I can see that you guys are tuning in. If you can hear my voice, please type voice in the chat, all right? Please type voice in the chat if you can hear me. In the meantime, I just wanna make sure that everybody can hear me. I will track my own handphone as well. Yeah, I think I can hear myself, all right? Let me see who is watching already. Okay, I can see some of you. Okay, voice using is here. Hi, good to see you. Who else is here? Sun Eileen is also here, all right? Chiwa is also here. All right, great to see everyone of you. Hi, good to see all of you now. Now tonight, it's gonna be super, super exciting because we are going to dissect a stock together and this stock can really potentially give you at least an email premium, at least 46% return if you want to invest for the long term, all right? All right, everyone is here. Bingo, bingo. Hi, good to see all of you. Fantastic. Now, before we delve into that, right, I also wanna share that, hey, this is a very, very special opportunity because at the end of the day, right, this stock has been really talked about by so many people but recently because of so many bad news, you know, like so many regulations here and there and that is why people have been really shining and are shining away from this stock. Can anybody make a guess? All right, can anybody make a guess what company am I going to talk about tonight? And in fact, if you've been following me be it in the Telegram channel, my own Telegram or maybe in Buffer Online School Stock Ideas, right? Actually, we've been publishing this report, this stock idea again and again because we really feel that this is a very great company to invest for the long time. Oh, well, some of you are already making that guess. Okay, some people guessing Alibaba. All right, Kel guessing Alibaba. Well, Joanne guessing about Facebook. All right, Baba Andrews thinking it's Baba, Oh yeah, well JD, JD, all right. Wow, a lot of names coming out. Well, actually a lot of you get it right, all right? And that stock that can really give you the potential 46% upside. Later on, I will also explain to you how do I come out with this calculation? Where does the 46% comes from? I can share it to you, right? In our opinion, it's very, very powerful. And some of you have already guessed it correctly and that is Alibaba, all right? I'm a long-term, in fact, I'm really a long-term fan of Alibaba and because of the reasons so many reasons involve, right? People are really trying to avoid this stock, okay? But one of the greatest news that we had lately about Alibaba is actually about the fine amount, right? Okay, anybody still remember the news? What is the amount that being fine? All right, by the Chinese government for Alibaba's and like monopoly, monopolistic behavior, all right? What, how much is the fine, guys? You guys remember, right? And later on, okay, even though this is a stock that I believe all of you know, but I just really think that a lot of people are not really fully understanding this business and how powerful it is. And that is why I actually did, you know, like a full day of research so that I can present to you what is the opportunity inside, all right? So that you can see and hopefully you also have more confidence to invest, right? Despite of the volatility because at the end of the day, you can be go out there, you know, chasing a lot of stocks that have, you know, increased by, you know, a lot due to, you know, even though despite of the recent tech correction, like a lot of tech has been really going up a lot, right? But the thing is Alibaba is actually one of the few rare opportunities that have actually beaten down because of so many bad news and regulations. And if you want to make profit from the stock market, you also need to start to learn to look at opportunities despite of the drop, all right? So $2.8 billion fine, all right? Cal, fantastic. Da, da, da! Okay, to be exact, it's actually $2.78 billion fine. And in fact, this is also the largest, all right? Anti-monopoly fine ever rolled out by the Chinese authority. And in fact, right, this fine amount is even larger than what was being fined by Google, to given to Google back then, all right? So Google back then was actually the EU also fined Google $2.7 billion for its, once again, it's like anti-monopoly, these kind of regulations, right? So that was what faced by Google. And now this fine is even greater than what Google used to suffer, all right? So how many of you are actually worried about this fine? What do you guys, I also wanna hear about your opinion. What do you think about this fine? Are you worried? If you're worried about the fine, about regulations, please type worry in the chat. On the other hand, if you feel that this fine is actually, well, pretty good ending, all right? Because at least this fine is given and maybe Alibaba can start to move on from here, then you can type move on. Let me know, let me know, all right? I need you guys to participate as well, okay? Eileen think, no, what do you mean by no? You say that the fine doesn't, no worries. Ah, okay, no worries. I see, I see. So can I say it's no worry. What about other people? Is anyone that is being worried about the fine as well? Google is too expensive, okay? Thank you. I'm not talking about Google here. I'm talking about Alibaba here, all right? Move on, move on. Not worried, but too bad, no cash to find, all right? Exactly, yes. So for this, right? Actually a lot of people out there when the fine actually come out, all right? To a lot of investors out there, they really feel that this could be a very good closure, all right? And that is why actually the stock price did pick up for a while, all right? Before it actually come down again. So in this, you know, this special sharing, I just want to really go through the near-term future of what is Alibaba going to be like and the long-term fate ahead.  In order to really help you guys to invest better, all right? So like I mentioned earlier, once this fine is actually given by the authorities, right? Alibaba share price actually rose 6%, all right? In the US pre-market. But once again, all right? You know, the stock has been really going, like going nowhere, ding dong, ding dong. Even after a fine pick up a little bit, now it just go back down, right? So to a lot of people right now, Alibaba is still suffering a huge loss, all right? If you invested back in like November last year, the high was almost 320. And now it's crashing back down to like 235, all right? And the lowest point, I think back then in Christmas time, all right? The lowest point that Alibaba went to, it's about 210. So that was about 30% fall. So right now, if you buy and you just hold on until now, you have suffered a 24% losses, all right? Which is indeed very painful, okay?  And I know almost 30% drop to a lot of people. It's pretty scary, all right? But is this a danger or is this an opportunity? What do you guys think, okay? Before I'm going to share with you, my finding I also want to find out about what do you find about that? If you think that the risk is too much and you are still unsure about the uncertainty ahead, maybe you can type danger. On the other hand, if you feel that this is opportunity because Alibaba, in your opinion, it's a great business and right now it has really come down a lot to you it is a good opportunity to invest, right? Then you can type opportunity, all right? Find it's okay, okay? So you say more worry about more whacking, yeah, yeah. So there's indeed some sentiment about like worries out there, right? So Eileen also think that it's future opportunities. Okay, who else? All right, so most people are thinking that this is an opportunity. Well, we shall see, all right? So I'm going to present to you some of my findings, but I just want to say that, all right? This is not the first time that Alibaba or any tech stock that is being whacked, all right? By the Chinese or by the authority, or be it it's Chinese government or be it it's like foreign governments, right? It's definitely not the first time it's been whacked, right? So let me give you some of the examples back then, all right? What kind of companies has been really fine and whacked by authorities, all right? Facebook, okay? If you still remember 2018, right? That is the first data leak. That was when actually Facebook share price fell by a lot, close to 45% drop, all right? Since 2018 back then when the news leaked out and then within just a short few months, from like July all the way to November about December, all right? Within a short six months, a stock price tanked by almost 45%, all right? Another tech stock, another tech stock, all right? It's called Tencent, all right? And Tencent once again actually is regular because it's a China company is regulated by the Chinese government. So in 2018, the company is affected why? Because the Chinese government didn't want too many of the teenagers, the youngsters in China to play so much game. And that is why they start to regulate and they actually really freeze and they stop so many new titles being released. And Tencent has actually been one of the largest, video company in the world as well by revenue as well. And because of this, the stock market really reacted to close to 50% drop as well, all right? So delisting risk as well, Alan is also worried about delisting risk. We can talk about that later. Chiwa like the sound of me how I say Alibaba, okay? So this is definitely not the first time that tech companies are being regulated. So, but my question to you as what happened to Tencent as well as Facebook stock price right now? Guys, this is 2018. The scenario I just presented to you was 2018. And despite of the fall, okay? It's like catching a falling like night, right? If you talk about it, what happened to both of their share price right now? Guys, anybody know? Back then Facebook dropped to 137 and then for Tencent, this is listed in Hong Kong, talking about the Hong Kong listed Tencent, back then was 344 Hong Kong dollar. Almost, actually lowest point is about 350 Hong Kong dollar. What happened to both of them right now? If you just go and Google it, you'll realize that it's freaking more than double. All right, has been more than double ever since the bad news come out. And if you just bought it during that time, your portfolio will have been like, wow, just by investing in these two great companies, your portfolio will have just been double, all right? But the question is, will Alibaba be the next Tencent, be the next Facebook that can really potentially have a lot of upside, all right? So this is what we want to analyze tonight as well. So anybody is excited? If you're excited, please start excited in the chat, all right? Because I am really, really excited to share with you my finding as well. So of course, when we are talking about whether is this company a great company that you can invest in a long run and so that the share price can really, given time increase, right? You have to make sure this is a good company. If you go back to the Facebook example or the Tencent example, both of them are brilliant companies, right? And that's how, given time, okay? When they recover from the bad news, right? That's how the market sentiment start to pick up because we realize this is a great business and that's why they will invest and that's how the stock price will go up, right? So when it comes to Alibaba analysis, you have to make sure you need to do your own deal diligence, all right? So many people are excited. You have to understand firstly, is this a good fundamentally sound company, all right? And second thing is, what are the risks involved? All right, some of you already pointed out some of the risks, later on we can discuss about that and thirdly is, is this the right price to buy right now? And fourth is, if you know how to use technical analysis for those who have tender OMI before, right? You will definitely know the importance of a technical analysis or so. So if you want to enter with a higher level of assurance, with a higher ROI in mind, then you also want to look at technical analysis. So we want to go through these four segments tonight as well and in fact, we actually go through very, very in depth because we do publish our own stock research report to our community inside BOS or Buffer Online School Stock Ideas.  