Analytic Continuation and the Zeta Function

Where do complex functions come from? In this video we explore the idea of analytic continuation, a powerful technique which allows us to extend functions such as sin(x) from the real numbers into the complex plane. Using analytic continuation we can finally define the zeta function for complex inputs and make sense of what it is the Riemann Hypothesis is claiming.

Links:

Chapters:
00:00 zetamath does puzzles
00:23 Recap
02:40 Bombelli and the cubic formula
08:45 Evaluating real functions at complex numbers
12:33 Maclaurin series
21:22 Taylor series
27:19 Analytic continuation
35:57 What goes wrong
48:19 Next time

Thanks to Keith Welker for our theme music.

[bastienmassion]

I've seen countless videos about the Riemann Zeta function, and it is the first time that analytic continuation is explained in some details, and also so clearly. Moreover, you finally made sense of the fact that as holomorphic complex function is once complex differentiable, it implies that it is infinitely complex differentiable, which my complex calculus teacher never managed to do. Thank you so much, this channel is pure gold 🏆

[amritawasthi]

These videos are criminally underwatched.

[saikrishnasunkam]

Best video on analytic continuation I've ever seen. Where was this 5 years ago when I was in Complex Analysis 😂

[ativjoshi]

This is one of the most well-made and accessible math videos I've watched on YouTube.

[nikita_x]

The best explaination of analytic continuation on YouTube!

[roygalaasen]

I don’t know why I haven’t subscribed to your channel already, but that glitch is now sorted. Brilliant content!

[manishprasad]

This is the first time in my life I actually understood analytical continuation. I study physics and I took a course on string theory where we saw the classic sum of all natural numbers in a calculation. And my professor just briefly told us about analytical continuation and substituted the value -1/12 for that sum and I always felt extremely uncomfortable with that calculation. I had to accept analytical continuation as this magical thing that somehow extends the domain of real numbers into complex numbers. But your explanation makes so much sense! I'm so glad I watched this video! Thank you so much for making this :)
For those who are curious, the calculation was in Bosonic String theory where we calculated the number of spacetime dimensions which turns out to be d=26 if you use this result.

[rahulpsharma]

As an engineer who just studied complex calculus as a ‘process’ to solve problems in book to pass exams, this video is truly enlightening. I m just a hobbyist now with no real goal to apply it in real life but the satisfaction I got after watching this video is amazing. Pls make more of these. It’s been a while since the last video was posted.

[trueDdg]

I've been looking for a video like this since the 3Blue1Brown video first introduced me to the idea of analytic continuation. The depth of detail and length of the video are just great. I ended up re-watching the previous episode and watching this episode and I was excited and engaged at every step of the journey.

[padraiggluck]

When I studied complex analysis many years ago we only had textbooks and handheld calculators. I cannot express how much I appreciate your animations and explanations. ⭐️

[anastasiagoold]

Are you telling me I can understand analytic continuation with very basic calculus?? Holy shit. This video is literally a game-changer.

[piercexlr]

I have never seen someone who rivals 3b1b in terms of clarity and introducing something intuitively. Thank you for such a clear explanation.

[r.w.emersonii]

These tutorials are phenomenal. The animations are gorgeous! And the commentary is slow enough that we have time to digest the material. In addition to asking "What?", you also ask "Why?" and even "Why not?"-- all of the questions we would like to ask but never do, as we race through a math book or class. These questions make the material accessible to intuition. You also warn us when you introduce things that are not intuitive -- "e to the pi times i", for example.

You explained the rationale behind analytic continuation, for example: How complex functions resemble polynomials! And then you provided an example where the Taylor expansion even allows us to continue a function to real numbers that break the initial function definition! An example just occurred to me: The Gamma function allows us to define negative factorials!

Only one minor quibble: I would like to see, in more detail, how the Taylor expansion changes, as we hop from circle to circle. How does the series in the destination circle differ from the initial series. Can't wait till you take on the elliptic functions!

[thisisnotmyrealname]

You're the kind of person who's voice makes me hungry

Amazing video btw, well done! I learnt something today

[alokaggarwal]

Thanks, this is a fantastic series which finally helped me understand analytic continuation and the zeta function.

[shmuelalexis]

Your explanation and presentation is unavailable good. I watched the entire video and it was really "out of the real", and still, I was able to understood every single bit.

Thank You for your amazing work. I scincirly appreciate it.

[MasterHigure]

21:25 In my opinion, one of the clearest hints that math screams at us that complex numbers are a thing, even when working purely within the reals, is if you try to do Taylor series of this at different points. Centered at 1, you get convergence up to sqrt(2) away. Centered at 3, you get convergence up to sqrt(10) away. Centered at k, you get convergence up to sqrt(k^2+1) away. That's the Pythagorean theorem telling you that there is some convergence obstruction that's one unit away from the origin, orthogonal to the number line.

[TheBasikShow]

What an excellent video! Definitely my favorite explanation of analytic continuation on YouTube. I will note that I’m surprised you didn’t talk about the analytic continuation of ³√x, which has the same problem as √x but is more interesting (at least to me) because it is ostensibly defined on all reals. That is to say, you don’t even need to loop around the singularity to get a contradiction! Still, this was an excellent video, well worth 50 minutes.

[ZuhairAlahmad]

أشكرك على تقديمك مع أن وقت الفيديو يستغرق معي أكثر مما هو مخصص وذلك للترجمة مع ذلك سأشاهد جميع محتواك

[diribigal]

This made me realize I need to go back to rewatch the previous video first. But that I means I get more minutes of math content from you, so it's a plus in my book.