series is a Taylor series but the condition I am going to state now it is a very weak condition it just says that there is a deleted neighbourhood of the singular point where the function is bounded in modulus you see the bounded function is a very weak condition okay but that is strong enough to make it analytic at the point. So that is the amazing power of the Rayman removable singularity okay so let me write that down f is bounded and of course bounded means in modulus in a deleted neighbourhood so I am using n B D for neighbourhood as an abbreviation of Z0 so this is the fourth condition which is by far the weakest condition on a removable singularity okay so you know why it is weak if you want if I might say a little loosely is that you know continuous function is always bounded but there are bounded functions which could be very highly discontinuous okay so the moral of the story is that boundedness giving rise to continuity is already something that is very hard to expect you should not expect that and in this case boundedness is giving me analyticity so you can imagine analyticity is a terrific condition because you know analyticity at a point means that you know not only that it is differentiable in the neighbourhood of the point including that point but it is also infinitely differentiable there so it is a terrific condition okay that you are able to get this from boundedness is an amazing thing that is what you should appreciate okay. So alright so now what I am going to do is I am going to you know prove that these various conditions are equivalent okay and in the process help you revise some basic complex analysis okay fine so let us look at these conditions so you know let us look at condition 1 and 2 you see it is very clear that 1 implies 2 okay if the analytic function extends to an analytic function at that point then certainly it extends to a continuous function at that point because an analytic function is continuous okay differentiability implies continuity so 1 implies 2 is very trivial alright and now so you can see so I am just trying to see which of these are very easy to deduce which of the equivalences are easy to deduce so 1 implies 2 is pretty easy okay and well I think if you look at 2 and 3 2 and 3 are equivalent okay 2 and 3 are equivalent that is very easy to see because you know you take the so let me so let me first tell you in words suppose you assume 2 suppose the limit of the function at as z tends to z0 exists okay then I can take the limit of limit as z tends to z0 in the Lorentz expansion and as limit z tends to z0 in the Lorentz expansion exists means that cannot be any negative powers of z-z0 in the Lorentz expansion because if you have a negative power of z-z0 in the Lorentz expansion it will be a term of the form An by z-z0 power n and that as z tends to z0 will go to infinity okay and you cannot get a finite limit. So the moral of the story is that 2 implies 3 is obvious and 3 implies 2 is also obvious because you know if the Lorentz expansion does not have any negative terms then I can take limit as z tends to z0 in the Lorentz expansion and essentially what I will get is a constant term okay as I if I take limit z tends to z0 I am just setting z-z0 equal to 0 z equal to z0 and then since it is already a power series in z-z0 if I put z equal to z0 I will get the constant term and that will be the limit okay. Now so 2 and 3 are clearly equivalent alright now there is only one technical point which I want you to notice because this is an advanced this is a course in advanced complex analysis the technical point is that you know how can you take a limit in the Lorentz series okay how can you take limit z to z0 z tends to z0 of a Lorentz series see so basically the we are arguing in the following way we are arguing as if we can take the taking the limit as z to z0 z tends to z0 in the Lorentz series is the same as taking the limit in each term and then summing it up okay that is the way we are arguing okay and why is that correct that is because the fact is because of the fact that you know the you see the Lorentz series as it is you know that is its convergence is uniform okay and whenever the convergence is uniform okay you can take a limit okay and so that is used okay you can take a term wise you take a series or you take a functional series okay and you take limit z tends to z0 of the functional series okay that is the same as taking limit z tends to z0 of each of the terms of the functional series and then taking the limit of the resulting numerical series this is allowed provided the functional series converges uniformly and basically I am just using the fact that if you have a convergent uniformly convergent series of continuous functions then the limit is also continuous okay that is all I am using so that is a little bit of technicality that is used when you want to prove 2 and 3 are equivalent okay and of course you know 1, 2 and 3 all the 3 will imply 4 because you know continuous function is bounded, continuous function on a compact set is bounded. So the difficult part is to go from 4 to 1 okay 4 is the weakest condition the condition 4 is the weakest condition is the condition that says that you just tell me that near z0 near that singularity the function is bounded and low behold it becomes analytic at z0 that is the most tremendous observation okay. So 4 implies 1 is the most difficult part that is the crux of the theorem okay which we will try to prove and essentially 1 will again essentially use Laurent expansions okay so let me write down all this so clearly 1 implies 2 which is equivalent to 3 and they all imply 4 okay so this is what we have seen the non-trivial part is 4 implies 1 okay so you know if you want 1 implies 2 basically uses analytic implies continuous okay and 2 implies 3 is going to use well uniform limit of continuous functions is continuous okay and of course 1, 2 and 3 imply 4 all these basically use the fact that the continuous function on a compact set is bounded okay continuous function on a compact set is bounded okay so the non-trivial thing to prove is 4 implies 1 namely that boundedness in a neighbourhood of the point gives you the analyticity at that point which is an amazing thing which is and interestingly the way you prove it is again using Laurent expansions okay you just use the Laurent expansion. So here is a proof recall that so let me go to a different colour recall that the Laurent expansion of f and z0 is f of z is equal to sigma n equal to minus infinity