Because every single time we publish our report, it's really true a lot of findings, a lot of research hours, all right? You put in our heart and soul inside, make sure you're going to read it, all right? So after this webinar, for those who have not read, go and read it if you are already inside our bossy community. So I just want to put things into perspective for all of you. Let's compare the two greatest giants in the field of e-commerce, right? All right, so we are talking about Alibaba, which is the biggest in China versus Amazon, all right? So while Aaron's reports are really good, thank you so much, all right? So let's take a look and really compare what is the positioning of Alibaba versus Amazon? Because you have seen Amazon, the stock price of Amazon just been crazy, right? Just keep on going up all the time, throughout so many years, all right? So if Alibaba is a great business, given time, I personally also believe that Baba's share price will increase, all right? So let's do some comparison. Firstly, this is 2020 data, which is just last year financial reports, all right? So let's take a look at Amazon followed by Alibaba's key ratio performance, okay? We are not going to go through nitty-gritty for everything, but some of the key facts I want you to pay attention to, all right? So the first one, it's called gross merchandise value. What does that mean as, because this is an e-commerce platform, people buy and sell things, so whatever things has been sold there, right? Your goods and everything, so they will, for example, this cup, right? Would definitely cost some money, let's say $6, right? So the gross merchandise value by selling this cup is $6, but there's so many things in the platform being buy and sold, sell every single day. And that's how, if you add up everything, all right, in a given timeframe, that's where you get your gross merchandise value. So the more gross merchandise value out there, that means you know that this platform, okay, more people are buying and selling from that, right? This is also a good sign. So let's take a look at Amazon, all right? For Amazon in 2020, the GMV was 335 billion US dollar. Now, my question back to you as, what do you think is the GMV for Alibaba? What do you think? Do you think it's less than Amazon? Because in our opinion, Amazon is like one of the largest, in fact, it's the largest in our opinion, right? Do you think it's less? If you think it's less, less. However, if you think it's more then type more. What do you think? Yes, correct, Ellen, it's by subscription basis, but we publish every single week, we actually generate a report for you, for you to really do your due diligence, okay? See how we analyze, and then if you invest together with us, given time, long-term, you will make money, all right? So Amazon cannot fight with Baba, C1C cannot fight with Baba in terms of GMV, more and more. Exactly, so many of you really understand Baba well, all right? In fact, it's like more than three times, in terms of US dollar as well, this is not in CMY, it's one trillion. This is one trillion in terms of GMV, it's crazy, it's crazy, right? And now let's take a look at monthly active users. If Amazon has 200 million monthly active users, do you think Baba is going to be more than that? The answer is yes, okay? Because it's China, China has the population is so many more times than US, and many countries combined, and that's why the monthly active users for Baba, it's also way more than Amazon, right? Now the next thing is net profit margin. You wanna see, does this business has a good margin, so that it means that it does have pricing power, right? So let's take a look at Amazon. So for Amazon, their net profit margin, it's 5%. Anybody can make a guess for Ali Baba, how many percent for the net profit margin? Anybody, anybody? In the meantime, for those who are watching, if you know that, hey, some of your friends, they will also get to benefit from that, because we are going to stretch a lot, cover a lot in depth about Ali Baba, a potential good opportunity to invest, then share this out like this video, share this, so that more people also get to benefit as well, all right? So what do you think is the net profit margin? So see when thinking about 30%, has Jonathan thinking about 30%? Very close, okay? Very, very close. Apple thing is more than 20, exactly. It's more than 20%, it's 22.5%. It's like, whoa, a whopping close to five times more than Amazon. Why is it so? Because at the end of the day, Ali Baba actually runs a different business model, all right? Amazon really, they do everything. It's like a vertical integration, right? But for Ali Baba, they mainly are just really through the self-platform. They are like a middleman. So they earn money through many through ads, as well as some buy and sell transaction fee. So it's very different from how Amazon actually runs, because Amazon is very, very K-Pax heavy. They have their own warehouse, they invest in a lot of things. And that is why at the end of the day, net profit margin for Amazon, it's very, very little as compared to Ali Baba, because they are two different business model. Now, let's take a look at Total Revenue, all right? So if you think, okay, let's take a look at Amazon. Amazon is 347.9 billion USD. Now, what do you think is the Total Revenue for Ali Baba? If the net margin is higher, all right?  What do you think will be the Total Revenue of Ali Baba? Can anybody make a random guess, all right? If you really cannot make a random guess, right? Maybe you can just type. Do you think it's more, or do you think it's gonna be less? What do you think? Okay? Once you see what it's thinking, oh, well, it's gonna be double, right? Okay, trillion, okay? Because this Amazon is in billion, right? So Ali Baba is going to be trillion. Let's take a look. Dun, dun, dun. Well, it's actually less than that, okay? It only had 86 billion in terms of Total Revenue. Why is it so? Once again, it's really because of the business model is different, all right? So Ali Baba, if you actually use Taobao to buy or sell before, right? Especially you as a seller, they don't charge you for any fee for listing your products inside Taobao, okay? Basically, it's free for you to lease out or lease anything, right? So how do Ali Baba make money? Really, majority of the revenue also, in terms of the e-commerce site, true advertisement. So if you have so many stores out there and you want to rank higher, you want to catch more people's eyeball, go into your store, go into your traffic. That's how they actually get people to pay so that their stores, everything can be placed in a better position, right? So that's why not everybody wants to pay advertisement dollar in order to get higher ranking. And that's how they also definitely at the end of the day will not be able to earn as much as money compared to Amazon because Amazon, they take in every single transaction, every single transaction, they actually take a cut. And apart from that, Amazon also does a lot of other things as well, right? They also sell their own products and everything. So Amazon has more revenue. So in terms of the size, you can still see that Amazon is really a giant, right? In terms of the e-commerce space for the revenue that is going to be earned, right? So now the next thing we want to look at is the market cap, right? How big is the company already being traded in the stock market? So for Amazon, it's already a trillion dollar company, 1.7 trillion dollars. On the other hand, Alibaba, right? Despite of the giant size as well, right? More users, more population, more GMV, right? The market cap, because of the recent drop, it also decreased a little bit. So now it's only 600, yeah, about $600 billion, right? So you can see that Amazon is almost three times the market cap of Alibaba. So if you think that, you know, Alibaba, if it has a long-term growth, at the same time, e-commerce is also an ever-growing space, right? Then at this market cap right now, actually, Baba, this is a first clue that you can get, right? At this current price, Baba can have a lot more potential upside as compared to Amazon as well. Because for Amazon to double, it need to have another 1.7 trillion. But for Baba to double, it's just another 600 billion market cap. So that's why this is in terms of the size, you can already have the first clue, which one can give you a higher potential in the long run. All right, now, let's take a look at the nitty-gritty in terms of the business, right? Exactly how Alibaba make money. And this is important because at the end of the day, you need to understand the business well before you decide to invest in anything, right? So you can see that actually 46% come from the China retail marketplace, right? And then the other 20% come from China commerce retail others, which is like some of the supermarkets, all right? While the majority, which is close to 50%, in this case, it's your Teemo, it's your Taobao, all right? It's your JuHuaSuan. I've never used JuHuaSuan before, but I think this thing is already integrated inside the Taobao app already because I've seen that before, all right? Then you have other part of the business segment, including your Lazada, which is considered the international commerce side, all right? You also have other things like clock computing, which is making up to be 8% of the revenue, all right? In 2020. And then it also does like digital media, which is like Youku. So if you want to understand what is Youku, it's more like the YouTube, right? The YouTube in China, because YouTube is then, all right? That's why they will use Youku for video browsing and everything. So Alibaba also has more entertainment. And recently you watch some movies, right? You also realize that Alibaba is involved in the production of the movie, right? But once again, the media segment is a very, very small percentage of the total revenue, right? The majority, the main bout, right? Actually over 60% is still in the retail side, which is the e-commerce side. Now, let's take a look at the financial performance, right? Is this really that strong, right? Has this performance of Baba been growing all the time? Because we as investors, we want to make sure it's keep on going consistent. And that's why we'll have a piece of mind to invest. Now, the first thing we want to look at is the GMV has it been increasing over time. And you can see that for the past five years, hey, basically it's double, right? In terms of the GMV, the cross-merchandise volume, so many people are keep on buying and selling from there. And the transaction volume, in terms of the dollar wise, also double, super, super high growth, all right? Another one, if you look at the total revenue of how much Baba is making from 2016, it was 101, now it's 509. It's like more than 5x, super crazy in my opinion. If you think it's crazy, please type 5x in the chat. I really think that the growth of Alibaba is just exponential, right? But later on, we're also going to look at the future growth as well, right? Does Alibaba still have future growth? And if they