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 :)
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
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!
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.
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.
No, they'll stand together, much as the two functions in the video. LaTex is a complex of TeX macros for writing papers. Manim formats and animats TeX formulas for the screen. They both rely on TeX.
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)! :)
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.
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!
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.
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 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
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 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 :)
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.
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.
@@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.
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.
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!!
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.
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.
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🤩
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.
A note regarding 7:35 Consider a polynomial with zeroes at A and B. The product form would then be (x-A)(x-B). So naively, I would expect the first "guess" for a third-order polynomial for sine to be (x-(-pi))x(x-pi), and not (pi-x)x(pi+x) as presented in the video. This not only scales the slope at x=0 by pi^2 but also flips its sign, which is why we want to divide the whole thing by -(pi^2). Since we only need to divide one of our new terms by -1, it's convenient to do so with the x-pi term, and rewrite (x-pi)/(-1) as (pi-x). This felt like a missing step and bugged me, so I figured I'd share how I explained this to myself.
Thanks a lot! We are in a world of relationships, not a world of linear scales! Therefore, a few years ago I realized that the “acceleration of gravity” is equal to Pi^2! (Rotabler, from Russia, True Physics) ____________ Спасибо большое! Мы в мире отношений, а не в мире линейных шкал! Именно поэтому еще несколько лет назад я понял что "ускорение свободного падения" равно Pi^2! (Ротаблер, Россия, Правильная Физика)
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).
I know the harmonic series because of music, but recently i learned how to mutiply and divide with the series, as well as do trig with it. I literally learned trig from the harmonic series in like 3 days.
This channel is one of my favorite Math ones! Each step follows cleanly from the last, kinda like a recursive formula! If textbooks were that way, I'd probably have had waaaaay less anxiety - if at all - back when I was in my former major (Physics). Content like this is what keeps me liking Math, despite my rather dramatic switch in subject area (I study History now). Tysm for your work, LTC! 🤗
Absolutely wonderful!!! I said this many times before and I will have to say it again. You belong to the BEST graduate school. You should be doing your PhD at MIT, or Harvard, or Cambridge, or anywhere you want. 🎉🎉🎉
on the subject of trig functions from other functions. if you define h(y,x) =x^y + 1/x^y then 0.5 h (i,x) is a cosine function with period 4. (so angle measure counting quarter circles). Do x*pi/4 for radians.
I love math I wish I was better at it this video is so fucking mind opening it’s crazy to see you only have 3 videos, looking forward to more from your channel please keep it up!
Very simple observation, in 0 the slope of sin(x) function is 45° so worth 1 so its the identity function x = x, so sin (x) = x, so lim x->0 of sin(x)/x = the slope = 1
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!
Agreed. The reflection formulae are actually simpler, in my view, when using the Π function. Γ(x)·Γ(1 - x) = π/sin(π·x) becomes Π(x)·Π(-x) = π·x/sin(π·x) = 1/sinc(π·x), which I believe is much more elegant. As for ψ(x) - ψ(1 - x) = -π·cot(π·x), you get a different formula. Note that Π'(x)/Π(x) = -γ + H(x), so we have that -γ + H(x) - (-γ + H(-x)) = H(x) - H(-x) = 1/x - π·cot(π·x).
8:52 How do we know that changing the order of the multiplication doesn't change the result? For infinite products, isn't commutativity invalid most of the time?
We don't! That's part of the reason this isn't a real proof. Actually, one might go even further and say that there _is_ no well-defined order up until that point, because there's no clear starting point in something that goes forever in both directions. But I think it's worth pointing out that if you interpret the order of the double-sided infinite product as "start with x, then do the terms on either side of it, then do the terms on either side of those, etc...", then we aren't rearranging terms at all. We're just moving them to more clearly represent the order we already had in mind.
