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 :)

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.

The legend is back ❤

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.

Bro your content in on an other level .Please don't stop uploading

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.

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'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.

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!

This is an absolute gem of a channel, I can't wait to have my mind blown even more in the next videos.

I genuinely hope this channel can grow as big as possible as quick as possible. Wonderful 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 watched 3 videos of yours back to back and I am still not bored. Great work. Thank you for making these videos.

I absolutely love these videos! It's crazy how fantastically intuitive the explanation and presentation are

so true, i can never remember if Γ(x) = (x-1)! or Γ(x-1) = x!

It's nice to see elegant connections between different math concepts. I like your videos, hope they continue.

I cannot express the happiness that I felt seeing you posted a new video! I'm so excited to watch this

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.

Very excited for more content! I loved the last two

With simple small steps.... climbing the mountain, is only possible with an excellent guide, The view is breathtaking! Thank you so much!

Keep up the great videos! Mark my words... You are going to be one of the greats! I'm calling it now.

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.

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

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.

This is simply beautiful. An awesome series of videos, great job!!

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!

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!

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).

Terrific!! Great videos man, please post more often then three videos in one year...please!!

Hi Lines! I love the captions! That's very good even for people who aren't hard of hearing.

Dude I was literally JUST thinking about your channel LMAO Glad to see a new video, keep up the good work :)

as usual, very interesting and beautiful video, can't wait to see the nexts !

Omg I absolutely love to see new uploads from you!

love your videos glad you’ve kept making more

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.

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).

Fascinating math and great animations too.

HOLY ANOTHER VIDEO, THIS MUST BE GREAT!

Bro, you did it again. Amazing video 🙏

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.

This channel is simply fantastic! ❤

your videos are so well made and entertaining!! ❤❤

Subscribed after the factorial video, I can say that Im not dissapointed! Well done

Great stuff, Welcome back

Your video is extremely enjoyable ! Thanks!!

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🤩

This was great! Thanks for posting

You are a genius of explanation! ❤

Bravo and applause! Thank you so much 👍🏽

Thank you for this amazing video! Your channel is amazing!

Incredible work, as always :).

I’ve been waiting for this video for so long!

Yes, king! We love lines that connect!

Amazing. It was one of the most amazing video I've ever seen

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. 🎉🎉🎉

Well put together, you earned a sub ❤

these videos are amazing! keep them coming :)

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!

Thankyou so much for making these videos

I was waiting for this, very nice !!!!

Wow!!So much intuitive.. ❤❤❤

great stuff, man!!!

Underrated channel

I love these videos!!

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.

I'd love to see the full proof made by you

Keep up the great work!

Your videos are always a joy

Omg this is the Best!! Thank you!!😊

Youe videos are absolutely great man 👌🏻 But just don't make me wait another 8 months for the new one 😊🤝🏻

This was actually like a cool video and it was about math!? Nice job dude

Awesome video! Keep up the good work

This is really good!

YES HE"S BACK I LOVE YOU

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).

this is the true definition of "it's all coming together..."

amazing topic!

Let’s goooo he’s back, had me worried there for a minute

Your videos are amazing!

This guy just appears making some of the best math content out there and then disappears from existence like nothing happened 💀

Amazing video once again!

This is helping me on my sick day

great video, as always

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.

BEST THANK ZOU FOR DOING MY DAY

This is great! Awesome!

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 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.

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

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