Extending the Harmonic Numbers to the Reals
HTML-код
- Опубликовано: 28 май 2024
- The harmonic numbers are the partial sums of the harmonic series - sums of whole number reciprocals. This video explores how we can extend their domain to the entire real line.
The animations for this video were made with the community edition of Manim (www.manim.community). Huge thanks to everyone who worked on the library, as well as the members of the Discord server who answered my many questions.
This is my entry for the Summer of Math Exposition 1.
All the music in this video is available on my SoundCloud: / lines-that-connect
00:00 - Intro
1:45 - Graphing the Harmonic Numbers
2:47 - A Recursive Formula
4:23 - Using the Recursive Formula
7:33 - The Super Recursive Formula
8:52 - Finding the Interval
11:27 - Example: H(0.5)
11:59 - Deriving the Solution
13:10 - Graphing the Solution
#SoME1
Thanks for watching everyone!
I obviously hoped to see a positive response, but I didn't think I'd actually get thousands of views on my first video! It makes me really happy to see so many people enjoying this topic. The Harmonic Numbers are the tip of an iceberg into some really cool math, and I hope to eventually share that whole journey on this channel.
I've been working on another (completely unrelated) video, but as I'm doing this between work and other personal projects, I can't give any estimates as to when it will be ready. I just wanted you to know that there definitely is more on the horizon!
oh yeah you gonna blow up. havent watched these kinds of videos in a while and youtube just puts you on my home page...pretty sure youre gonna have a huge spike of views (or just had and the algo still loves you)
been a pleasure to watch too
keep it up w these kind of videos, rlly good
You definitely deserve it, production quality is what matters in these kinds of explorations
Keep making content man, i will absolutely watch it all... love this kind of videos, already subscribed. Good job! 👍
Are you telling me the points don't follow a logarithm? I'm so disappointed.
Guys, get ready, we are literally witnessing the birth of a new legend in the math-educational RUclips scene. This is going to be great, i can feel it
i agree
I feel so, too. It reminds me of when I first went crazy for 3b1b
I agree. He has everything 3blue1brown has.
no truer words were spoken.
This aged like milkshake
This was excellent. I've watched a lot of the SOME1 videos and this is easily one of the best. Do you expect to release any more anytime soon?
Thank you!!
I can't really make predictions on how soon, but I definitely have more in the works!
6:00 that pun was so good! sneaky, unobtrusive, and perfect in context. subscribed.
I plugged the formula into Desmos on my phone and it crashed lol
Nice video. One very small point I would make would be at 10:55 on your justification of using a straight line for the interval, as opposed to some squiggly up and down one. I believe it would be better to justify it under the intuition that the harmonic functions extended to the reals should be a strictly increasing function, rather than the rather loose "it is the most natural. It's probably being overly pedantic, but mathematics is all about rigour
Even if you add the argument of strict increase, it is not necessary for the function to be a straight line, there still is some intuition to it. The argument probably should be that if f(x) does not change much also every derivative of f goes to 0. And even this is hard to proof as you can not do the limit of the difference quotient because you do not have infinetely close points for which the function is defined.
Anyways, as we are looking for a function that behaves kind of natural in some way, we can just take that as a definition of "natural": that every derivative disappears on the interval between arbitrarily close points.
(Now as I am writing this I wonder, if you could proof that under the condition, that every derivative should be continous in any point - but I am not enough of a mathematician to try that myself.)
@@ery5757 well for the really big numbers in order for it to be monotonically it needs to be close to a straight line, otherwise it would go above the second value/below the first. It may not be exactly but as you get bigger and bigger it gets closer and closer. This is expressed by the statement that for any x
lim[N->inf] H(N) - H(N + x) = 0
@@JGHFunRun Well the thing is that this does not hold for any x, but only for x
@@ery5757 yea um x was (heavily) implied to be finite when N explicitly goes to infinity. You’re being extremely pedantic and I’m not even sure why - for any finite x, x inf] N since N >> inf. Heck in the video he explicitly said something equivalent to “for very large N in relation to x” N>>x. Do you think the statement of “very large N in relation to x” stopped applying at some point? Because if so tell me where it stopped applying. I’ll wait
Oh, and if you really want to be pedantic it’s only if |x|
Yeah that part bothered me a bit too
I thought it was just a coincidence the harmonic numbers looked like a discontinuous logarithm, but actually, the natural logarithm of x can be expressed as the integral between 1 and x of the reciprocal function. So the nth harmonic number is kind of like an approximation of the integral definition of ln(n), using integer width areas.