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 :)
6:30 this is really cute, you can actually see it just by taking the O(x^3) part of both sides. LHS=x-x^3/6+O(x^5) RHS=x-x^3/pi^2*sum(1/k^2)+O(x^5) sum(1/k^2)=pi^2/6
For those who want to see the reflection formulae for the Pi function and the Harmonic numbers function, instead of the ugliness that are the Gamma the Digamma functions, then: •Π(x)·Π(-x) = π·x/sin(π·x) = 1/sinc(π·x) •H(x) - H(-x) = 1/x - π·cot(π·x).
0:35 "to the trig functions we all know and (hate)..." Well luckily I have you, 3b1b, and other indian TH-camrs that helped me with my advance math courses in my highschool xD
It's relatively straightforward to prove the reflection formula for the Gamma function using Liouville's theorem, rather than the sine product formula.
Love the presentation, esp. the graphics, reflecting Euler (the natural log lover and master manipulator)'s work including 0.577, Basel equation and its sum with -1 as a part of the denominator, all the way to prove that the number of primes is infinite like the product and the sum of all primes. (Similar conclusion by Euclid 2000 years earlier).
It’s not a disappointment at all that you didn’t include the rigorous proof of the sin formula! Because you put a link to a great proof in the description
I honestly get tired of "You cAnT tAkE LoG Of a NeGAtiVe". K, you take ln (-5). You end up with ln5 + pi*i. pi*i is a constant. When you differentiate it, it goes away. This strikes me as an old, similar objection to the cubic formula because sometimes you will dip into complex numbers for a few steps before calculating your 3 real roots. Ln is one of if not the nicest function we have, and trying to box it in only for it to be like nope.avi is poetic.
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?
I like how the window of Wikipedia was dragged into the screen
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
@@wyboo2019 the equivalent of Fourier not giving a shit about the fast Fourier transform and never publishing it
@@person8064wasnt it gauss
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.
Please make more videos on these intuitions feel stuff
The legend is back ❤
I love that you pivot from frustrated that the digamma function is defined with x-1 to thankful when things canceled out xD
Awesome! It's incredible how seemingly unrated mathematical ideas come together in unexpected ways!!
"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.
No, they'll stand together, much as the two functions in the video.
LaTex is a complex of TeX macros for writing papers.
Manim formats and animats TeX formulas for the screen.
They both rely on TeX.
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)! :)
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
EVERY YEAR, HE UPLOADS A BANGER
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!
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.
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.
@@angeldude101 1/x in an identity with the harmonic series? no way!
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
This is an absolute gem of a channel, I can't wait to have my mind blown even more in the next videos.
This guy just appears making some of the best math content out there and then disappears from existence like nothing happened 💀
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!
Am I the only one who got bored of 3b1b? I dunno, something in his way of teaching just makes his videos so... 😴
I genuinely hope this channel can grow as big as possible as quick as possible. Wonderful content❤️❤️
I watched 3 videos of yours back to back and I am still not bored. Great work. Thank you for making these videos.
Euler is my hero. His accomplishments have always amazed me
It's nice to see elegant connections between different math concepts. I like your videos, hope they continue.
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 :)
Dude I was literally JUST thinking about your channel LMAO
Glad to see a new video, keep up the good work :)
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.
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.
With simple small steps.... climbing the mountain, is only possible with an excellent guide, The view is breathtaking! Thank you so much!
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.
I was putting off watching this video, until I saw your channel name under the title! I’m glad you’re still posting!
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!!
Keep up the great videos! Mark my words... You are going to be one of the greats! I'm calling it now.
I absolutely love these videos! It's crazy how fantastically intuitive the explanation and presentation are
I cannot express the happiness that I felt seeing you posted a new video! I'm so excited to watch this
Very excited for more content! I loved the last two
Very well done!! I love the intuitive Euler’s proof. I hope you make more videos.
it has been one freaking year for this channel to make its 3rd video but the quality is gold
One of the fastest clicks of my life. I'm glad to see you upload again!
Hi Lines!
