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 🏆
Yes and it's great that someone finally explained the whole -1/12 business without just saying it's obviously not true and pointing out the incorrect methods used to get it. I also feel like none ever related it to the zeta function which seems wired since to very obviously is very related
@@edwardzachary1426 Actually Mathologer explained that very well on a RUclips a couple of years ago. What is curious, the Ramanujan summation directly relates this -1/12 result to 1+2+3+... and that perfectly coincides with the analytic continuation of Riemann zeta. And this is not the random occasion: Ramanujan summation also relates 1+4+9+... to 0, in the accordance with the fact this is trivial zero of the zeta. Actually using the Ramanujan technique of summation of this kind of series series we will always arrive to the corresponding values of Riemann zeta function. And yet, the analytic continuation path taken seems to have no relation to Ramanujan summation of the divergent series! There must be a hidden deep connection between so much different math, which is well beyond my understanding.
@@nikitakipriyanov7260 What you wrote here aligns well with my understanding, and I link to that mathologer video in at least one of the videos in this series. It is an excellent source. At least to the best of my knowledge, the hidden deep connection to which you refer is hidden from everyone, e.g. no one knows how to make sense of this connection.
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. ⭐️
After watching this whole video, my one nitpick that |x| isn't a great example for showing the rigidity in complex analysis, because it's not differentiable at 0. I feel like a key fact is that complex functions that are differentiable in an open set are locked-down in a way that differentiable real functions aren't. But if we give up differentiability then we can mess things up in the complex plane just as easily as on the real line. But everything else was really good, and this is the most accessible introduction to analytic continuation I've seen that doesn't try to hide the things that could "go wrong". Great video!
There is always a complicated balance in how much detail I want to get into about the formality of what is going on. I thought the absolute value example hinted at this without having to delve into the technicalities of what it means for a real valued function to be analytic.
This is the first time it has been years I am trying to understand seriously the true meaning of analytic continuation principles and technics, (all others who made videos like you, are like scary that we understand really the true meaning of Riemann hypothesis and we may prove or disprove it.) You have shown in this video exactly what does it mean: Expanding the zeta function with Taylor series principles to get zeta(z) defined in the whole complex plane, and the different outputs that we will get if we pass by thier singularities in the complex plane. I personally the first time I wished that this video never end when the video ends! why ? because simply you are a true teacher what deserve all my respect for the insight what you gave to us that dismiss the whole darkness in this way.
What an excellent video! Definitely my favorite explanation of analytic continuation on RUclips. 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.
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.
Thank you so much for this incredible video. Everything is explained in a very comprehensible yet detailed manner. I am currently writing my final paper on the the Riemann Zeta Function and the connection between the values of its analytic continuation and those of (divergent) inifinte series, and this video has really helped me get started.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
Sir, your videos -- while different in style than Grant's over at 3Blue1Brown -- are equal in clarity and educational quality. Bravo! You deserve several orders of magnitude of more subscribers.
This video is just amazing! It explains analytical continuation like no one has ever been able to do so. Not only that, it explains the nature of complex numbers and functions especially the illusive logarithm which i had a great difficulty understanding when i took complex analysis in the university! The explanation is so deep that after the 50 min presentation one is able to retrace the whole argument of analytic continuation from really scratch in ones own mind! I sometimes wonder why such easy concepts were taught to us in the university in such a convoluted and incomprehensible way!!! And I also wonder how much time was put to produce this beautiful video. The fact is that many people will appreciate this effort tremendously!!
Thank you so much for this wonderful explanation !! Really eye-opening for me. I have never thought of analytic continuation in this way. I thought I knew but I did not. The best I have seen so far. Thanks again!!
This is by far the best video on this topic I have found. In fact, this is the best presentation on this topic I have seen. I'm really looking forward to the next video! I wish I was able to watch this during my PhD, I would have benefited greatly! Please, please keep up the incredible content!
Thank you so much for the kind words. I've really tried in these videos to explain things in the way I eventually came to think about them, in the hopes of giving people the scaffolding to wrap their minds around the more formal treatments of these topics. I'm glad it works for you, I'm really enjoying making these, and I can't wait to release the next one!