Also ln(x) is approximately equal to H(x) minus Euler-Mascheroni constant, for large values of x.
Surely the limit of the slope of this function also is 1/x like the ln
@@meraldlag4336 Yep, if you look at Sir_Isaac_Newton's comment, ln(x) is approximately H(x) - the Euler-Mascheroni constant, so it's derivative (slope) is 1/x
holy crap that's genius
When I first learned about sigma notation and the Riemann Zeta function, I spent the next 5 nights playing on Wolfram Alpha, and it was fun. It was very similar to what you've done here! Thanks for the connection with the digamma function, too, I never knew what that was.
Wau, that's pretty cool
(Another name for the archaic Greek letter "Wau" is "Digamma")
5 nights at wolfram alpha
Next step: generalize it to complex numbers
Then find all the zeros
The same formula works
Great visualization, great pacing, interesting topic!
12:28 Math is crazy in a way that you can have an good intuition about every separate fact but when combined, knocks you back to the start and realize you don't really comprehend the whole story.
It was hard for me to silence the voice in my head screaming "It's just a natural log! Use an integral approximation!", but this was definitely worth it! Great video :)
"Just a log"?
Euler-Mascheroni-constant: "Am I nothing to you?"
It's in the name: integral _approximation_
Amazing first video - I can't wait to see what's next!
Not taking into account the reasons why you'd prefer one notation over another, I think it's a bit curious though how computing the first numbers gets more and more difficult as you get further into this video - I mean at the start it is just evaluating fractions, and by the end you have to calculate an infinite sum!
Thank you!!
The infinite sum is great because you can take derivatives and integrals, and in general a bigger domain means more potential to find interesting patterns.
But if you ask me for H(20), there's no chance I'm adding infinite terms when I only need to do 20! (Plus, the sum converges pretty slowly, especially for larger numbers)
Oh! Can you can do calculus on sums? I always thought that was kind of a dead end!
It's still pretty nuts how do calculate the Nth harmonic, you have to add the first N reciprocals. Coming from a computer science background, I feel the best way to implement this would not be one of the mentioned formulas; a lookup table is too sparse, the classic definition gets too slow too quickly, and the summation takes too long to converge. I'm kinda baffled by how many ways there are do calculate this one thing!
@@zanzlanz i'd personally see whether the integral of the geometric series was viable (depending on how expensive non-integer powers are)
@@LinesThatConnect Well, of course you only need to calculate 20. After all, the sum telescopes for positive integer inputs.
@@zanzlanz You can do calculus on sums; you just have to make sure there aren't any shenanigans going on. It's like with mixed partial derivatives; usually they're equal, unless the function is going nuts at some point. (For more eye-glazing details, look up the Monotone Convergence and Dominated Convergence Theorems.)
As for calculating harmonic numbers, there are asymptotic expansions which work pretty well. For example, H(n) is about ln n + gamma + 1/2n - 1/12n² + 1/120n⁴, where gamma is the Euler-Mascheroni Constant. You can continue extending this approximation, but unfortunately for any finite n the terms start getting larger after some point (this is a general problem with asymptotic series). However, the computational advantages are enormous. Finding the billionth harmonic number by summation would take literally billions of calculations and only give you about eight places of accuracy past the decimal point; the above formula (with a previously calculated value of gamma) yield over fifty places after the decimal point (the first omitted term is of order 1/n⁶, or one part in 10⁵⁴).
Damn, I am a high schooler who is interested in maths. Gamma-function, Digamma function, harmonic numbers and extension of series from integers to real(and complex) numbers are definitely one of my favorite topics. Honestly, this is the best video about harmonic numbers I’ve seen so far.
Dude, this video is one of the best produced math videos I’ve seen in a while! You have the elegant animations of 3Blue1Brown while also touching subjects I’m more interested in, so bravo, honestly! Keep up the good work, I’m excited to see what you have in store in the future!
