The Trig Hiding Inside the Factorials (And the Harmonic Numbers)
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- čas přidán 16. 05. 2024
- In this video, we build on my last two videos by exploring connections between the gamma function (the extended factorials), the digamma function (the extended harmonic numbers), and trigonometry. We derive Euler's Sine Product Formula, which we then use to prove the gamma and digamma functions' reflection formulas. Finally, we derive a related formula for calculating cotangent.
Watch my previous two videos here:
Extending the Harmonic Numbers to the Reals: • Extending the Harmonic...
How to Take the Factorial of Any Number: • How to Take the Factor...
An Elementary Proof of the Sine Product Formula:
www.researchgate.net/publicat...
The animations in this video were made with Manim: www.manim.community/
Music credits:
Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Night Music by Kevin Macleod
Space Chillout by penguinmusic
river - Calm and Relaxing Piano Music by HarumachiMusic
Surrealism (Ambient Mix) by Andrewkn
... And a couple of my own songs:
/ the-fog
/ thanks-for-watching
Chapters:
00:00 Intro
0:43 Background and Notation
3:24 The Digamma-Cotangent Connection
5:09 The Gamma-Sine Connection
6:04 The Sine Product Formula
9:59 Proving the Gamma-Sine Connection
12:22 The value of (1/2)!
13:07 Proving the Digamma-Cotangent Connection
14:21 The True Logarithmic Derivative
15:52 An Infinite Sum for Cotangent
17:46 Final Thoughts
Some notes/clarifications that didn't make the cut:
2:29 It was a toss up between the Gamma function and the Pi function. (Pi(x) = Gamma(x + 1), so Pi(n) = n!) I much prefer the Pi function, but I've only seen it used a handful of times, while the Gamma function gets all the attention. I eventually decided it would be better to introduce people to the version they were more likely to see in the wild rather than using a notation they might never see again, so I went with the Gamma function.
5:21 You can get Greek letters and other LaTeX symbols in Desmos by typing them in a regular text editor (e.g. "\Gamma") and copy/pasting them in. That's how I got the Gamma there.
7:33 I put the pi's before the x's, even though the other order might at first be more natural -- (x - pi) instead of (pi - x). However, with (x - pi), the graph would be flipped upside down, which would be easily fixed by dividing by -pi instead of just pi. But I jumped straight to (pi - x) to save a bit of time.
15:00 I went back and forth between a few interpretations of the logarithmic derivative.
If we had used the complex logarithm, everything would have just worked. But the background is a bit too complicated in my opinion.
Another way would have been to use ln(|f(x)|), which still has all the nice logarithm properties, but it's less interesting and it doesn't extend to complex numbers very well.
16:57 Technically, we also need to show that the result converges uniformly to prove that this step is valid. But I'll leave that as an exercise for the viewer :)
Nice commutative diagram at the end.
5:26 i actually found this while messing around with this on desmos as well, i was looking into approximations of the factorial function and ended up trying 1/x! which i found to have a derivative everywhere (other than negative integers) so i messed around and got the sinc function (sin(x)/x) then shifted one of them over to get the sin function, i was so shocked by this discovery and was really confused but never really looked into it for some reason, really happy i finally got an explanation to it
I have a question for you. Do you have an email I can contact? Thanks!
With the Pi function, the reflection formula becomes 1/Π(x)Π(-x) = sin(πx)/πx. While the left-hand side is _way_ nicer than the Gamma version, the right hand side looks like it might not be... _at first glance._
Let's look back at the sine product formula: sin(x) = x ∏k=1→∞ (1 - (x/kπ)²)... wait. What's that x doing out front? Let's move it to the other side and see what happens: sin(x)/x = ∏k=1→∞ (1 - (x/kπ)²). The product now looks nicer, but the left-hand side... _That looks just like the reflection formula!_ sin(πx)/πx. Just to see what it's like, lets sub in πx to the product formula, since it looks like it should be able to cancel some thing: sin(πx)/πx = ∏k=1→∞ (1 - (πx/kπ)²) = ∏k=1→∞ (1 - (x/k)²). Beautiful. This function, sin(πx)/πx has its own name: the normalized sinc function. Often just written as sinc(x), or sometimes sinc(πx), where sinc(x) is instead the unnormalized version sin(x)/x, so you could call the normalized form "nsinc(x)" instead, but that's just something I made up.
If you want a separate example of the sinc function (normalized or not) being useful on its own, _I dare you to take its Fourier Transform!_ If you haven't seen it before, you probably won't predict it.
Hi, can you pls find any connections between amount of isomers of alcanes for instance, i only found out that ratio between two adjacent numbers in line that does not include stereoisomers is suspiciously approaching exponent, and if including stereoisomers it is approaching π
This channel is one of the best things SoMe has spawned! As a mathematician myself, I really do appreciate the way you explain things, intuitive and visually beautiful but still with healthy respect for (and remarks about) the mathematical rigour! Love it!
What is SoMe?