does, then where does the future growth life as well, lies as well, right? Now, the next thing we want to look at is apart from the cross-merchandise volume, apart from the revenue growth, right? We also want to look at are there more people using Alibaba after all, right? And you can see that the annual active consumers, all right, in the China retail side, which is your Alibaba, which is your like T-more, your Taobao, it has also been growing very, very consistently, plus, all right, the mobile monthly active users has also been doubling, all right, from 2016 to 2020 last year, it has also doubled. And why is mobile MAU so important? Because in our day and age, all right, every single time that we just browse anything, we are using mobile, right? And this is definitely the trend that Alibaba is going to target as well, continue to target, continue to get more people onto its app. Now, the next thing is apart from the revenue, apart from growing in terms of users, we also want to make sure the operating cash flow is also growing all the time, right? So for this, you can see that it's also a very healthy growth all the time. And why is operating cash flow so important? Because you want to make sure these business, all right, on top of the top line, which is the revenue, right? Does this business really generate money from its day-to-day operations, right? And operating cash flow is a very clear sign that Alibaba is very healthy and is growing business as well, right? It's almost like, yeah, almost four times, almost four X. Now, the next question you need to ask is, well, Alibaba is so strong, so powerful, right? But what is the reason behind that Baba can be so strong? This is also the test time for you guys, for those who have studied investment before, you know the power of economic mode, right? What economic mode do you think Alibaba has? Okay, what economic mode do you think Alibaba has? Let me see, okay? Crypto learn, wow, we have a crypto guy inside, okay? He's very excited for five X, all right? But apart from, yeah, so what do you think is the economic mode of Alibaba? Let me see, let me see your comments, okay? Network effect, Aaron said network effect, okay? Let me just remove this. In the meantime, okay? Jack Ma. Chia Ying saying that Jack Ma, Jack Ma is the, Jack Ma is the reason why he's so strong, okay, okay? Okay, economic mode, well, Jack Ma is not part of the option, but well, it's possible, he's a very, very strong leader. But right now, actually, Jack Ma is not so much involved in a day-to-day operation. In fact, he's not so much involved in Alibaba that much, right? So network effect, whole ecosystem, all right? Very good, all right? A lot of you know about network effect. Indeed, all right, like Joshua, like Edmund talked about, right? So Jonathan also talked about it. It's really about the network effect, right? And why is network effect so powerful? Because the more people buy from Alibaba, then it will attract more sellers to come in, right? And as more sellers come in, there's more variety on this platform, then more people, more buyers will also come in. And that is one of the very simple understanding of how network effect will help to continue to grow Alibaba in the future, right? And the second thing, yes, like Vera mentioned, right? It's also one of the biggest e-commerce in China. It's leading in the world as well. So in terms of the branding power, it definitely stick to your mind, right? Whenever you talk about, hey, I want to do e-commerce shopping in China, the first platform that you ever thought of is will be Alibaba, right? So it has a really branding power. It's like Coca-Cola. Immediately you think about soft drink, you will think about Coca-Cola. You think about fast food, you think about McDonald's. So when you talk about e-commerce in China, it's Alibaba, right? And it does have a very strong brand. And people, that is why, right? Whenever big organizations, they want to penetrate into the China market, the first one they want to collaborate is also Alibaba. And that's how, you know, the product offering get more attractive, more customers come in, right? When more customers are here, more sellers will be there, right? And they really have... I also have a pretty good, in my opinion, very, very solid ecosystem, right? I don't have time to go through that, but inside our bossy report, I did write about that before. So let's continue. Let me just stop share, because, hold on, huh? And then... And I reshare again. Okay. So yeah, so these two, in my opinion, is one of the two largest intangible assets, the two largest economic mode that Alibaba has. The first one is Network Effect, okay? Followed by the intangible assets, which is the branding, right? Which is like what you mentioned, like you guys, a super, super strong, powerful brand that Alibaba has established over the years, right? And if you want to talk about the market share in China, in terms of e-commerce, even though there are other platforms like Rising, like JD, like Pingduo Duo, but you can see that Alibaba still dominates, right? And that is why, when it's able to dominate, there is a reason why people want to continue to use it. There's also reason why brands want to continue to really make sure they have a presence in Alibaba, because 58% of the market share, right? Once they tap into Alibaba, they can literally tap into the majority of the China consumers out there. And people in China are just getting more wealthy. They have more disposal income, and that's how they will buy more online as well, right? So this is the place, all right? We have talked about how Alibaba has done well in the past. But from now on, we want to talk about the future growth of Baba, right? Because in order to make sure we as an investor, our money will continue to grow, then we have to make sure that this company is definitely a growing company, not stagnant, but growing as well, right? But before I share with you, in my opinion, what is the future growth potential, all right? I would really appreciate all of you to like this video, all right? In the meantime, share this out as well, right? Because I really hope that through this education, more people will be inspired to really start investing. And most importantly, don't invest blindly, really through proper system analysis. And this is what we want to cover as well, right? And in fact, Baba has a great opportunity, so I hope more people will also get to understand that, hey, this is what they can also start grabbing opportunity to start investing, all right? And for those who have not followed on my telegram channel, you can also choose to follow my telegram as well, because I do post very, very frequent insights from inside there as well, all right? So with that done, okay, for those who are done, please type done in the chat. Please type done in the chat. I would love to have all of you help to like and share this, all right? So with that, let's keep going, but let me drink some water first. Okay, now, future growth potential. Where is it? Where is it? Here it is. This one, okay. In fact, this is what we published inside our bossy report as well, right? Firstly, the first opportunity, the first growth potential actually lies in China itself. And then you can see that Alibaba right now, right? The digital economy, GMB, is one trillion US dollar. But if you compare it with the China consumption, China consumption has like close to six times of what Alibaba is getting right now. And if you look at the China GDP itself, right? It's 14.5 trillion. And that is why the CEO of Alibaba, Daniel Zhang, also believed that Baba's digital economy grossed merchandise value, right? It's still a fraction of what is the Chinese consumption right now. And that is why it has a lot of room for growth, all right? So you have Baba, non-listing of N group is a great disappointment. Yeah, but maybe we can never say for sure, maybe in the future the N group will come back. Well, but this is something we cannot predict. But once again, even though this N group is not being listed, it actually doesn't impact the business of Alibaba, right? So if you're a shareholder, then understand this business even more so that you have more confidence to maybe add on if opportunities is given in front of you. Now, the second growth opportunity acts. Smart phone penetration rate in China, right? Right now, what is the percentage? You can see that in 2020, the percentage was about 55%, which is about slightly more than half. So what does that mean as, wow, there's still half of the opportunity, the population out there that has been untapped even within China itself. Do you think this is powerful? If you think this is powerful, right? Please type powerful in the chat. Seriously, if when I look at this data, I was like, I was mind blown as well. I was like, what? Only half of the data, half of the population owns smart phone. So even in China itself, when more people have more disposable income and more people have more internet connection assess, that's when definitely Baba will also get to benefit as well. So thank you everyone for helping to like and share this video very powerful, right? Fantastic. Now, this is just one of the future growth potential. We still have more to cover. The second thing, actually lies in the cloud, all right? And the sky is the limit. Why? Because if you study Alibaba business, Alibaba also does cloud, all right? And let's take a look at the global scale of how cloud computing, this economy itself, right? It's going to evolve over time. So this is one of the research I found online, all right? So according to this research, the cloud computing market, all right, was estimated to be 321 billion in 2019. And then by 2026, which is like five years down the road, all right, it's gone to reach 1025.9 billion, which is like almost five times as well. Okay, like three times, three x, almost three x. And the compounded annual growth rate, all right, for cloud computing market, it's 18% year on year. And Baba is also inside these business to tap into the future potential growth of how cloud computing is going to be like. And in fact, it's very true, right? Like a lot of things because of COVID, so many businesses, they start to realize the importance of cloud, right? It starts to realize that they need to, you know, work from home, a lot of things they need to store in the cloud. And this is definitely one of the future going forward. And domestically, Alibaba, it already holds a dominant market share, right, within China itself. You can see that Alibaba cloud dominates about 40% within China. Other competitors including Tencent, including Baidu, including Huawei cloud, right? But Alibaba is the leader. And how about internationally, all right? Internationally, this is the chart and the few very, very important competitors that you also need to be aware of, right? The first one is definitely Amazon, AWS, right? That's why Amazon is a very, very solid and strong business. It's close to like 33% dominant, right? In terms of the cloud space. And then Microsoft is really growing. In fact, Microsoft is also one of my favorite business as well. And then Google is also gaining more and more market share. And you can see that the orange line, Alibaba is also gaining momentum as well. In fact, it actually overtook IBM, right? IBM's cloud is like going downhill while the rest are either maintaining or it's not actually going uphill as well. So this is a very good thing. And based on the annual report that Alibaba wrote, right? Alibaba said that there are cloud computing revenue growth. It's 62% year on year. Super, super high growth. And if you still remember the chart I showed you