I discovered this while "discovering" the stamp collector's problem for myself. There's a fun approximation that's useful if you're trying to find how long it takes to get items from a random draw.
Wow! I used essentially this idea to find a continuous extension for the factorials! I know the gamma function already exists, but doing it this way gives you a different formula that happens to give you exactly the same thing as the gamma function. I never considered doing the same thing with other functions, so cool!
I didn't even notice 15 minutes have gone by. That's how good you are at explaining things. Awesome, man! Keep going! The world needs more lucid explainers like you!
I am so glad for 3b1b's Summer of Math Exposition, great videos are popping up everywhere !
Nice! Though we *can* get H(0) = 0 from the original sum-of-reciprocals definition too: for that case the sum has no terms, and an empty sum equals zero (likewise an empty product equals one). Don’t be afraid that the sum is “from 1 to 0” in that case-that’s equivalent to being from 1 inclusive to 1 exclusive, which then makes more sense to have zero terms in it.
Excellent video! Watched through the end, I was basically hypnotized by the quality of the manim animations!!! So cool!!! Keep up the good work
Absolute brilliant video, I love those exploration videos that take you through a journey of discovery. And this video does it perfectly. Can’t wait for the next video
As someone who's interested in somewhat niche generalizations like this, this video was really interesting! It was very well explained and visualized and easy to understand
I love your teaching style! I hope you enjoyed making this video as much as I enjoyed watching it, cause if you keep up this level of quality, you WILL find success here 😁
Got excited that I'd found a new cool maths channel. Got disappointed when I realized it's only your first video. Got excited again when I realized it's only your first video. Please keep it up!
This is amazing, and I'm really excited to see you're upcoming videos
Beautiful animation, great explanation, fantastic video!
Can't wait to see more of you
I love this kind of "extension" video. Keep them coming
Size does matter, doesn't it?
Thanks mate, It was really informative and the way you presented it felt quite intuitive. Keep up the great work. Waiting for the rest of your videos.
the quality of this is astounding, i've never subbed to a channel this quickly.
I'm usually a casual viewer when it comes to math videos but man... videos like this makes you appreciate how beautiful math is. Really cool video, hoping to see more! :)
I'm very happy that the summer of math is/was a thing there are so many really exellent videos emerging recently (and especially also so many new great people doing interesting educational videos). I really hope more people see this as I find it very well made
Wow! What a great watch! Thanks so much for putting this together!
Wow, I rarely find a new channel to which I’d like to subscribe, but I’ve «never» been so quick to hit the bell icon too. 0: Extraordinary success lays just over the horizon for this channel. Keep it up!!
Great start to your channel. Interesting topic, top class qualify. Subscribed and notification set in anticipation of the next one!
Great video! Really informative! I’m excited to see what you do in the future!
You are a great narrator, I never was excited about sums that much before.
Really great video. Good luck with SoME!
Amazing video! Good job, I can’t believe it’s your first one! Keep it up man, I’ll be coming back for more!
Wonderfully made, You sir have earned my subscription, I really hope to see more from you
Great video! Honestly this was presented so well I decided to go to wiki, and start deriving some of the stuff that was presented there as well as following the steps that were taken in this video. Really well done :)
Great video! Looking forward to seeing more from this channel.
Amazing video! Extremely clear explanation and a very well chosen topic. Simple yet extremely slick arguments. Subbed!
That was very well done. I'm looking forward to going back and watching more of your other videos.
Amazing presentation, looking forward to more stuff!
Excellent video. Looking forward to future content
This is an excellent video, well presented and well explained. I'm looking forward to your future videos.
Very nicely done. Looking forward to more!
That's a great observation about it extending the domain. Seen that formula many times and literally never thought of it that way.
This is great content, I hope your channel grows!
great work. can’t wait for the next one!
You deserve more subscribers, this video is extremely well made!
Absolutely amazing video!
So amazing explanation!!! Thanks for share it
Wonderfully explained, math is so beautiful. Looking forward to your new content, you'll surely make it big!
I can see that you took great effort in making sure that every step and intention behind it is clearly conveyed. When you're an expert yourself, it's hard to know what steps would be difficult to digest from a novice point of view. I enjoyed watching this video very much.
