When CAN'T Math Be Generalized? | The Limits of Analytic Continuation 

Hey, thanks for watching! This video was made as part of the 3Blue1Brown third Summer of Math Exposition (#SoME3), a so far annual contest to encourage more math content online. If you've ever considered making a math video or other piece of math content, there's hardly a better opportunity to do so than this event. More details here: 3blue1brown.substack.com/p/some3-begins

Even if mathematicians couldn't generalise the series, they made sure to generalise the notion of ungeneralisable series 😂

When? When I need it to the most

I wish animations are more widely used in school. Visually seeing relationships and interactions really helps me understand and learn topics. This would've been helpful in my Complex Analysis class.

I'd be really interested to see a proof / deeper intuition for Fabry's and Polya's theorems in your style. This was a very nice introduction

About the end: I encountered meta-mathematics and reverse mathematics quite early in my undergraduate studies and it kept me forever. Making mathematics itself an object of mathematical reasoning is - yes - complicated at times because you have to keep track of two “mathematicses” - the one you study and the meta one that you use to do the studying -, but it has so many interesting and thought-provoking results, even early on. If it sounded complicated, think of a toolmaker: A toolmaker uses (meta-)tools to make (object-)tools; a meta-mathematician uses one set of (meta-)axioms to reason about some (object-)axioms. Maybe confusingly, these axioms can be the same, as a toolmaker can use some tool to make another one of that tool. What is reverse about reverse mathematics? The idea is to reverse the reasoning process: Normal mathematicians (blindly) accept some set of axioms, they find examples for things, observe patterns, craft definitions, formulate propositions they hope to be true and then - hopefully - prove those propositions true, that is, logically derive them from the axioms. Reverse mathematics takes propositions and attempts to answer the question: How fundamental are they? This is the same question as: What axioms are necessarily needed to logically derive the proposition. The fewer and the more fundamental (in the eyes of the human) of axioms suffice, the more fundamental the proposition is. I have a rather simple example for you: In classical logic, for every proposition _A_ it is true that _A_ or not _A._ (In formula: A ∨ ¬A.) Also, in classical logic, not not _A_ implies _A_ (in formula: ¬¬A ⇒ A) and in fact, if you take away any single these axioms, the other follows from it and the rest of the logical axioms as a theorem. Both of these are somewhat controversial, and you’ll understand why by me giving you a mathematically infallible investment strategy: Pick a stock and a time frame; after the time frame, if the stock is above its current value, buy it; otherwise, short it. The problem is, the stock will indeed be above or below (that is: not above) its current value, but - you can’t really know until then. The strategy I outlined can only look mathematically sound because of _A_ ∨ _¬A._ The logic without both of them is called “intuitionistic logic” and it can only derive statements that are computationally valid, that is, if you find a fool-proof investment strategy using intuitionistic logic, it can actually be followed. Of course, in intuitionistic logic, _A_ ∨ _¬A_ cannot be derived for every _A,_ but for some; and it is equivalent to _¬¬A_ ⇒ _A,_ but that is due to the assumption that from a contradiction, anything follows: For every statement _B,_ if we can derive _A_ and _¬A_ (a contradiction), we may conclude _B_ (in formula: A ∧ ¬A ⇒ B). If we do away with this as well, we have an logic system called “minimal logic” and it makes not assumptions about falsehoods/contradictions/negation. In minimal logic, finally, _A_ ∨ _¬A_ and _¬¬A ⇒ A_ are not equivalent anymore. Here, we have that _A_ ∨ _¬A_ for all _A_ follows from the assumption _¬¬X_ ⇒ _X_ for all _X,_ but _¬¬A_ ⇒ _A_ for all _A_ cannot be derived from _X_ ∨ _¬X_ for all _X._ That is, one of them is more fundamental than the other, but you need to visit a weak logic to see it. I can give you a little motivation for meta-mathematics: The “ordinary” axioms of mathematics are - generally - the axioms of Zermelo-Fraenkel set theory (ZF) plus the Axiom of Choice (AC), called ZFC. In ZF, i.e. without AC, one can prove that it is equivalent that (a) every family of non-empty sets has a non-empty Cartesian product (this is trivial for finite families, but provably not provable in ZF without AC) (b) every set can be well-ordered (again, this is trivial for finite sets, but not infinite ones, but provably not provable in ZF without AC) If you need a rephrasing for (a) it is this: Consider sets _A₀, A₁, A₂, …_ that are not empty, i.e. there is _a₀_ ∈ _A₀, a₁_ ∈ _A₁, a₂_ ∈ _A₂,_ etc. in the Cartesian product _A₀_ × _A₁_ × _A₂_ …, there “obviously” is the element (a₀, a₁, a₂, …), however, in ZF without the axiom of choice, we cannot actually prove that the tuple exists! Both, (a) and (b) are actually equivalent to AC, they are as good as the axiom itself. However, (a) looks like something that’s “obviously true,” like, how could a family of non-empty sets have an empty product? On the other hand, (b) looks like something that shouldn’t be true since - of course _some_ sets can be well-ordered - why should every single set admit a well-ordering? As humans, we have no choice in the logical consequences of axioms, but we do have choice in what axioms we base our reasoning on. (In a sense, given a fixed, agreed-upon axiom system, you cannot meaningfully ask “why” some theorem is true - the answer always is that it’s derivable from the axioms -, but you can meaningfully ask “why” this or that axiom is part of the axiom system because here, human intention and choice is involved. Now my personal conclusion is, if an axiom system (ZF in this case) derives the equivalence of two statements and one “obviously true” and the other is “obviously false,” then the axiom system is not good. It’s not that it’s inconsistent, that’s a technical term and subject to proof. What I mean is, maybe there’s a better system of axioms out there. One in which the “obviously true” is true and the the “obviously false” is false. Without being exposed to meta-mathematics even good mathematicians don’t have thoughts about it. In introductory courses such as analysis or linear algebra and most of what sits on top of it: complex analysis, functional analysis, probability theory, and even numerics, students are generally assumed to work in ZFC almost religiously. We laugh at Blaise Pascal for not considering other potential gods in his famous wager, but most of today’s mathematicians work in ZFC without ever considering working in a different axiomatic system. If you ask for one, there are other set theories, but there’s other foundations of mathematics that don’t take sets as their primitive notion; one of them is called Homotopy Type Theory (or HoTT for short); I found it really interesting to read the introductory HoTT book, homotopytypetheory.org/book/ (it’s 100% free) in fact, I read it multiple times, and there’s complicated stuff, but after a while, it finally snapped; I don’t think it’s too important to understand everything you read in there, except for the first chapter (which explains the basic concepts and notation) and maybe the second, but I got very far in my first read where I didn’t really understand most of the second chapter.