to infinity An z minus z0 to the power of n where An is 1 by 2 pi i integral over gamma fw dw minus z0 to the power of n plus 1 this is the Laurent expansion okay and where gamma is a simple closed contour which goes once around the point the singular point z0 in the anticlockwise or positive sense okay we can very well take gamma to be a circle centred at z0 sufficiently small radius okay. So now the point is that the point is to you know let me explain the idea of the proof see what are you trying to show you are trying to show that the function is analytic one way to show that the function is analytic is that its Laurent series actually a Taylor series because you know a Taylor series always represents an analytic function okay a convergent power series within its disc of convergence always represents an analytic function okay. So incidentally that is also that is also probably used in one of the earlier equivalences okay that is something that you have to remember if you have a convergent power series okay then it can any power series converges in a disc okay the disc could have possibly infinite radius in which case it is the whole plane as happens in the case of a polynomial or an exponential function and in that disc the convergence of the power series is always normal namely it is uniform on compact subsets okay and it is also absolute okay. So this is something that you should have come across in the first course in complex analysis where essentially you make use of the Weistra's M test okay so and a convergent power series by Abel's theorem is that it can be differentiated term by term and what you get is again a power series is the same radius of convergence and that is the derivative of the original power series and this is one way of showing that the derivative of a power series exist and it is gotten by differentiating it term by term the term by term differentiation is justified because of the uniform convergence okay and if you apply this ad infinitum what you get is that a power series is infinitely differentiable it is actually analytic and it is infinitely differentiable. So a power series whenever you are looking at a power series inside its disc of convergence you are actually looking at an analytic function and what has the analytic function to which it converges got to do with the power series this power series is nothing but the Taylor expansion of that limit that limiting function. So you start with the power series it converges within its disc of convergence to a certain function that function is an analytic function and if you expand it as a Taylor expansion at that point you will get back the power series. So the moral of the story is that whenever you are looking at a convergent power series you are actually looking at the Taylor series of an analytic function okay and what is that analytic function it is exactly the function to which this power series converges okay. So what we are trying to show is that we are trying to show that this point Z0 is basically a point where the function is analytic so what do you expect is that you expect the Laurent series should actually be a Taylor series that means all the coefficients of the negative powers of Z minus Z0 in the Laurent expansion should be 0. So you try to show that all those coefficients are 0 then you are done okay and how do you show those coefficients are 0 the coefficients are given by integrals and integrals can always be estimated by the so called ML inequality okay. So what we will do is we will show that all the negative coefficients in the Laurent expansion they are all 0 and we are done okay that is exactly what I am going to do. So what is An? An is 1 by 2 pi i integral over gamma fw dw by w minus Z0 to the power of n plus 1 where you know the pictures like this you have Z0 and you have gamma a simple closed curve going around Z0 sufficiently close to Z0 going around once okay. Now you take for take gamma to be the circle okay you could the shape of gamma really does not matter because of course it is theorem actually take gamma to be the circle mod Z minus Z0 is equal to epsilon for epsilon positive small enough so that the circle on the circle and inside the circle except at the point Z0 the function is analytic okay and well you know then you can so the equation Z mod Z minus Z0 is equal to epsilon you can use that write it as a parametric equation and do an integration okay. So this is the same as writing it as Z mod Z is equal to Z0 plus epsilon e to the i theta where theta varies from 0 to 2 pi okay and so this integral becomes so this integral so if I calculate the modulus of An mind you I am trying to show that the modulus of An is 0 I am trying to show that the An are 0 for negative N this formula is valid for all values of N okay I am trying to show An is 0 for all negative N that is good enough to say that the Lorentz is the Taylor series and that will tell me that the function is actually analytic at the point okay. So I have to calculate mod An mod An is going to be modulus of this integral okay and then you would have come across this estimation formula in a first course in complex analysis which is used all the time which says that the modulus of the integral is less than or equal to the integral of the modulus so this is less than or equal to integral over gamma of if I take the modulus I am going to get 1 by 2 pi mod Fw mod Dw by mod W minus Z0 to the power of N plus 1 okay and this integral becomes therefore just integral 0 to 2 pi because now I changed the variable of integration from W to theta mind you you should always remember that whenever you write an integral then you have an integrand and you have a variable of integration okay and the integrand is a function of the variable of integration and the variable of integration should always vary on the area on the region of integration okay in this case the region of integration is the curve gamma which you have taken to be a circle so your W is actually varying on the circle okay so W should be written as Z0 plus epsilon e power i theta okay. So what I will get is I will let me put this mod F Z0 plus e to the i theta and then I will have to write out so I should change this Z to W so it will become W equal to Z0 plus e to the i theta Dw will be i epsilon e to the i theta D theta and if I take mod Dw I am going to get epsilon D theta okay and then I am going to get here I am going to get epsilon mod W minus Z0 is epsilon e to the i theta its modulus is epsilon so I will get epsilon to the