I Proved Liouville's Theorem (Or Did I?) 

@kgangadhar5389 Thanks so much for your support!!! Please tell me what you liked and what you didn’t specifically, so that we can publish more of what you guys want and what interests you! 😎

@@dibeosOnce again, Thank you for the reply. I like videos related to complex analysis. I am into partitions, modular forms, combinatorics, abstract algebra, etc. I like to see videos where problems are solved on board or using any other tools, which gives me time to navigate with the author.

@@kgangadhar5389 Nice! Thanks for telling us. We will definitely post more proofs and general explanations about these subjects you mentioned here on the channel. 😎

Hi Tletna, thanks for the comment! Let me address each of your points: 1. Let’s say that this proof (and others in the channel) are standard. But I did explain “deeper” some parts of it, especially the intuition behind them, so that the pre-requisites to follow along would be the bare minimal possible. I did my best. 2. This proof is not original, but since mathematically proving something requires being rigorous enough, and since I’m a human being who does mistakes, I might have been sloppy somewhere. I don’t know. That’s why I ask the audience, so that I learn math and you guys learn math. And if I do any mistakes, please point it out so that I can get better. 3. The extension of 𝑟 to infinity is a theoretical tool used to understand the behavior of the function over the entire complex plane (not only locally). When we extend 𝑟 to infinity, we are not distorting the shape of the circle but rather examining the function's properties as we consider increasingly larger circles. Its circular shape is preserved, only its radius changes. When we take the limit as r→∞, we find that the bound on |f'(z_0)| must approach zero, implying that the derivative is zero everywhere. This shows that the function must be constant. I don't know if it is clear, but the concept of extending 𝑟 to infinity is just a way to capture the global behavior of the function and use the properties of holomorphic functions to draw conclusions about their nature. The circle's definition and its role in the integral remain intact; what changes is our perspective, allowing us to apply the bound universally, rather than locally. Let me know if it is clear now, and what parts of the video you found confusing, so that I can explain better next time.

@@KELLY-hn8mk yesssss, I was hoping somebody would ask!!!! 😎

@@RU-vid_username_not_found I just looked it up. It is quite interesting, but how does it relate to Liouville’s Theorem? 🤔

@@dibeos No, That was just a suggestion of something I would like you to prove, if you please.

@@RU-vid_username_not_found aaaah got it hahah sorry, I didn’t understand. Yeah, it’s a great idea! I’m gonna include in our list of ideas right now!!!

@@Satisfiyingvideo-uu9pw no, I never heard about it… what’s that?

@@dibeos you can search it in youtube. You can watch video about it. It is a mystery of number. And it has no proof.

chill

@@jay31415 hi Jay, of course I’m not trying to claim I’m the first one to prove it. It is just a cool title structure I thought of for this new series of videos in my playlist “Proofs”. I hope you liked the content as much as the title 😎