as someone who just finished a complex analysis course, this video felt like a nice application of a lot of things I learned

The first series of powers of 2 "equaling" -1 reminds me of the usual representation of -1 in computers as many 1s, which is effectively saying the same thing!

Although I'm not super familiar with complex analysis (it's been quite a while since I studied it in uni), the explanations were very clear and intuitive, and they subtly gave away where they were going to go next, making it very pleasant to actually see you explain it after you were wondering about that. Great job!

2:26 "No, we're not going to talk about p-adics today." Does that mean you plan on making a video on them later 👀? I'd love to see a 3rd science/math channel talk about them (1. Eric Rowland, 2. Veritasium).

Awesome!!! It took me until graduate complex analysis before it really dawned on me that you can have these dense boundaries of singularities that would prevent analytic continuation, so seeing this in such a clear format is a treat!

"Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective.

Hey, I’ve never seen someone explaining the murky concept of analytic continuation so neatly, well done! I’m subscribing 😅

I had this problem as a homework assignment in my complex analysis class, very cool to see it animated here and explained so simply!

The animations are great! When I first studied complex analysis, conformal mapping, etc., one kept one eye on the z-plane and the other on the w-plane. We all looked like Marty Feldman after the final exam.

Wow! The animations really helped visualize the equations so well. Studying math, I had to do this in my head and it wasn't always feasible. Thanks for reminding me how cool math can be. Keep shining!

I can't believe it was only about 8 months ago I watched last year's SOME2 round-up and had a look at these videos. Great channel!