earlier, cloud computing the cloud business right now, right? It's about 8% of the revenue, right? Total revenue is contributing about 8%. As these side of business continue to grow, I definitely will foresee that these 8% will start to take a larger and larger significance portion inside the total revenue. And one thing that I'm also very, very happy to see is the core commerce, right? Which is the majority of the revenue right now. It's still growing 35% year on year. And that's why this is a very, very strong business, right? So Chiwa also said Alibaba has so much room to grow, right? The same level as Amazon, right? I also agree both of them. It's very solid, very, very steady company. But you talk about growth rate, Alibaba definitely has a lot of growth rate as well. But when we talk about growth, we also need to look at the risk level as well, right? What kind of risk is Alibaba facing? And we all know, right? Firstly, it's this Chinese regulation from the government, right? And then just now we were talking about the fine, yeah? About 2.7 billion USD. And I'm so happy that most of you are not worried, right? Because this is what I also share inside my Telegram channel. I said, guys, you need to look at things into perspective, okay? 2.7 billion dollar looks like a lot. But if you really look at the overall picture, right? In the big picture, it's only 4% of Alibaba's domestic sales, right? And not even talking about international combined, it's just the domestic sales, it's only contributing to 4%, right? So think about it, right? If your portfolio is down by like 5%, are you really worried about that? Or you are actually okay, right? So in this case, I really think that Alibaba, despite of the fine, they can really pay off so easily and they're just part of their, just 4% of their total revenue for last year, right? And that is why I even compared to, you know, 11-11, the single day that Alibaba really makes famous of. In fact, right, it's even way lesser, right? Way lesser than like the single day, make 20 more times in sales, right? Compared to the fine. So once again, whenever you read this kind of news, right, always see things in perspective.  So that is a very, very dangerous thing. Make sure you always see things into perspective, all right? And the second one that many people are afraid of, it's actually the delisting threat, right? Because the U.S. saying that if it doesn't comply with our rules, if you have any affiliate with the, maybe the Chinese government, this is where they are going to delist you. So this is something that, you know, a lot of China company as a result has been tanking as well. And Alibaba just one of them, right? So what do you guys think? Do you think this threat is like really, really real? If you think it's very real for Alibaba, right? Please type real. But if on the other hand, if you think that Alibaba probably it's unlikely, unlikely to be delisted, then please type unlikely as well, okay? What do you think? All right? Let me know what you think before I share with you. What do I think? All right? Let me see. So Chiwa say, is there any specific risks related to Alibaba? I think one of the specific risks we are talking about this is because, well, Alibaba is, if you're talking about these risks about the monopoly risk, Alibaba is definitely one of the top player being target in China, right? Because it's indeed the leader, the global brand, the China's global brand is Alibaba. So that's why the Chinese government is definitely scrutinized, going to scrutinize this company more. And when it comes to find, they're going to find more, right? On Alibaba compared to other companies, right? But once again, I think from this lesson, Alibaba also learn, right? They definitely don't want it, don't want their company to be continued to be in this metal of having to really deal with the Chinese regulations so much. So they're going to comply, right? And that's why, according to the find, as well as the final rules, right? They do need to report to the Chinese government occasionally to tell them about what kind of practices are they doing? And I think it's a good thing because finally, Alibaba will accept what kind of regulations that Chinese government require them. So basically, if I'm not wrong, they can no longer ask anybody to say, hey, you can only list on my platform, you cannot list on anyone else's platform. So this is something in many people's, in the Chinese government's eyes, it's not fair. And that is why they ask Alibaba to stop doing that. So even though Alibaba stopped doing that, I don't think the business is going to be impacted because based on my research, the companies, right, as being affected by this one platform only rule, it's like less than 10,000, right? As compared to so many companies that has been listing on Alibaba, only 10,000 of them is really being impacted. So I really don't think these 10,000 clients will impact so much into Alibaba's bottom line. Yeah, so unlikely, right? A lot of you think it's unlikely, but on the other hand, there's one person think that maybe it's possible, right? So I think at the end of the day, in my opinion, I don't think it's very likely, okay? Because I do think that Alibaba also want to comply to the rules and regulation that the US government gives, right? But even if it does happen, what's going to happen to any US share that's going to be listed from the main market is they will go to this market called OTC over the counter. So when this company is being sent from the main market, go to the OTC, right? OTC to a lot of people, it's where the low liquidity comes in, right? Not so many people are very interested in the stock and that is why OTC generally has a lower liquidity, has a lower volume every single day. But it doesn't mean that you cannot invest in OTC counter. You can still invest in OTC. In fact, Tencent, right? It's also a company that it's listed in the Hong Kong stock exchange. At the same time, it's also in OTC. So Tencent is not going to be affected for sure, right? But on the other hand, even if it's delisted, Alibaba can still be inside OTC and I believe people who understand the power of Alibaba, they will still continue to buy the OTC or trade in the OTC counter as well. So I'm not so worried about that. And one more thing is Alibaba also has its Hong Kong presence, right? It's also listed in Hong Kong. So your shares can also be converted into Hong Kong shares as well. So at the end of the day, I am really not too afraid about Chinese stocks being delisted. As long as I know that this is a great company, I invest in the long term, even an OTC counter can make me money, which happened to Tencent as well. I also am a shareholder of Tencent and Tencent stock price, if you just check, right? Later, you just check. It's been increasing over the years as well. So it's a beautiful company even though it's inside the OTC exchange, right? So we have talked about the growth, we have analyzed the fundamental of the company, we have analyzed the risk as well. Now, let's go into the valuation, right? Is this the right price to buy right now? And like I mentioned before, there is a potential 46% upside. Where does that come from, right? I just want to share with you one of the simplest, right? Really, one of the simplest way for you to do valuation yourself is to see, okay, before you see or make sure you do your deal diligence commercial time, please follow Telegram channel if you have not done so because I'm going to share with you more tips on the go. Okay, but once again, whatever things I have shared with you right now, it's not buy or sell recommendation, yeah?  That's how you're really going to familiarize with yourself, how can you analyze the company better and what are the key information that you need to know, right? Before you invest in Alibaba or any other company, super crucial, all right? Now, let's take a look at the valuation. This is the one, all right? One of the simplest way to see whether is this undervalued, overvalued, right? As long as this company has been making money all the time, which in fact Alibaba has been, right? So you just need to look at the PE ratio, all right? And you compare the PE ratio, the current PE ratio to the past five year PE ratio. And what does PE ratio means is how many times is the market, right, willing to pay for Alibaba at this current earning per share, right? So the more times that the market willing to pay, that means they are generally, like, pretty positive about this company because I don't mind paying you higher because I know that in the future, as this company continue to make more money, all right, the growth rate catch up, that's how, okay, as a shareholder, I will make money. So basically it's a very, very simple way to understand the market sentiment, all right, towards the stock. So if you look at the past five year, the market sentiment means this market is willing to pay 39 times, all right, to the EPS of Alibaba, 39 times. And currently, because of all these negative news, because all these fine, the regulations, the delisting news, suddenly, right, the stock markets, for Alibaba, right, the stock price really tank. And that is why at this current level, Baba still continue to make money from its earning per share, the earning per share. In fact, it's growing, right, it's consistent. But the stock price drops so much, that is why the PE ratio suddenly becomes smaller. And if you buy right now, that means you are buying it cheaper than how many times in the market it was willing to pay for the past five years, right? You are buying cheaper than average, right? And where did the 46% upside come from, right? If you just use the 39 divide by a minus of your 26.65, you just divide, right? Your potential upside, it's 46%. Because given time, if it's a good company, eventually, right, this company is going to, all right, what's going to happen is the share price will increase, right? When the share price increase, that's where the PE ratio will tend to go back to its average price, right? So average was 39. So if you just give it time, 26.65 will start to increase, increase over time, go back to 39, or maybe sometimes even more, right? So that is your potential upside. And if you think about it right now, Alibaba, what price is it? I think 23 something, right? 46% increase, it's over $300, right? And I totally think that Alibaba, if you give it time, I personally believe, once again, it's not by yourself recommendation, but I really feel that given time, Alibaba will increase to more than $300, right? That's in my opinion, right? So, but once again, this stock market is always, you know, very volatile, especially in the short term, all right? Guys, if you look at short term, right, what does the trend looks like to you? What does the trend looks like to you? Is this a downtrend? Is this the uptrend? Is this a sideways? What does it look like to you? Very obvious, it's actually a downtrend, right? And it has been downtrend like since, since November last year, right? So it has almost been half a year, all right? Almost been half a year going downtrend for Alibaba, right? So, ding dong, ding dong, ding dong. So now, if you really talk about it, it's approaching, seems to be approaching the resistant line, right? For those who have done OMI before, you will know what I'm talking about. Basically, they are reaching the roof, right? Reaching the resistant. So as any possibility that Alibaba is going to go down even more, if you look at the short term technical analysis, yes, all right? Because if you look at the downtrend, it does have more room to fall, okay? It can even fall, let's say to $180. It's possible, right? But once again, all right? Whenever, when we go to investing, right? Do we want to focus on short term or do we want to focus on long term? Guys, what do you want to focus? Short term or long term? Because if your focus is very short term, then definitely your decision is going to be very short term as well. But once again, let me bring you back to what happened to Facebook back then. Short term is a downtrend, right? If you zoom in, zoom enough, right? It's a downtrend. And if you are too afraid to invest, right now you wouldn't even have that upside because Facebook took about six months to go down 45%, right? From the peak all the way to the bottom of the bottom in the 2018 data leak crisis, right? 