I love the captions! That's very good even for people who aren't hard of hearing.
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.
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.
Incredible, may you have a long and prosperous career!
Finally a new video after months! I really like your videos, please be consistent on your channel.
2:23 the way I remember that it's -1 is to remember that the gamma function has an asymptote at x=0
This is simply beautiful. An awesome series of videos, great job!!
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🤩
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.
Let’s goooo he’s back, had me worried there for a minute
Terrific!! Great videos man, please post more often then three videos in one year...please!!
A note regarding 7:35
Consider a polynomial with zeroes at A and B. The product form would then be (x-A)(x-B). So naively, I would expect the first "guess" for a third-order polynomial for sine to be (x-(-pi))x(x-pi), and not (pi-x)x(pi+x) as presented in the video. This not only scales the slope at x=0 by pi^2 but also flips its sign, which is why we want to divide the whole thing by -(pi^2).
Since we only need to divide one of our new terms by -1, it's convenient to do so with the x-pi term, and rewrite (x-pi)/(-1) as (pi-x).
This felt like a missing step and bugged me, so I figured I'd share how I explained this to myself.
so true, i can never remember if Γ(x) = (x-1)! or Γ(x-1) = x!
but Γ(x+1) = xΓ(x) 😅
Thanks a lot!
We are in a world of relationships, not a world of linear scales! Therefore, a few years ago I realized that the “acceleration of gravity” is equal to Pi^2!
(Rotabler, from Russia, True Physics)
____________
Спасибо большое!
Мы в мире отношений, а не в мире линейных шкал! Именно поэтому еще несколько лет назад я понял что "ускорение свободного падения" равно Pi^2!
(Ротаблер, Россия, Правильная Физика)
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).
I know the harmonic series because of music, but recently i learned how to mutiply and divide with the series, as well as do trig with it. I literally learned trig from the harmonic series in like 3 days.
Omg I absolutely love to see new uploads from you!
This channel is one of my favorite Math ones! Each step follows cleanly from the last, kinda like a recursive formula! If textbooks were that way, I'd probably have had waaaaay less anxiety - if at all - back when I was in my former major (Physics). Content like this is what keeps me liking Math, despite my rather dramatic switch in subject area (I study History now). Tysm for your work, LTC! 🤗
this is amost as good as 3blue1brown, really nice work man
Absolutely wonderful!!! I said this many times before and I will have to say it again. You belong to the BEST graduate school. You should be doing your PhD at MIT, or Harvard, or Cambridge, or anywhere you want. 🎉🎉🎉
HOLY ANOTHER VIDEO, THIS MUST BE GREAT!
I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.
This was actually like a cool video and it was about math!? Nice job dude
on the subject of trig functions from other functions. if you define h(y,x) =x^y + 1/x^y then 0.5 h (i,x) is a cosine function with period 4. (so angle measure counting quarter circles). Do x*pi/4 for radians.
0:03 "Connect That Lines" LOL
Subscribed after the factorial video, I can say that Im not dissapointed! Well done
I'd love to see an extension of the arithmetic, geometric and weighted means to the real and complex numbers.
Best things in math:
1-“Let’s put a box around it”
2- cleaning up and simplifying a mess
I love math I wish I was better at it this video is so fucking mind opening it’s crazy to see you only have 3 videos, looking forward to more from your channel please keep it up!
Would love to see videos/ a video series about Dirichlet's theorem about arithmetic progressions in prime numbers
Very simple observation, in 0 the slope of sin(x) function is 45° so worth 1 so its the identity function x = x, so sin (x) = x, so lim x->0 of sin(x)/x = the slope = 1
This channel is simply fantastic! ❤
I’ve been waiting for this video for so long!
I love these videos!!
Amazing. It was one of the most amazing video I've ever seen
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!
as usual, very interesting and beautiful video, can't wait to see the nexts !
Bro, you did it again. Amazing video 🙏
Yes, king! We love lines that connect!