Thanks RU-vid for this recommendation! This one particular video us one of the best math videos I've seen, in terms of math-noob-friendliness. Love it!

I paid so little attention in my Complex Analysis class, to the extent that I can't even assess whether my lack of motivation was my own fault (quite likely) or that of the lecture content. Since then I occasionally come across a video/article/discussion that highlights just how conceptually fascinating it is, and this is my favourite so far - thank you! I'm excited to begin relearning it properly some day. Alongside some of the other fields that feature heavily in modern approaches to number theory, I feel it's one of those subjects that can make an algebraist love analysis too 🤩 You asked whether there would be interest in proofs of the theorems you mention, and my answer is definitely yes for completeness' sake, but beyond anything else I would love you to keep focusing on aspects of the subject that lend themselves to great intuitive dives like this one, whatever those might be.

I had done this back in uni 2 years ago, but I was able to somewhat get the gist after watching your video. Damn, makes me want to pick up math again, but this time with no exam pressure, just pure interest

Honestly, quite the amazing video! Will be rooting for you in SoME3!

Convergence (syntropy) is dual to divergence (entropy) -- the 4th law of thermodynamics! Finite (localized, particles) is dual to infinite (non localized, waves). Waves are dual to particles -- quantum duality. "Always two there are" -- Yoda.

I love the symmetry of Fabry's and Polya's theorems

very good video. my understanding of "analytical continuation" had always been very handwavy, but you put it very simply, and were able to apply it in a way that made it make a lot of sense.

Thanks for this video, I had heard about this before but had never seen an actual concrete example of when it happens!

What a great video. Wonderful explanations and beautiful + helpful animations! Well done!

I think this problem also leads well into a discussion about why analytic functions are conformal maps.

What a fantastic video. Being unable to thread the needle past the boundary was awesome. Thanks for making it!

Holy smokes that was truly insane (especially 3 theorems at the end)! I gotta check out complex analysis sometime...

Just went over analytic continuation for graduate econometrics! This is right on point, great video!

Thanks so much for this video. I was trying to settle this in my mind for a long time and finally I can put the pieces together.

7:09 an easy way to think about complex differentiability is that if you have f(x+iy)=u(x,y)+iv(x,y) being differentiable, then the jacobian matrix of f looks like multiplication by a complex number.

Thanks for the primer on analytic continuation! I've wondered about this before, as it is often mentioned when the Riemann Hypothesis is discussed, but sadly my mathematics education hadn't quite reached this area.

This was a great video! I will watch any complex analysis videos you decide to make.

the arrow visualization is also a really nice way of seeing that even though the series diverges, it's imaginary part converges, which i think is neat!

Thank you! I have struggled with Titchmarsh's excellent Theory of Functions book for years now. They have exercises about these gap series that I never understood how to solve. You have helped me to solve them.

This is fascinating and very well explained! Thank you for making this video!

Thanks for the clear graphical explanation of this interesting topic!

Subscribed for the high quality explanations and visuals!

I have to say, I love the visualization of complex power series you use at about 12 minutes in. It makes it so clear how complex power series are related to Fourier series. Specifically, the Cauchy integral formula can be thought of as saying that if a function f is holomorphic on a disk D centered at a, then the Taylor coefficients centered at a are equal to the Fourier coefficents of f restricted to the boundary circle of D. That is, complex Taylor series are really the same thing as Fourier series. The visualization here makes this painfully obvious, especially when paired with some of 3Blue1Brown's videos on Fourier series. This also makes it a bit clearer why the gap series would be badly behaved. We're taking a function with well behaved Fourier coefficients, but dropping the contributions from many frequencies. Having very few high frequency Fourier coefficients all with the same weighting is also likely to lead to pretty sporadic behavior, as you would have large amounts of very fine structure but not enough to expect useful interference.

THIS IS CRAZY MAN never heard about the function wanting to be "the center of mass" of the values it oscillates between

Thanks for the video, I dont usually catch this kind of glimpse in complex analysis, it was pretty to remember things about power series.

Oh my god, that transformation of g(z) into a chaotic tangle was one of the most beautiful things I've seen in math in a long while. It just grabbed some deep part of me.