45% down, right? In six months time. Another 10 cent, it took about nine months for 10 cent to go from the peak all the way to the bottom as well. So if you use the same kind of deduction for Alibaba, so Alibaba, can it go down further? Possible, right? How many more months to go? Maybe another three months to another half a year. It's possible because for a stock to go from the peak all the way to the bottom, from Facebook's case, from 10 cents case, it can be six months to nine months to maybe sometimes one year as well. It's really, really possible. But once again, if you just zoom out, right? Like what Sir Key said, right? If you just zoom out and focus on the long term. Guys, let me show you the long term trend. What do you think right now? What do you think right now? If you just zoom out, this is five years. I'm not super long as well, just five years. What is this trend? This is a very obvious uptrend, right? It's a very obvious uptrend despite of the drop right now, right? Despite of that you couldn't even see it, right? You feel like it's just a very normal channel, like just being bouncing between the resistance, the support line on the long term uptrend. So I wouldn't be surprised that if Alibaba continue to fall further to maybe $200 because if you look at the long term support, right? This blue line, which at the bottom, adds a good support, which is about $200. I wouldn't be surprised if Baba continue to fall to $200. If you have money, if you want to invest more, this could be a good time for you to invest further. But would I want to wait until a really drop to $200 before I invest? I wouldn't do so, right? Because once again, this is a good company has been really under value, all right? If you just keep on waiting and waiting, you can never, okay? You can never catch the bottom, right? So once again, once again, if you want to invest right now, right? Then do position sizing, okay? We don't have time to talk about position sizing in this webinar, right? Inside the bossy report, we did talk about that. But once again, you need to enter by tranches, right? You can enter by tranches, you can enter a bit here. If it does drop further, like Wendy say buy more, right? If it doesn't fall further, it just go up, right? At least you didn't miss the boat. I think that is very, very important. So in my opinion, is this a falling knife? Yes, it does look like short term, but am I willing to catch it? Because I am a long-term investor and I really don't mind catching a falling knife because once again, I don't know when it's the bottom. So I just want to buy a bit when the market give me opportunity. And when it go further down, as long as it's still within my portfolio position sizing, I will buy some more, right? So this is a very, very simple approach. And if you just zoom out a little bit to look at Bali Baba's long-term trend, right? It's very, very solid bull run all the time, right? So now let me ask you one more time, right? Do you think this is a danger? Or do you think this is an opportunity, guys? What do you guys think? Do you think this is a danger or opportunity, right? So once again, one of the things that we share here, it's really for you to do your deal diligence, right? Cannot just because Chloe say, oh, Baba, very good, then I need to go buy, right? Because once again, you will never have my conviction, right? So make sure you do your deal diligence, go and really study it, okay? Go read more reports to find out, right, for yourself so that you can build up the conviction for yourself as well, right? Very, very important. So Sir Key says it's opportunity, Carol says it's opportunity, Jay also thinks so, right? Hanh also thinks so, fantastic, right? So once again, this is an opportunity, but you need to make sure you do your own deal diligence and position size properly, okay? Don't go all in, let's say you have a $10,000 portfolio, then you go all in, Baba. Once again, it's very dangerous because you need to make sure you do know how to diversify your portfolio as well because we can never say for sure how fast will Baba recover? What if it takes another half a year, right? Then do you want your portfolio to tank for the next half a year? You don't want the 100% portfolio to be tanking, right? Because you are going to miss some opportunity costs as well. You could have used some of the money to invest in companies that are uptrend right now and then you can continue to make money regardless, right? But I'm very happy to see a lot of you seeing that this is an opportunity. Indeed, in my opinion, I've already added positions in Baba. Baba, I think I really maxed out. I cannot add any more. Maybe if it dropped to $200, I would be like, why I find so early? But once again, well, I thank everyone for joining this webinar. I hope that you find it helpful. And for those who are really wondering, for example, just now I talked about the bossy report, right? You are thinking about, hey, exactly how is this bossy report going to help you? Tomorrow, in fact, I'm just conducting a very, very short webinar about one and a half hour, all right? I'm going to share with you how we analyze so that you can learn how we analyze so that you can become a better investor. So this is a buffered way of investing system. I'm going to share with you step by step. And I'm going to also dissect another real case study for you so that you can learn our thought process to improve on your own investment with thesis, all right? And thirdly, I'm also going to share with you some of the ways for you to cut short your learning curve and really, really tap into opportunities that we are here to share with you every single week. So for those who are keen, then just go to this link, rebrandly.ly.ly slash bossy webinar. So in the meantime, let me see if I have any more questions, all right? If yes, must join the webinar. Thank you so much, Nicholas, for sharing this link inside the chat as well. Investors, almost 15% of my portfolio, gee, wow, we are very close, we are very close, okay? I'm almost there as well, slightly more than you. Opportunity, Charlie Munger, yes, yes, exactly. Aaron also mentioned about that. In fact, recently, Charlie Munger actually went to buy Alibaba shares, all right? So he himself, he's also a long-term investor and he also see it as a very great opportunity, all right? So that's why you can take it as a reference as well. But Alibaba options as well, right? So Nicholas, if you know how to do options, well, you can go and do options. But once again, you have to make sure you are not paying too much for the premium, especially if you want to do strategy X, right? Because volatile, you may end up paying a lot more. So just make sure you check your OMI rules, okay? But before you enter, all right? So shares plus options for Alibaba, same here, I also do both, all right? So that's all I have to share. It's an hour already. Hope you guys really enjoy it. And yeah, let me see if there are any more questions. If not, I'll just put this link over here so that all of you, if you want to join in, then yes, join me tomorrow as well. It's gonna be a very short and sweet webinar, one and a half hour, all right? So on, thank you, Chloe, thank you, everyone. So for those who have not joined my telegram as well, make sure you do so that you can also follow some of my investment insights, which I post quite often, almost every single day as well. So if not, okay, thank you so much, Erin, and thank you everyone for joining Hang Hall as well. So I hope everyone have a great night and we will see you guys tomorrow as well. See you, bye-bye.


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OK. So we continue, and we use the notation we assumed so far. We take this power series summation ak z minus a. We assume that the radius of conversion is r. We also denoted by f1 of z, the series obtained in this way, ak z minus a, k minus 1, k greater or equal to 1. We already observed that the two series have the same radius of convergence, so also the second one has the radius of convergence r, and then in some sense we decided to split fz as snz plus rn of z, where sn of z is the expansion up to the order n, so it is a polynomial, and the remaining part, rn, is the tail in the power series expansion.  So we observe as well that if we define s prime of z to be a1 plus 2 a2 z minus a plus n an z minus an minus 1, we can also, sorry, this is s prime of n, right? You can also define f prime f1 of z to be the limit of s prime n of z, okay, by definition. So as I said, we wanted to prove that starting from a power series function which is defined as a power series expansion with the radius of convergence r, this function is complex differentiable. So that we consider the incremental ratio, and this is given z0 such that when first of all, as I said, we can take a equal to 0, and this is without loss of generality, then we can take z and z0 to be smaller than rho which is smaller than capital R, the radius of convergence, okay, so everything, if z and z0 are given with these assumptions, this function here is well defined for z different from z0. For any z, this is defined, this is a number, this is a complex, so this is the difference of two complex numbers, and this is the quotient of two numbers, okay. On the other hand side, we want to prove that this becomes very close, in the topology we introduced, to f1 of z0, and so we can consider the difference, and we have to show that the modules of this difference tends to 0 as z tends to z0. And to do so, we recall that f of z0 was split into two parts, the final part up to the degree n and the infinite part, the series from n plus 1 to infinity. So we rewrite everything like this, and then we divide everything again, so this fraction in this way, then we subtract and add s prime and z0. So here what is left is the following, rn of z minus rn of z0 over z minus, okay. Because of the triangle inequality, the modules of this is the modules of this, but is smaller or equal to the sum of the module i, this plus the modules of this plus the module d. And if we show that these three modules i are smaller than epsilon, epsilon over 3, of course, if you want to be very precise, so you show for z tends to, for z close to z0, then we show that this incremental ratio has a limit as it tends to z0, f1 of z0, okay. So we are left to conclude the following. So this is smaller or equal to, with the notation so far introduced, what I said first is, okay. This is, in real calculus, formally the definition of the derivative, and this can be made smaller than epsilon if z is close to z0. This comes from, this is also infinitesimal as n tends to infinity, since we are considering what that f1 of z0 is the limit as n tends to infinity of s prime n. The difficult part is this, which is an infinite sum, so it is the series. We have to see if this series converges to something which can be considered small. But remember from the definition, rn of z is the sum of nc to the power n, sorry, k, with k greater or equal to n plus y. So it's, as I said last time, it's the t, so the part, the remaining part, not the polynomial, so that if we consider this difference, this is in, evaluated in z0, so if we consider the difference, and this is what we are doing, we obtain the following, so rn of z minus rn of z0 is in fact summation k greater or equal to n plus 1 ak z minus, sorry, z to the power k, correct?  