@@zetamath Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
There is a video on RUclips in which a Professor explains the gyroscopic effect to his students theoretically and exemplifies it experimentally after that. One of the comments reads : " I was watching this when my mom entered my room and I had to switch to porn because it was easier to explain." This is one of the most explosive jokes I have ever heard.
Wow! I have seen soooo many absolutely excellent videos related to this subject, I really didn’t think anybody could ever improve them any further. And yet here it is. This video has performed an analytic extension of my brain! 😄👏👏👏👏👏
Wow. Just wow. You lay out the concepts of analytic continuation clearly and succinctly. The "ah ha!" moment for me was pointing out how the absolute value function has a singular analytic continuation, but that continuation does NOT align with the evaluation of absolute value for negative numbers. This helps to clearly define what analytic continuation means: it's kind of a hypothetical, where we assume that the function has a valid definition on the complex plane, then there is only ONE valid definition, even if that definition is different from the function as defined for the reals, or even if the function is undefined on that part of the reals.
I am so interested in seeing the math behind analytical continuation of Zeta, but I am afraid I will get lost in the math without Zetamath to guide me.❤
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
I've been watching youtube for 20 years and I have never liked any video ever, until now! Thank you for putting in such effort, time, and sincerity in your explanation. Btw, if you don't mind me asking, what tools did you use to make this video?
We're really glad you liked it! These videos are a labor of love. Apart from off-the-shelf video and audio editing software, our videos are made chiefly using a mathematical animation tool called manim. There's a link to their website in this video's description!
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.
This is really outstanding work. I never did complex calculus in college so my only source of continued education on these topics is RUclips and Wikipedia. I got a sense from other videos of the rigidity of complex valued functions but this explanation is intuitive and insightful. Really curious why some functions are single-valued or multi-valued and can’t wait for the next video. This is very cool.
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.
This was very helpful for my thesis, which touches briefly on analytic continuation. I was very confused about the topic and couldn't find anything that explained it clearly until I stumbled upon your video. Thank you! :)
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
I want to echo many of the comments I've seen below. Your video is spectacular in the sense that it approaches analytic continuation in far superior manner to (all) the other videos I have watched regarding the Riemann zeta function.
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.
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!
Thank you so much for this fantastic work. I'm a nuclear engineer, but I have a love for math and physics. I casually want to learn more about the zeta function, not to win the millennium prize, but because analytic number theory is just really interesting. Please keep up the great content!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
This is fantastic! I first learned about the concept of analytic continuation from 3b1b and you helped clarify what it actually. Absolutely fantastic work here!
Seeing Euler's Formula and (more generally) rotation being found using analytic continuation was soo beautiful! I fololw the CTC channel almost everyday and have seen your (rather lovely) puzzles been featured. Today, I was curious about what zeta functions were and found your math channel and have absolutely loved another creation of yours today :D Edit: Just wanted to comment more
Looking for some help... Can anyone explain the answer to this question. "The local zoning ordinance requires that any property with impervious coverage that discharges more than 100,000 gallons of water in a 4-inch rainstorm must have a retention pond to handle the water. The proposed subject property will have 260,000 sf. of impervious surface. A cubic foot of water contains 7.48 gallons and retention ponds can hold no more that 500,000 gallons for safety reasons. There is no retention pond an the plan. What will this property discharge in a 4-inch rainstorm. Trying to use an HP C12 calculator... thank you.