Edit: nvm! Summer of math exposition!
@@therobertguy2436 Its SoMething amazing
your feelings are irrational
some?
The legend is back ❤
I’d never FORMALLY heard of Euler’s product formula for sine before… but I’d actually wound up discovering it for myself. It’s always nice to know that I would’ve been a trailblazer if not for that meddling Euler.
Haha, I did the same thing but for the product representation of sin(x)/x. Feels good huh
Same
i have probably a dozen, completely-full 5-subject notebooks of math ive done, and i'm kinda hoping someone digs it up after im dead and theres something nontrivial in there lmao
Math should always be presented like this; it's thoughtful, intruiguing and simply aweinspiring. The derivations lined up so nicely that it doesn't even feel abstract anymore.
Awesome! It's incredible how seemingly unrated mathematical ideas come together in unexpected ways!!
I love that you pivot from frustrated that the digamma function is defined with x-1 to thankful when things canceled out xD
EVERY YEAR, HE UPLOADS A BANGER
I love how this builds on your first two videos without being too overwhelming. Great job with the explanations (and I can’t wait to see what you cover next)! :)
"And a couple of my own songs" Naturally. Amazing video; textbook quality. It is work like this that's going to turn Manim into the new LaTeX.
I love these connections between weird mathematical functions that seem to come out of nowhere, so I can only imagine my excitement if I somehow managed to discover that reciprocal product being a sine wave after watching your previous video. This is amazing, thank you for these videos man.
Bro your content in on an other level .Please don't stop uploading
I love your channel! Not only do you present interesting connection, but the way you present those connections is really unique and ”simple”. What a great find :)
This is an absolute gem of a channel, I can't wait to have my mind blown even more in the next videos.
I absolutely love these videos! It's crazy how fantastically intuitive the explanation and presentation are
I genuinely hope this channel can grow as big as possible as quick as possible. Wonderful content❤️❤️
This is incredible! Better than 3b1b, because it’s straight to the point. I’ve been waiting for you to post a video for ages. Keep up the good content!
I really appreciate the subtitling, the sound design, and the visuals. I never had to adjust the volume, the music never overpowered your voice, the animations were very easy to follow, and the subtitles were spot on. Oh and the maths was kinda mindblowing!
It's nice to see elegant connections between different math concepts. I like your videos, hope they continue.
Very excited for more content! I loved the last two
I've always heard that Euler's identity is "the most beautiful formula in mathematics", but after watching this video, I have to say I think the digamma reflection formula is a serious contender. It's one of those surprising connections that's obvious in retrospect that make me love math. Thank you for such a great video.
Since the gamma and digamma functions were both offset from their discrete counterparts, I decided to check the reflection formula for the extended harmonics and got H(x) - H(-x) = 1/x - πcot(πx).
The left-hand side is quite a bit nicer than the digamma version, but the right hand side not so much given that extra 1/x.
I cannot express the happiness that I felt seeing you posted a new video! I'm so excited to watch this
I watched 3 videos of yours back to back and I am still not bored. Great work. Thank you for making these videos.
I need these videos more than once a year-ish. I also think it would be cool to explore the gamma constant more in a future video, but you do you. I'm just enjoying the ride.
Keep up the great videos! Mark my words... You are going to be one of the greats! I'm calling it now.
Your videos really are great. You explain things very well, and at a very good pace. You always conclude the video reaching very satisfying and unexpected results. I love it.
I love your channel, I like that you derive the formulae. I also like that you use white text on black background, it doesn’t hurt my eyes like the inverse does. Also I like that you center the equation you’re working on, it makes it easy to see even on a small screen. Your voice is nice and I like that you say what the names of the symbols are as you’re showing them
Dude, i'm RE watching this two weeks later, after having watched it multiple times in a row as soon as it came out (subscribed with notifications hehe) -- it made perfect sense on the first watch because your explanation was so smooth and intuitive, but because there are SO MANY HIDDEN LAYERS TO IT! I've been doing this stuff for at least a decade.... and I *still* learned things from this video, and then learned more every time I re-watch to unearth more. Please please please, keep it up!!
This is simply beautiful. An awesome series of videos, great job!!
With simple small steps.... climbing the mountain, is only possible with an excellent guide, The view is breathtaking! Thank you so much!
Finally a new video after months! I really like your videos, please be consistent on your channel.
Awesome! You've managed to connect so many dots in my head that I'm finally able to grasp how all these concepts fit together. Thank you!
Would love to see something on how to calculate the incomplete gamma function (integral formula for gamma from 0 to x rather than 0 to infinity). Key applications: Gamma probability distribution, machine learning, and many more.
These functions are important when dealing with fractional calculus, such as finding the half integral of 1, which is 2sqrt(x/pi). The half integral of 0 is surprisingly 1/sqrt(pi*x).
Note that this is only really important because we have defined it to be that way. Cauchy's integral formula is true for analytic functions, and asserts that we can infinitely differtiate (and integrate) any analytic function. It happens to use factorials because of its iterative nature, which lends to using the gamma function.