n plus 1 okay so this is what I am going to get alright and you know what I am going to do next you see whatever I assumed I assumed condition 4 condition 4 is that the function is bounded in modulus in a sufficiently small neighbourhood so you know this term mod F of Z0 plus e power i theta I am going to remove that and put an M because mod of F Z is going to be less than or equal to M in a sufficiently small disc and I am assuming that epsilon is small enough so that this circle lies in that disc in that deleted neighbourhood of Z0 okay. So what I am going to the next step is I am going to get rid of this mod F Z0 plus e to the i theta I am going to pull that out and instead of that I am going to put an M I am going to get it I am going to get oh I forgot 2 pi this is 1 by 2 pi outside and then so you know what I will get is I will get basically I will get M times epsilon times 2 pi divided by 2 pi times epsilon M plus 1 okay mind you when I integrate 0 to 2 pi D theta I am going to get 2 pi okay and that 2 pi is going to cancel the 2 pi outside so basically what I am going to get is I will get mod An is less than or equal to M times M by epsilon power M this is all I am going to get assuming that by 4 mod F Z is less than M for all Z close to Z0 where B assume also gamma lies okay. So I am going to get this now watch see epsilon is a small quantity okay epsilon can be made as small as I want I can make epsilon smaller I can make epsilon 10 to 0 okay. Now if N is negative if N is negative mind you I am trying to show that the An's for N negative are 0 I am trying to show that all the negative terms in the Lorentz expansion do not exist okay so all the negative Lorentz coefficients are all 0 I am trying to show so if N is negative that is the case I have to look at then this epsilon power N will go to the numerator so I will get a small quantity to a positive power okay and if I now let the small quantity go to 0 okay its positive power will go faster to 0 so the numerator will go to 0 and since this is valid for all epsilon greater than 0 mod An has to be less than or equal to 0 and that will force mod An is 0 and that will force at An is 0 okay and we are done okay and that is the end of the proof okay. So let me write that down now if N is negative then epsilon power N tends to 0 as epsilon tends to 0 and so mod An equal to 0 implies An equal to 0 okay so this implies that the principal part in the Lorentz expansion is 0 that is what we have actually proved is that we have proved that you know 4 implies 3 in fact you have actually proved 4 implies 3 and so 4 implies 3 alright and of course 3 mind you is equivalent to 3 implies 1 because if the Taylor see if the Lorentz expansion is a Taylor expansion okay namely if it has no principal part then it is a power series it will converge to a function and that function is going to be equal to the given function outside that point and therefore what happens is that you extended that function analytically to that point also okay. So let me write that down there is a little bit of there is a little bit of technicality so let me write this down we observe that 3 implies actually 1 for if 3 holds we have sigma N equal to 0 to infinity An Z minus Z naught power N converges to G of Z and converges to G of Z in mod Z minus Z naught lesser than epsilon and G is equal to F for Z naught equal to Z naught okay so see so let me repeat this see if you assume 3 what does 3 say it says that F of Z has a certain Lorentz expansion in which there are no negative terms if F of Z what does it mean it means that you have first of all a Lorentz expansion which converges to a function that function to which it converges is none other than F of Z okay and this is valid whenever Z is not equal to Z naught but what is the Lorentz expansion that converges to F of Z it is actually a Taylor expansion but you know namely it is just a convergent power series and you know the convergent power series is actually Taylor expansion of the analytic function to which it converges so there so you take only the Lorentz expansion which has 0 principle part it will converge to an analytic function call that function as G of Z now that function is going to coincide with F outside Z naught by definition because you already know that the Lorentz expansion also converges to F outside Z naught so in principle what has happened is you have found an analytic function G of Z which is analytic at Z naught and outside Z naught it coincides with F okay so finally this proves 4 implies 1 alright and that completes the proof of the Riemann removal singularity theorem that is the end of the proof which I will signify by putting a usually in books you see that people put a shaded square I will put something like this to indicate end of proof okay but there is a remark that I want to make so the remark is to fix some loose ends in the statement of the theorem in the so I am going back to the first condition okay the first condition namely the definition of a removal singularity what is the condition the condition is that the singularity is removable namely that there is a continuous function there is an analytic function to which this function converges what I want to say is that you see the condition only says that the function converges to an analytic function at that point but it does not say that this analytic function is unique okay so the first condition which is just the definition our definition of a removal singularity is that your function can be extended to an analytic function at that point okay but I say it can be extended to an analytic function I am not saying it can be extended to a unique analytic function and the fact is that it can be extended to a unique analytic function and the reason is that there is a deeper theorem behind us suppose it extends to 2 analytic functions at that point okay then you use the identity theorem which you should have studied in a first course in complex analysis which says that if 2 analytic functions coincide on an open set non empty open set or for example even if they coincide on a sequence of points which has a limit point at which both of them are analytic then they have to be identically equal. So that identity theorem will tell you that the if the singularity is removable then the function to which the analytic function to which the given function extends at the singular point is actually a unique function okay so there is the identity theorem there okay that you should remember okay so I will stop with that.