Therefore, when considering this quotient here, we divide the factor z minus z naught, because z is different from z naught, it's taken different from z naught. And this is summation of ak this, correct? So now I take the modules of this, this is the modules of this, and become smaller or equal to the modules, or to the sum of the module i, right? And so this is summation sum of ak, and then I have here, how many, do you know how many summons are in this parenthesis, k elements, right? So I have that module of z and module of z naught, remember, are both smaller than raw, and raw is smaller than r. So when I consider the modules, modules of z and then the modules of z naught, I can substitute because I'm considering an inequality here, instead of z, modules of z and modules of z naught, modules of, sorry, raw, okay? And notice here that the power of the module i is precisely k minus 1. So it becomes smaller or equal to raw, k minus 1 times k, correct? Okay, okay. And this is precisely the kth element of f1 at raw. And this is convergent, f1 is convergent. So this number here can be made smaller than epsilon, cannot divert, it is actually convergent to something which is zero, because raw becomes very small. So this is more than epsilon. So summing up these considerations, we can say that actually it's true that for any power city expansion, for any power city expansion f of z, and when considering the, what we say, the derived power expansion, we have the following that this is smaller than epsilon as z minus z naught is smaller than delta and n is sufficiently large, right? Good. So first, a first important step in comparison is made. So assume that we consider as a class of function, we are interested in the class of function which are expressed in power series expansions. These functions turn out to be differentiable in this, in this sense of the limit of the incremental ratio of this, and this is finite. Good. Furthermore, we have this. From the derived power series, we can continue the same trick and consider the derived of the derived power expansion. There is no reason to consider this as the last. And applying the same argument, we conclude that actually it is, again, differentiable. And then we apply the same trick and argument to the, the right and so on and so forth. So the only power is z infinity. And there is an expression, if we use this to be the kth derivative, this is the kth derived series. So this, evaluated at the center of the power expansion, at a, this is now to be what? Well, thinking, we have an example. If we evaluate the function at the center, we have only a naught. If we evaluate the derived function in a, in a, we obtain a one. If we evaluate the function derived, derived in a, we obtain two a two. In general, we obtain k factorial ak, right? Okay. So this is k factorial ak, which is also important to know. And in general, what we have, in fact, is the following. That fk, so the kth derived series, derived series, has these expansions, something like factorial ak plus k plus 1 factorial ak plus 1, c minus a plus. This is also important because when we apply this to our consideration, for instance, to the function exponential, we obtain the following, important example. Remember that a of z was the function. One of the example of power expansion of, of complex, say, complex value functions defined in terms of power series expansion with radius covered in infinite. And the coefficients were given as follows, 1 plus c plus d square over 2, factorial plus zm over m factorial. The coefficients were real, but well, it's just by chance for some sense, okay? Now we can say, well, this function is differentiable in the complex sense. And we know that, that is, it's the derived function is the function which has the following expression. 1 plus z plus d square over 2 factorial plus zm over m factorial, okay? So we actually have that function exponential solves this equation, differential equation with initial condition, like in the real case. Well, do we have other examples of function with the real coefficient which can be, can be extended to the complex case? Well, for instance, why not? Let's try with trigonometric phi, see? Cosine sign, we remember that cosine of x was 1 minus x square over 2 plus x4. By the way, x is real, right? Sorry, two factorial, sorry. Yes, the coefficients were real, so again, just by similarity, we can say, well, we extend to the complex case and put this way.  And similarly, we find sine of z to be z factorial plus, okay? This is by definition another function. So I invite you to verify the following. First, verify that the radius of convergence of, so that r, radius convergence sinus and cos sinus is infinite. Again, and furthermore, verify that the derived series of cos is sin, sorry, is minus sin and of sin is cos. Like in the real case. So this is an exercise. Now we have a relationship between cos, sinus, sine and exponential in the complex case. Nothing similar is in the real case, okay? Another similarity with the real case, the form would take cos square of z plus sin square of z, and this is equal to 1. What we don't have is the following. You consider e i z plus e minus i z. Remember, the i and minus i is the property that i square. So for all positive powers, odd or even, they are negative positive with i in front or minus 1, right? So I take this and divide by 2 and this is, prove it, cos of z. Prove, simply substitute in the definitions of the two power series, okay? And similarly, 2i e i z minus e i should be this, right? This is, and from the two, I quote here, we conclude the following. We have first that e i z is 2 cos z minus e. We have e i z is 2i z plus e i. Then I sum the two relations and I attended the e i z as cos z is very important to know. In particular, when we restrict z to the reals, you obtain the famous Euler relations. E i t is cos t plus i sin t, okay? Very important. So what is difficult to prove also in the real case? In general, what is difficult to prove when you start dealing with power expansions is the fact that when you multiply, you have a mess. But what is surprisingly true is that, in fact, for some power series expansions, like the exponential, you have good properties. So that the coefficients are mixed, but they are mixed well. This, as a consequence, implies, for instance, that if you take e to the power a plus b, this is equal to e power a times power b, following the traditional power rules, okay? Rules of, you learn in primary school, right? But it's difficult to prove it in general, all right? You have to make some calculation, to make some consideration about the convergence, so on and so forth. So what is not surprising is that in the complex case, you actually have the same property. But in order to prove this, I want to make some more general consideration about derivation in the complex world. You will see that this property has a consequence of the properties of derivative and derivation in the complex setting. So let me, first of all, use this notation, which will be the same for the entire course, which is quite standard in many books. So we have a complex value function of complex variable. So we assume that f is defined in an open set u of c and its values are in c, okay? So we take as variable z, which is, since c is like r2, also written as x plus iy. This is the real part and why the imaginary part of the complex number z, correct? So similarly, the function f has two components, the real component and the imaginary component, right? Therefore, we write f of z to be the sums of two real valued functions of complex variable. For them, typically u and v.  Sometimes you can also find this, just to use the same symbols for the variable and for the components of the function. Now, after using these notations, we can consider the function from u into c as a function from an open set of r2 into r2 with two components, instead of calling them as you normally do in the real case, f1, f2, we call them u and v, right? Now, let us assume that f has complex differentiable z0. This means that the limit as h tends to 0 of f z0 plus h z0 over h exists and is finite. And normally denoted by f prime of z0. If you want, there's a definition. We say that f is complex differential of z0 if the limit of the incremental ratio exists and this is finite, nothing different from the real case if you want. Pardon me? You? Yeah. Yeah, sure. If you assume that f is defined in u, z0 has to be in u, otherwise there is no meaning. So you start from a function defined in an open set. You take a point inside this open set and you consider the limit. Okay, since h tends to 0, we can assume that h is in a smaller neighborhood of z0 in such a way that this smaller neighborhood is containing u and this is not a restriction because we are assuming that h is so that this is defined, right? Do we have examples of functions which are differentiable in the complex sense? Yes, we do. All functions which are defined as power series. So we have examples. But what does this definition imply? This definition implies many things. First of all, there is no assumption about the choice of h. H can be any number tending to 0. So remember that on the real line, when you assume that x, a real number, tends to a value x0, you can approach x0 from the left or from the right, correct? So in fact, we normally define the left limit and the right limit and we said that if the two-linear limits coincide, then this limit can be different. In the complex plane, z0 is a point in the plane and we have infinitely many directions along which you can converge to this point. So we are not assuming that a privileged direction is assumed, nothing is assumed. So for instance, we can restrict our consideration to an incremental ratio along the reals or along the pure imaginary numbers. So let us take as h tends to 0, h real. In the definition of conflict differential ability, nothing is assumed about h. The only hypothesis you have to take is that h is becoming closer to 0. So in this particular case, we have that considering f of z0 plus h and using the notation real and the imaginary part of z0 to be x0 plus iy0 and z in general x plus iy, we have the following. This is f of x0 plus h plus iy0, but remember that f was also as u, when we consider the two functions u and v, the real and the imaginary part of f, be dependent on z or if you want to look at the real structure behind the complex numbers depending on x and y. So that if I take the incremental ratio with the assumption that h is real, we have the following.  