That's such a cool trick to understand intuitively, you made it very simple. I hope you plan to make more videos like this!
Please make more videos like this. I want to know the mystery behind why graphs look the way they look for particular equations. This is amazing
This is excellent! And the first video on this channel? This sort of thing has a sizeable audience, I'm sure, and it's distinct from what I'm seen elsewhere. Keep it up!
Just found you today and I honestly loved the video, I hope to see more in the future.
Favourite SoME1 video I've seen so far, really really good video 🥳🥳🥳🥳
Wicked animations. Loved the video!
Damn this video is so impressive! Especially for the first video, it feels like the product quality matches 3b1b. I look forward to your future videos!
This was great. Pretty sure this is the first time I've subscribed to a channel that has only one video!
Very well explained. I am amazed about the influence of infinitesimals in modern math.
You did a very good job with this video, nice work man.
That was really nice! I hope new videos are coming
This is the best summer of math video! It deserves to win
Great video, best math content I've seen in a while
I read the title and thought about it for a day and came up with a different extension H(x)= integral of (1-t^x)/(1-t) for t from 0 to 1. And then I learned the completely different approach from the video. Time to try proving they are equal haha...
Btw, a missed opportunity of actually computing H(0.5). I think it is 2-log4
This thing was so dope. You sir are really cool
Fantastic. I subscribed, can't wait for more!
What a great video! Enjoyed every second
Nice work. Can’t wait for next episode…
That’s it, I’m binging this entire channel.
This war very well done! Please continue making these nice math videos! 👌
You have explained the generalization of the Harmonic function so well that I can't wait for your explanation of the Riemann zeta function.
But take your time and do it right.
I loved this video, everything about was just perfect 🙂
Hi,
I have watched nearly all of 3blue1brown's videos, and yet I still think yours was one of the best I have ever encountered. I am *begging* you to upload another one, you could easily match 3b1b's videos in quality, and you in fact already did, if not better. RUclips lacks great content like yours
I was searching exactly this video! I loved it, tysm
this is now my favorite math video!
Wonderful!! More videos, please! :)
Great video. Thanks a lot!
It's beautiful, the intuition behind generalized harmonic numbers which is explained in video is pretty cool. Especially the idea that going to sufficiently large N and then noting the fact that H(N+x)=H(N) was really lovely. I am really curious to know the source of this.
Amazing video! Great job!
Very interesting and well-done video
this video and this channel are amazing, from france : congratulations for your job
The highlight of the beauty of his articulation skills in this video was the part where he explained how there is a well-defined value for x=1/2 at very very large N since there is a smooth line meaning things are approximately the same (especially knowing about limits from calc) then he visually showed taking regular intervals back from the recursive relation meaning we have a value at every step of the way. It’s amazing! I have gained a lot from this experience. I now see mathematics more as a puzzle where we are trying to think about a very clever way to construct our function in a way we can think of. It blows my mind to see ingenuity with full clarity and appreciating the trick he pulled because usually textbooks pull tricks and it isn’t full understood/appreciated bu I think the visuals and simple words he used really helped here. Wonderful video! Wonderful! Amazing! So cool! I like this much much much more than 3Blue1Brown which usually feels too fast and cramped and confusing.
I even liked my own comment. Wonderful video! Very easy to digest!
Thanks for a cool, well made video. It is great to see high quality math pedagogy on channels like this one and 3brown1blue. When I saw the strange behavior for x
You're a legend. Kudos to you!
This video was great! On par with (or maybe even better) than a 3blue1brown vid!
Neat👍😎
Amen to that
This video is beautiful!
Gg mate. Insanely good and interesting video with an excellent explanation !
Cannot believe this channel is so underrated. Keep working, and you'll be famous.
Very nice video and explanation!
Def one of the better math channels in youtube. keep it up! it would be cool to do some computer science videos :)
Yup, definitely subscribing. Will still watch the other 2, of course. :)
On paper, I knew how to derive this infinite sum. But this video did a fantastic job of making it much more intuitive. Well done.
enjoyable video, hope to see more in the future
First SoME1 video I could actually understand!