Extending the Harmonic Numbers to the Reals
ฝัง
- เผยแพร่เมื่อ 28 มิ.ย. 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
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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 TH-cam 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
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
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
Great visualization, great pacing, interesting topic!
Next step: generalize it to complex numbers
Then find all the zeros
The same formula works
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.
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.
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!
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).
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 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.
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_
I am so glad for 3b1b's Summer of Math Exposition, great videos are popping up everywhere !
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!
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
Excellent video! Watched through the end, I was basically hypnotized by the quality of the manim animations!!! So cool!!! Keep up the good work
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
Beautiful animation, great explanation, fantastic video!
Can't wait to see more of you
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
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.
Wow! What a great watch! Thanks so much for putting this together!
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 😁
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
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! :)
the quality of this is astounding, i've never subbed to a channel this quickly.
Really great video. Good luck with SoME!
You are a great narrator, I never was excited about sums that much before.
Amazing video! Good job, I can’t believe it’s your first one! Keep it up man, I’ll be coming back for more!
I love this kind of "extension" video. Keep them coming
Size does matter, doesn't it?
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 work. can’t wait for the next one!
Great video! Really informative! I’m excited to see what you do in the future!
Great start to your channel. Interesting topic, top class qualify. Subscribed and notification set in anticipation of the next one!
Wonderfully made, You sir have earned my subscription, I really hope to see more from you
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.
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.
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.
Amazing video! Extremely clear explanation and a very well chosen topic. Simple yet extremely slick arguments. Subbed!
Amazing presentation, looking forward to more stuff!
You deserve more subscribers, this video is extremely well made!
Favourite SoME1 video I've seen so far, really really good video 🥳🥳🥳🥳
Excellent video. Looking forward to future content
That was very well done. I'm looking forward to going back and watching more of your other videos.
This is great content, I hope your channel grows!
Wicked animations. Loved the video!
Absolutely amazing video!
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.
This was great. Pretty sure this is the first time I've subscribed to a channel that has only one video!
Great video, best math content I've seen in a while
This is the best summer of math video! It deserves to win
This is an excellent video, well presented and well explained. I'm looking forward to your future videos.
What a great video! Enjoyed every second
Wonderfully explained, math is so beautiful. Looking forward to your new content, you'll surely make it big!
Nice work. Can’t wait for next episode…
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
Fantastic. I subscribed, can't wait for more!
That's such a cool trick to understand intuitively, you made it very simple. I hope you plan to make more videos like this!
You did a very good job with this video, nice work man.
Just found you today and I honestly loved the video, I hope to see more in the future.
Very well explained. I am amazed about the influence of infinitesimals in modern math.
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So amazing explanation!!! Thanks for share it
this is now my favorite math 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!
I was searching exactly this video! I loved it, tysm
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.
this video and this channel are amazing, from france : congratulations for your job
Cannot believe this channel is so underrated. Keep working, and you'll be famous.
You're a legend. Kudos to you!
I loved this video, everything about was just perfect 🙂
This was a joy to watch.
Wonderful!! More videos, please! :)
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!
This video is beautiful!
Gg mate. Insanely good and interesting video with an excellent explanation !
That was really nice! I hope new videos are coming
Amazing video! Great job!
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.
That’s it, I’m binging this entire channel.
First SoME1 video I could actually understand!
Def one of the better math channels in youtube. keep it up! it would be cool to do some computer science videos :)
Very interesting and well-done video
enjoyable video, hope to see more in the future
Brilliant man, you are not showing math but how to think and investigate thats more important than the math fact itself
Please make more...
This is really great stuff!
That is something that I had been wondering for a long time
Can't wait to see more
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. TH-cam lacks great content like yours
Finally someone who make video about this topic. And that method is awesome 👌
Great
Waiting for more videos like this one
Yup, definitely subscribing. Will still watch the other 2, of course. :)