So the study is reduced to the real case because this is a real value function and these are the two components of the real and imaginary part of the complex variable. Here we have a ratio of real numbers and this is represented in the real calculus by what? The derivative with respect to x at x0 over the function u. So if you want to use this, this is notation. And here I have i times well the same derivative with respect to x of the component, second component and evaluate it if it's not. For the sake of simplicity, since we will have many derivatives with respect to x to y and so on, I prefer to use this as a shortcut in our considerations. I put an index x and with this I mean derivative with respect to x if you don't mind. This is somehow quite common especially in differential geometry in order to avoid all these notations. So the next step is of course to consider what if instead of taking as I said h to be real h a purely imaginary. So in this case I have something different if you think we have the same pattern in some sense but this time the increment is made along y instead of along x. So I take h tilde to be something like this, i times h and h real. In this case the increment ratio starts from is the following which becomes using the notation so far introduced that as follows x0 y0 plus h remember that we have h tilde in the denominator plus i v x0 y0 plus h minus v x0 y0 2 over h tilde and h tilde is i h in other words this becomes okay i and i cancels. So the second part becomes what you expect to have so the function so the increment ratio of the function v with respect to y what about the first term minus i very good minus i because we have well 1 over i is in fact minus i so I have minus i and this is the same h okay here and and I call this appears and it's cancelled by the i here and h in the denominator here gives you this so when I take the limit as h tends to 0 so also h tilde tends to 0 this is the only way to make h tilde tends to 0 right correct. So I have that first terms tends to okay minus i you will expect to y x0 y0 plus v y sorry x0 y0. But the limit exists so the limit as h whatever h you take as soon as it goes to 0 the limit exists and is the same right independently on the choice of the way you are approaching 0 so the two expression have to be the same so we have this important relations and the calculation with h real we obtain it f prime of z0 when h tilde was i h and h real we have obtained what f prime of z0 is v y of z0 minus i u y of z0 since the two the limit is the same these two expression are the same and we obtain this important u x at the z0 is z0 sorry for z0 and u y z0 is minus these two conditions are called Cauchy-Riemann equations we have to so this is the real part and this is the imaginary part so u y is minus v x so we can summarize what we have done so far in the following proposition if f is complex differentiable that's enough then f satisfies Cauchy-Riemann equations that's enough. So the real and imaginary parts the two real valued functions associated to f have partially with respect to x and to y such that these two condition are satisfied okay well important consequences of this first of all here you see we have made a direct calculation and obtained this property from this we also say we go and can also say that basic rules of their derivations in the complex settings are inherited from the real case so that if you sum two differentiable function at z0 then the derivative is the sum of the derivatives right because of the the factor here we are derivating with respect to x one real value function okay then we have also liveness rule chain rule everything works fine this is very important so that we don't have to somehow we don't have we don't have to to to to get bothered with you know the the standard of stuff okay so we can use the same techniques we know from the real case probably most of you already know everything also in the complex case but well this is a next simple explanation now I want to characterize complex differentiable functions we have examples so the class is not empty which is also important to now you know on the empty set there are several theorems very beautiful theorems but normally we have a theory with something inside you're too young to appreciate what I'm saying but there are some cases in which you have very long statements and so the only condition was simply implied that there is nothing to to talk about okay okay so the the next task is the to prove the following to prove that any function complex value functions with real and imaginary part which are differentiable so in the in the real in the real sense which are the C1 C and which satisfy Cauchy Riemann equation they are in fact complex differentiable so that we can characterize complex differentials are those functions which satisfy Cauchy Riemann equations okay so for the time being we have the following power city expansion so the general functions as considered by ancestors our ancestors okay they were proved to be complex differentiable and we also know that they are still power city expansion and we can apply several times that we have important relation but okay this guarantees that we have something concrete to work with now starting from a complex differentiable function from this very simple calculation we have shown that in fact they satisfy some equations so the real and imaginary part satisfy differential equation which are called Cauchy Riemann equation now we start from the opposite side take two complex take one complex value complex valued function and consider the real and imaginary part to be sufficiently differentiable at least see something okay C1 so that they have the derivative and the derivative are continuous and assume that they also satisfy Cauchy Riemann equation can we prove that they are complex differentiable yes we can by using all this stuff we are we are characterized in several ways conflict differentiability okay so let me let me continue by saying the following now assumed f from an open set u of c into c components as I the real and imaginary part to be u and v to be such that u and v are c1 so continuous partial derivatives and satisfy Cauchy Riemann equations now I want to prove that this in fact implies that f well this is done at one point okay we assume that the functions are c1 and enable to c0 okay and the Cauchy Riemann equation are satisfied as c0 we want to prove that function f is completely differentiable at c0 okay now first of all assume assume take this incremental ratio at the point x plus i y so x y okay this is purely real this is a real valued function of two real variables x and y how can you express this difference in terms of linear part and the rest can you do this yeah okay how ux so h so you use the linear part right and so if you want this is different application of the notion of differential of gradient okay of a real valued function so we have that this is in fact ux x y times x by time h sorry plus ui x y sorry k plus call them call the function okay and this function here as a property that this tends to zero as h and k tends to zero good but we can say something more we can say that as h plus i k tends to zero so it is it tends to zero faster than h plus i k so it means okay the the speed is okay similarly vx plus h y plus k minus v x y as vx x y h plus v y x y k plus psi h k with a similar property as h k tends to zero remember this is well a node how do you say small o of h and k right or the modulus of h and k now we have these two relationships for the real and imaginary part of the function f we have to consider the following minus f of c over this is the complex increment we are considering i don't know how can we write it could be well w w has real part h and imaginary part k okay good so this is as before x plus h y plus k plus i vx plus h y plus k minus u x y minus i v x y over h plus i k correct so this minus this is something which can be expressed in terms of a linear part and the rest as we did before this this two u x plus h y plus k minus u of x y is in fact u x h u y x y k plus v h k right and similarly with an i in front we have the difference we evaluated before v x plus h y plus k minus v x y so this is plus i v x x y h plus i v y x y k plus sorry say h k over h plus i k and so far we have only used the regularity of the function not the fact that the functions u and v satisfy consumer equations but now we split the two well we don't consider five sign because they're tending to zero as h and k tend to zero so we're not interested in this we're interested in this part here and we have u x x y h and then I remember that v y u x and v y are the same right right so I can factor out u x and take i k h and i k and then I have also that u y is minus v x correct so u y is minus v x so here I have minus v x plus i v x h plus five sign sign h plus i k but this is not important because this will tend to zero right now minus v x is also plus i squared so I factor out i and I obtained finally that this is u x x y I simplify this and this then I have plus i and I collecting i I have v x h plus i k v x x y h plus i k over and then what is left is here in other words when consider this incremental ratio which is the general setting with the assumption that u and v real and the minor part of the function have satisfied Cauchy-Riemann equation of course they have to be at least with partial derivative otherwise you can even consider to solve the the to be to be satisfying the Cauchy-Riemann equations we obtain here that as h and k tends to zero or as h plus i k tends to zero this tends to zero and what is left what remains here is precisely the derivative of f as in the previous expression so the derivative of the real part with respect to x plus i the derivative of the imaginary part with respect to x so we I consider concluded that this proof and then any complex value function with sufficient to regular real imaginary part satisfying Cauchy-Riemann equation is in fact complex differential and this is one important consequence of the definition another important consequence is the following oh yeah assume that the function f as real imaginary part turn out to be of class c2 u okay so f is if i sorry u i use capital u for the open set and normal letter u to be okay i'm sorry but well this means a set and this means a function in any case u and v both of class c2 so they have also second partial derivatives with respect to x and with respect to i there were two y now consider this a second derivative of u with respect to x is also this according to my notation okay in short and also this for instance okay will be the notation i use so far correct x and y x and y okay and of course all the others y x and y y all right what is this let me see if you are familiar with this this is the laplacian of what sorry ah sorry yes true the laplacian of you sure sure sorry the laplacian of you is normally denoted by okay this symbol this is the laplacian this second order differential operator is called laplacian this is just a name how are called the solution of a laplacian equal to zero harmonic functions very good so this is completely real stuff right okay what is one of the important consequences of the fact that u and and being the real imaginary part of a complex value factor they satisfy Cauchy-Riemann equation those are consequence we obtain that these two functions are in fact harmonic functions how can we see this but remember that we have this relation u of x is v of y okay all right the Cauchy-Riemann equation once again and u of y is minus v of x if i'm not mistaken right therefore if i take u x x this is as v would be y x okay and if i take u y y this is like minus v x y if i sum up the two on the left hand side i'll paint this u x x y y on the right hand side i obtain v y x minus v x y however if you are sufficient to regular that is to say c2 is enough they have to be with partial derivatives which are continuous that's not then schwarz schwarz schwarz-lehm the theorem implies that the order of derivation is not important for the problem okay so the partial derivatives do not depend on the choice of the order of x and y so this is zero this is the same how do you call the theorem i call it schwarz theorem it should be short or something oh there are several schwarz in mathematical history there will be also schwarz-lehm we'll see in in complex and otherwise it's not it's not i don't think it's the same schwarz by the way the