Fractional derivatives and integrals aren't some innate thing (though they have their uses), but are a product of some fancy magic.
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Glad to see a new video, keep up the good work :)
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Hi Lines!
I love the captions! That's very good even for people who aren't hard of hearing.
Terrific!! Great videos man, please post more often then three videos in one year...please!!
I was putting off watching this video, until I saw your channel name under the title! I’m glad you’re still posting!
Incredible, may you have a long and prosperous career!
as usual, very interesting and beautiful video, can't wait to see the nexts !
it has been one freaking year for this channel to make its 3rd video but the quality is gold
love your videos glad you’ve kept making more
first time seing this channel, but i must sub. Great job with explanations. I loved how you stoped at some points, and explained the things that weren't so obvious.
I have watched all the tree videos and though I was aware of almost every point, I was deeply impressed by the constructive approach of the author. The interconnection between various functions looks very pronounced here. Moreover, the formulae may be used not only for some analysis but also for computational reasons (at least to estimate rate of convergence).
HOLY ANOTHER VIDEO, THIS MUST BE GREAT!
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Yes, king! We love lines that connect!
I’ve been waiting for this video for so long!
Fascinating math and great animations too.
Great stuff, Welcome back
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Subscribed after the factorial video, I can say that Im not dissapointed! Well done
I love Fourier analysis. It connects music and number theory, trig, analytic continuation, and differential equations. I’m working on a Fourier transform that can take a musical phrase and output a mathematical equation.
If you can, will you create an open source software for such transformation ?
@@PMA_ReginaldBoscoG of course. I plan on creating the transform in both directions so you can translate between math and music. If you assign a value theta and phi for every pitch and rhythm of a given phrase, then the magnitude of the dot product of the theta and phi vectors equals the resultant amplitudes. Thus, integrating across these dot products should output the phrase structure of that measure. Thus, you can take in a musical phrase and output an integral across an inner product space or take an inner product space and output a musical phrase. I use angles for pitches and rhythms because music is cyclical and repetitive.
Your video is extremely enjoyable ! Thanks!!
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Maybe it's because of me being in high school - you said it in your first video thats your vides are targeting ones like me - but I find your videos incredibly interesting, finally telling me very desired answers for my "mathy" questions. I would be grateful if you'd make more of them😀❤. I am sure that you are going to make a big and happy community of viewers🤩
Amazing. It was one of the most amazing video I've ever seen
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so true, i can never remember if Γ(x) = (x-1)! or Γ(x-1) = x!
but Γ(x+1) = xΓ(x) 😅
Underrated channel
Incredible work, as always :).
Thankyou so much for making these videos
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Well put together, you earned a sub ❤
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YES HE"S BACK I LOVE YOU
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I love these videos!!
BEST THANK ZOU FOR DOING MY DAY
This was actually like a cool video and it was about math!? Nice job dude
Let’s goooo he’s back, had me worried there for a minute
For an even neater representation of those sine terms in the denominator, one can use *secant* (sec) and *cosecant* (csc). They're defined as sec(x) = 1/cos(x) and csc(x) = 1/sin(x), which turns those fractions in the reflection formula into products.
This is an excellent video, it really utilizes the strengths and abilities of the medium (video) in order to more clearly explain things. The (literal) video, editing, and narration all reinforce each other in a way that is almost Kubrick-like :) Just add a Wendy Carlos soundtrack and you're basically there.
Did I understand the video? Nope. Did I learn things? Yup. Did I learn a lot more than I expected? F Yeah!
There are certain fields where I find it extremely useful to dip my toe in the deep end or "get comfortable with being uncomfortable." I learn a lot in the present, I'll sometimes have a drive to "catch up" on certain subtopics, and my brain has already started figuring out the "hard" stuff, or it's laying a foundation, whatever metaphor you want to use.
Thanks!
Youe videos are absolutely great man 👌🏻
But just don't make me wait another 8 months for the new one 😊🤝🏻
im crying this is beautiful
amazing topic!
great stuff, man!!!
Make more vids bro😢
This is really good!
Your videos are always a joy
Best things in math:
1-“Let’s put a box around it”
2- cleaning up and simplifying a mess
Thank you 🎉🎉😊
RETURN OF THE KING
I'd love to see the full proof made by you
Keep up the great work!
this is amost as good as 3blue1brown, really nice work man
Beautiful.
This is helping me on my sick day
Awesome video! Keep up the good work
This is great! Awesome!
Your videos are amazing!
great video, as always
Math is so elegant. Your explanations and animations truly highlight the beauty of maths. Thank you very much. Looking forward to your future videos as much as I was waiting for this one :)
You likely know this already, but your formula for cotangent is very reminiscent of Eisenstein's approach to trigonometry. I first encountered it in Remmert's magnificent text Theory of Complex Functions, which is quite possibly the finest textbook I've ever read.