Incredible video. I have a correction to make though. Around 27:00, you said that "every value of a complex function carries information about other points". However the correct statement should be that **analytic** complex functions have this property. This sentiment is again repeated at 39:00, but is seemingly contrasted with the real function |x|. You claim that real functions can be modified in one part without affecting other parts. However, |x| is a piecewise-defined function (it's equal to x if x is positive and -x if x is negative). If you similarly defined a complex function piecewise, e.g. f(z) = z for Re(z) ≥ 0 and f(z) = -z for Re(z) < 0, then you could again draw the exact same conclusion. In reality, the problem is that none of these functions are complex analytic, as |x| is really just √zz* (z* being conjugate) and it is easy to directly show that conjugation doesn't satisfy CR equations and is hence non-analytic. The main reason I stress this point is because there is a very tangible class of real functions that have the property of varying everywhere as a result of a change to any single point: polynomials! Every polynomial of any degree can be extrapolated by simply knowing a continuous segment and it's degree! And that continuous segment, strangely enough, can be as tiny as you want. This is exactly as you said for complex analytic functions, because polynomials are trivially analytic as they are just truncated Taylor series. If you know you have a quadratic polynomial and you have a small snippet of the curve around one of the roots, you can easily extrapolate this out to infinity. This actually works for any size segment, located anywhere. In fact, instead of needing a whole continuous infinitum of points from a segment, a polynomial of degree k can be fully recreated by k discrete points. We've dropped a whole ℵ class down! This set is generally chosen to be the zeros of the polynomial, but any k distinct points are sufficient. With this minimal set of information we use to describe a polynomial, it should be clear that modifying a single root will produce a different polynomial of the same degree. Back to the discussion of |x|, I believe the overall idea you are trying to convey is that you are trying to show the distinction between real analytic functions and complex analytic functions, but this example does not show that. |x| is not even real analytic. All that being said, incredible video. Extremely keen to watch this whole series, and especially excited to see a discussion of branch points and branch cuts at some point.
When I made this part, I definitely thought "This is going to upset the experts." But part of providing condensed explanations is allowing some degree of imprecision. Talking about "nice" functions does well enough for most purposes. Were I to talk about this to an analysis class, I would say : The difference between real and complex functions is that for complex functions C1 implies Analytic where as for real functions that is very much not the case. By the way, I think it is great when people put some of the technical details I omit in the comments, so well said!
@@zetamath I agree, and if I were in your position I would've said something like "nice complex functions cannot be modified in one place..." with a snippet explaining what "nice" means shown on screen for those who are curious that fades away after a brief period. The only reason it even struck to me comment is that I didn't recall any sort of asterisk being highlighted, so I just wanted to bring it up in the comments instead.
Bro, I'm just a 14 year old boy from central Europe and I'm very grateful for these youtube tutorials, nice work I think this video deserves to have more views than it really has. I have a question, where do you animate these videos, is it just python or some special program, I ask because I want to do some math videos also. Thanks for the answer.
Of course you can define a complex valued function to take on whatever values you please on the complex plane. It is only analytical functions that possess these rigid properties. Yes, it's semantics, but it's an important distinction
Why people use complex numbers. Because they relate to waves like sine wave cosine waves and merge etc. They can be represented as numbers as convenience. So number series was discovered. But waves are so many different types and so description is difficult. Some may have solution depending on sine wave or other harmonica is used. Like discrete waves pulse waves Etc. Zeta is specific to number line.
I love this approach. One learns a significant slab of real math with great clarity, and why it's important. Complementary to the equally excellent 3blue1brown style, I think.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions. Real is dual to imaginary -- complex numbers are dual. The integers are self dual as they are their own conjugates. Injective is dual to surjective synthesizes bijective or isomorphism. Elliptic curves are dual to modular forms. Subgroups are dual to subfields -- the Galois correspondence. Addition is dual to subtraction (additive inverses) -- Abstract algebra. Multiplication is dual to division (multiplicative inverses) -- Abstract algebra. Integration (summations, syntropy) is dual to differentiation (differences, entropy). Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics! The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. Points are dual to lines -- the principle of duality in geometry. Perpendicularity or orthogonality = duality in mathematics. "Always two there are" -- Yoda.
I am familiar with the concepts here. What isn't so easy, is actually how to use numerical methods to achieve analytic continuation itself. And, even if you can approximate it numerically, how can you prove that a particular zero lies exactly on the critical line? That's what I hope to learn from this series.
Great videos, thoroughly enjoyable and a lot of fun - probably the nicest introduction to complex analysis I have found. Why wasn't the series continued? What happened?
This is an excellent question, and ultimately not one I'm intending to get into in this series I'm afraid. To even state as a theorem what is being claimed here in a way that isn't a tautology requires a lot of nuance that would really derail the series. But it should not derail you from learning it! Any book on complex analysis will discuss this, but I personally am very fond of Gamelin as far as introductions to complex analysis go.