same man i mean in in germany it means black so it's a very common family name anyhow we conclude this for you and similarly i invite you to prove that this is also true this implies okay this is Laplacian Laplacian equal to zero is a consequence of fishery manipulation so in particular we can say that f complex differentiable implies its components harmonic functions as a matter of terminology complex differentiable is quite a modern way to say this to indicate this in fact for several reasons for historical reasons they are more often called holomorphic function okay so holomorph a holomorphic function means composite differential so that's the the standard terminology used in complex analysis harmonic function harmonic and harmonic functions are instead say a real calculus real analysis terminology okay harmonic and the also the the origin of this name this specific name is has some reasons okay related to property of a true use the the harmonic analysis you can you can find some well something related to wave equations and so okay question so far so you might ask why why all this long very long privilege to to very direct proof of something related to exponential and complex cosine and sine well because we also hope to use the tools we have so far to do so and in analysis without derivation without this specific tools you can do very little in fact this tells us a lot of things you are the entire theory related to the derivation all right well for instance so important consequence in real analysis is that from the study of this of the study of the in the increasing or decreasing of the function depends on the positivity or negativity of the derivative right because it is a real number it can be positive negative or zero unfortunately we cannot hope to have anything like this in the complex case because well complex numbers are not positive negative or but they can be zero can be zero so what if the function is such that it's it it is complex differentiable so it's harmonic and it has its derivative equal to zero what's your opinion about this yeah you know in an open it's constant like in the real case can you can we prove it directly without making many calculations no it's connected not convex yeah sure sure okay normally and this is quite natural to think of the sets as open and connected normally functions which are holomorphic are supposed to be defined in a domain domain means open and connected subset of c why is it quite natural well so far we have very few example but if you think that power series functions define functions which are holomorphic the radius of convergence can be big large or very small but it doesn't matter the the set of definition is a circle is a disk therefore it is an open and connected set yes so since we are using this property to be assumed for for the domain opening open set is natural because we have to to deal with the derivative so we have to have a neighborhood of a point right why connecting this is important as well well because probably the argument relies upon something which is topological okay so let us see it in detail if you don't mind or do you have do you have another another way to think about it any opinion well we should also try to find a shortcut remember the derivative the compass derivative is expressed in terms of the u x plus i v x right so if it is identically zero there is no increment of the real part the imaginary part of the function along the real axis but it is also minus u y sorry sorry minus v y so v y minus i u y so it is also constants along the imaginary axis and any two points can be joined by a polygon by a path with vertical or horizontal lines as segments this is also proof you are right but this is a step too too long for the moment yes all right yes but you have to prove the the cushy integral formula first right yes but in fact in in in a few hours say we have this special to the cushy integral formula because as a consequence of the cushy integral which is very peculiar in complex analysis will resemble something which is say the opposite way what we have done we will consider power set expansion and prove to be the complex differential or homomorphic now we start from the output and equivalently to have to have a really imaginary part satisfying cushy Riemann equation now we start from function which is complex differential or satisfied cushy Riemann equation is it true that it is complex analytic so it has a power expansion and it's true in fact but it spends some cushy integral formula so we introduce this important tool so I don't know maybe tomorrow or the day after tomorrow so but if you if you you Sure. OK. Thank you. OK. Your colleague is trying to use another argument. That is the very detailed argument to prove that the function is counter-differentiable and its derivative equal to 0 is in fact constant. He said, why don't we use the following argument? We consider the level set. We take a point inside this open and connected set. Consider the value of the function. So if I'm not mistaken, please tell me. So we consider u to be open and connected. Take a in u, f, of course, holomorphic, and assume that f prime, so the derivative is 0 for any z in u.  So I think that, well, a is, of course, not empty because there is at least one point. And since it is a level set, it's the inverse image of a function which is, I didn't say it, but of course it is continuous. We have partial derivatives as a component, so that it is continuous. Continuity is a consequence of differentiability also in the complex case. Because, well, I didn't say, I didn't write it explicitly. If you consider the incremental ratio and you know that the limit existence is finite, then, of course, f of z tends to f of z naught, as z tends to z naught, right? It cannot be anything else. Otherwise, the limit is not finite. Anyhow, since it is continuous, this is also, in short, f minus 1 of a. So it is the inverse image of a point, and it is a closed set. So a is closed. What is left to prove? If we prove that a is also open, we have a non-empty opening closed set, so it is the entire u, which means that for any z in u, f of z is equal to f a. So it discounts, and we're finished. So to prove this, we have, well, to prove that a is open, thank you for your observation, we do the following. So we have u, we have a, and we take an open ball of radius r centered at a, and with a small radius in such a way that it is contained in u, OK? I want to take z in b a of r and prove that z is actually in a. If this is done, I'm finished, because I've proved that any point in a has a neighborhood in a. And so a is also open. So how we conclude? So this is the last part of the argument. OK, I take z, and I take a. And I take the line, the segment, connect z to a. So remember that z and a can be considered as vectors. So I can use, well, standard stuff like this. Well, what is it? T z plus 1 minus t a, right? And t real, actually t real in between 0 and 1. So when t is 0, this number here is a. When t is 1, this number here is z. So this is the segment connecting a to z, right? Then I consider f of t z plus 1 minus t a. And call it g of t. So a and z are given. I take z in the ball. a is the point we'll consider from the beginning. So what varies here is the parameter t, which is real parameter. So g is a complex value function defined on 0, 1. Good. Now what can I say about, well, one point I forgot to say. Why can I take t z plus 1 minus t a to be a good variable for f? Because I'm considering z to be in the ball centered at a, right? It is complex, correct. So the entire segment is in the ball. And so f is defined for any t. So g is well defined. Now I can see that g of t minus g of s over t minus s. So this is incremental ratio. Is it for g?  Then I notice that if I write t z plus 1 minus t a, which is the first variable for f in here, then I subtract what is in here minus s z plus 1 minus s a. I obtain, notice that I have a and a and a minus in front. So n, a and a cancels where I have t z minus s z. And then I have minus t a plus s a. So I have t minus s z minus a. Are you with me? Just doesn't matter. It's very simple calculation.  And then I multiply for the same amount, and I do I have a t minus s. Right? So this cancels this, and this is t z plus 1 minus t a minus s c plus 1 minus s a. And this is an incremental ratio for f. Now when t tends to z, this tends to 0. And this tends to what? To f prime at s z plus 1 minus s a, which is 0. Correct? Good. So in the last few minutes, I conclude by saying that as t tends to s, we have the g t minus g of s t minus s tends to f prime of s z plus 1 minus s a times z minus a. But this is 0. Because this point is inside. It is in u in particular, but it is 0. Which means that this function here is constant. And g of 0 is g of 1. So g is constant. And g of 0 is what? f of a, and g of 1 is f of z. That is what I wanted to do. And this concludes very good. Because we are assuming that the set is open and connected. So we found an open and closed set inside a connected set and no name. So it is entire. Now how can we apply this to have the relationship we were looking for? So consider this function as a function of z. z is a complex number. And e, the power z, is the complex exponential. So this function turns out to be a complex differential. And this turns out to be a complex differential as well. Correct? Because, well, you just shift the center of the power expansion of u of z. So we have all the hypotheses. So what about the derivative? Is it 0? Are you all? It's e to the power c. Correct? But not because we are using the property. Because we made a calculation. I'm not cheating you. I'm saying, well, if you want to be very formal, use what we know, know what we want to know. You're right. But it's just a test just to verify that we can do this. Now we apply Leibniz's rule. Because we have a multiplication of complex differential function. And we apply Leibniz's rule. So it means that we have to differentiate first this times this function. And then this function times the derivative of the other. But the derivative of e of z, for definition, is e of z. So it's e of z times c, c minus z, plus, then I have e of z, the function, times the derivative of this. Remember that, as I said, we can use standard rules. So we have a composition of functions. Here we have minus 1 in front. Minus 1 is a coefficient. And then e c minus z. So it is 0. So the function is, in fact, constant. And in particular, as I said, 0 is e c minus 0. And from 0, it is the value that 0 is c of the power c. So we have this relationship. And this concludes today's lesson. Then we have that e z times e c minus z is e c. I call z equal to a, c minus z equal to b. So c is a plus b, correct? So we have a e a times c d, right? So in particular, just if z is x plus i y, e z is e x plus i y, and it is x times e i y. But e i y is, this y is, remember real number, as x, times cos y plus i y. And once again, we obtained something interesting. We obtained that the modules of v of z is e x to the power x, which is a non-negative real number. And y represents the argument. Pardon me? Pardon me? e to the power x? Sorry, it's strictly positive, right? Correct, correct. So e of z, correct. Sorry, sorry. Right, sure. That's what I want to say. e of x is strictly positive. And therefore, we can say that e of z is never 0, never vanishes. And as you will see, this is somehow the extremal properties. The only function, well, we'll see what happens in the future. But it's the only value which is missed by the function e to the power z, complex play, right? So I think we can stop here and see you tomorrow morning. I prepare the copies for you tomorrow, if you don't mind.