Very interesting stuff! Thanks for the great content!

I appreciate your video！I never learnt about Analytic Continuation before but my number theory course asks me to read a paper. I picked one about prime number theorem and this term is in it. Now I get the idea! Thank you again!

Wow! Fascinating and fully understandable. Great job!

i love your style, thanks for making videos like this! 😄 i was able to understand most of it and get some new insights, but also, watching graphs "misbehave" is just inherently funny!

I was so excited to watch this! I love it.

This is such a good video on the intuition of analytic continuation!

Perfect. The right mix of motivation and rigor. It’s hard to get it right, but you did.

Truly excellent explanation, thank you for making this.

i love this channel so much… keep it up!

This is probably the coolest video I have ever seen on RU-vid. And nobody at my physics faculty get‘s why this is so interesting to me

Beautiful video, looking forward to more from you !

Man these animations make my mouth water, thanks this was a great visual explanation

Visualization is the best way to understand anything. Great work 👏

I am glad you mentioned p-adics there.

this one gonna be ranked well in the SOME contest, I can feel it

PURE GOLD THANK YOU FOR THIS GREAT VIDEO!!

Thank you very much Very amazing and philosophical topic 🎉🎉❤❤

I studied a few of the concepts used in the analysis on my Control Engineering classes, always from the engineering pov. Looking these concepts from the mathematical pov make things sound much more reasonable.

Great video! Please make one about the gap theorems!

One of the best math explainers on the internet.

Another fantastic video! This was really interesting!

I’d love to see a video on the other gap theorems!

This filled in some gaps I had after taking complex analysis so thanks!

Really Elegant! Thank you so much , grateful

this is great❤ i would 💯% love to see proofs of those "gap" theorems of analytic extensibility! keep up the great work

Loved the video thank you so much for your content. Can we see a proof of the theorems next please :))))) i'd love that

can we just appreciate how slick the transition at 5:08 was?

That was mindblowing. Amazing video.

Thank you for this beautiful video❤

Great job! Would love to see as many visual proofs as possible

This was really an interesting topic. Thanks.

Amazing video. Thank you!

I’m in love with this channel. Hi from Colombia!

Very nicely explained. Good job !

This is beautifully executed!

This pretty much summarizes many techniques used in physics such as the usage of power series analysis in QM or rotations in the imaginary plane etc etc

Great video! Thank you!

I love this channel so much oml

Thks & Well-Explained

Really helpful, thank you

Keep doing what you are doing, man! This is great math insight nicely presented! I'm an old retired engineer and wish to learn more math so perhaps I will prevent dementia in the future. 😁😁😁😁

THE best submission to #SoME3 yet.

Outstanding explanations -- instant sub

Now I'm waiting for the video about extending fractional calculus to the quaternions.

this is really well-made!

Really nice, I'm still wrapping my brain around it. At first sight, it seems weird that removing terms from a well-behaved series causes a problem.

Brilliant!!!

This sort of series is called lacunary, from "lacuna" (gap). If you compute g(ω), where ω³=1 and Re(ω)=-0.5, it diverges to -∞, at half the speed it diverges to +∞ at the power-of-2 roots of unity. At the first seventh root of unity, g(z) diverges to ∞ at an oblique angle. At other roots of unity, it diverges in other directions. I've been studying a function I call khe (խ(z)), which cannot be continued past the imaginary axis. It turns out to have a lacunary series. The loops of g(z) as you feed it a circle close to the unit circle look very much like the loops of խ(z) when I feed it a line close to the imaginary axis.

I would love to see the proofs of Fabry’s and Polya’s theorems! 😎 great vid!

that was great, I understood the analytic continueation much better than with the 3b1b video

Congrats for HM! Great video

Thank you this video changed my life

Yes! Do a follow up, please

We love your content. Can we ask what next? Deep dive into an ocean of function progressions and series with all criterions and proofs? Or mountain hike to BirchSD conjecture? Or someone may ask to camp near statistics forrest?

thanks for the content, and yeah please follow up with the proofs and intuitions behind them!

Yes we are interested in more.

"...if there is enough interest..." I, for one, am very very interested... thanks for his content...