Aha. I finally see why sqrt(pi) shows up in the gamma function!
If you want to bypass the weirdness of the gamma function, the Pi function (denoted by a capital Greek pi) is exactly x!, no shifting necessary
Agreed. The reflection formulae are actually simpler, in my view, when using the Π function. Γ(x)·Γ(1 - x) = π/sin(π·x) becomes Π(x)·Π(-x) = π·x/sin(π·x) = 1/sinc(π·x), which I believe is much more elegant. As for ψ(x) - ψ(1 - x) = -π·cot(π·x), you get a different formula. Note that Π'(x)/Π(x) = -γ + H(x), so we have that -γ + H(x) - (-γ + H(-x)) = H(x) - H(-x) = 1/x - π·cot(π·x).
8:52 How do we know that changing the order of the multiplication doesn't change the result? For infinite products, isn't commutativity invalid most of the time?
We don't! That's part of the reason this isn't a real proof. Actually, one might go even further and say that there _is_ no well-defined order up until that point, because there's no clear starting point in something that goes forever in both directions.
But I think it's worth pointing out that if you interpret the order of the double-sided infinite product as "start with x, then do the terms on either side of it, then do the terms on either side of those, etc...", then we aren't rearranging terms at all. We're just moving them to more clearly represent the order we already had in mind.
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 :)
6:30 this is really cute, you can actually see it just by taking the O(x^3) part of both sides.
LHS=x-x^3/6+O(x^5)
RHS=x-x^3/pi^2*sum(1/k^2)+O(x^5)
sum(1/k^2)=pi^2/6
For those who want to see the reflection formulae for the Pi function and the Harmonic numbers function, instead of the ugliness that are the Gamma the Digamma functions, then:
•Π(x)·Π(-x) = π·x/sin(π·x) = 1/sinc(π·x)
•H(x) - H(-x) = 1/x - π·cot(π·x).
Great stuff, Welcome back
0:35 "to the trig functions we all know and (hate)..."
Well luckily I have you, 3b1b, and other indian TH-camrs that helped me with my advance math courses in my highschool xD
Since sine is an entire functions, the Weierstrass factorisation theorem can be used to show the sine product formula is valid
This feels like it could connect to the reimann zeta function somehow, interesting
YES HE"S BACK I LOVE YOU
Youe videos are absolutely great man 👌🏻
But just don't make me wait another 8 months for the new one 😊🤝🏻
I not even have watched the video, but i know its gonna be cool. I lof ur videoz ❤
Fascinating math and great animations too.
It's relatively straightforward to prove the reflection formula for the Gamma function using Liouville's theorem, rather than the sine product formula.
d/dx ln(x) where x=-2 is still 1/2 because the negative side is just ln(-x)=ln(x)+ a complex constant i pi
The natural logarithm is not well-defined for complex numbers.
this is the true definition of "it's all coming together..."
I always suspected this kind of connection (trig and factorial) playing around with gamma and digamma functions
Love the presentation, esp. the graphics, reflecting Euler (the natural log lover and master manipulator)'s work including 0.577, Basel equation and its sum with -1 as a part of the denominator, all the way to prove that the number of primes is infinite like the product and the sum of all primes. (Similar conclusion by Euclid 2000 years earlier).
It’s not a disappointment at all that you didn’t include the rigorous proof of the sin formula! Because you put a link to a great proof in the description
love your videos glad you’ve kept making more
I do not understand any of this, but still i enjoyed watching good content men :')
I honestly get tired of "You cAnT tAkE LoG Of a NeGAtiVe".
K, you take ln (-5). You end up with ln5 + pi*i. pi*i is a constant. When you differentiate it, it goes away. This strikes me as an old, similar objection to the cubic formula because sometimes you will dip into complex numbers for a few steps before calculating your 3 real roots.
Ln is one of if not the nicest function we have, and trying to box it in only for it to be like nope.avi is poetic.