These videos are top-notch. Great graphics, explanations and story-telling. To get more subs, you should put out some "short stories" that are around 15 minutes long. Little math tidbits and interesting things. You can use a lot of your existing material. It's really a shame you only got 6.36K subs! (I'm a non-mathematician interested in the subject.)
I'm overall very happy with how the channel has been growing, but I certainly have been considering what sorts of things I could do as short videos that people would find meaningful. I'm always open to suggestions!
We're glad you liked the video! I have no idea why you can't Super Thanks it, it should be enabled. Just RUclips being weird I guess 🤷♂️ Maybe see if you can Super Thanks one of our other videos? Thank you regardless!
this is the best video I've seen on the subject ! very engaging, and all information was presented in a manner which really gave a good understanding and intuition of the basics :)
This is a wonderful video with very clear and intuitive explanantions of stuff and pulling in someone who has detatched for math a long time ! Thanks a lot for all the effort in this content creation.
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 🏆
Yes and it's great that someone finally explained the whole -1/12 business without just saying it's obviously not true and pointing out the incorrect methods used to get it. I also feel like none ever related it to the zeta function which seems wired since to very obviously is very related
Agree with that once complex differentiable implies infinitely differentiable point!
If you're calc teacher NEVER taught that how bad were they lol isnt that a fundamental property
@@edwardzachary1426 Actually Mathologer explained that very well on a RUclips a couple of years ago.
What is curious, the Ramanujan summation directly relates this -1/12 result to 1+2+3+... and that perfectly coincides with the analytic continuation of Riemann zeta. And this is not the random occasion: Ramanujan summation also relates 1+4+9+... to 0, in the accordance with the fact this is trivial zero of the zeta. Actually using the Ramanujan technique of summation of this kind of series series we will always arrive to the corresponding values of Riemann zeta function. And yet, the analytic continuation path taken seems to have no relation to Ramanujan summation of the divergent series! There must be a hidden deep connection between so much different math, which is well beyond my understanding.
@@nikitakipriyanov7260 What you wrote here aligns well with my understanding, and I link to that mathologer video in at least one of the videos in this series. It is an excellent source.
At least to the best of my knowledge, the hidden deep connection to which you refer is hidden from everyone, e.g. no one knows how to make sense of this connection.
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. ⭐️
This is one of the most well-made and accessible math videos I've watched on RUclips.
The best explaination of analytic continuation on RUclips!
After watching this whole video, my one nitpick that |x| isn't a great example for showing the rigidity in complex analysis, because it's not differentiable at 0. I feel like a key fact is that complex functions that are differentiable in an open set are locked-down in a way that differentiable real functions aren't. But if we give up differentiability then we can mess things up in the complex plane just as easily as on the real line. But everything else was really good, and this is the most accessible introduction to analytic continuation I've seen that doesn't try to hide the things that could "go wrong". Great video!
There is always a complicated balance in how much detail I want to get into about the formality of what is going on. I thought the absolute value example hinted at this without having to delve into the technicalities of what it means for a real valued function to be analytic.
Isn't that the point of choosing abs(x)?
@@baronvonbeandip abs(x) isn't a "differentiable real function". So it doesn't show what's special about differentiability in the complex setting
This has to be one of the most lucid, clear, and accessible explanations on analytic continuation I've ever seen. Thank you for this video.
This is the first time it has been years I am trying to understand seriously the true meaning of analytic continuation principles and technics, (all others who made videos like you, are like scary that we understand really the true meaning of Riemann hypothesis and we may prove or disprove it.) You have shown in this video exactly what does it mean: Expanding the zeta function with Taylor series principles to get zeta(z) defined in the whole complex plane, and the different outputs that we will get if we pass by thier singularities in the complex plane.
I personally the first time I wished that this video never end when the video ends! why ? because simply you are a true teacher what deserve all my respect for the insight what you gave to us that dismiss the whole darkness in this way.
What a nerdy guy! On the other hand, what a fantastically clear explainer!
I’ve always wanted to understand analytic continuation and this video did it for me
You're the kind of person who's voice makes me hungry
Amazing video btw, well done! I learnt something today
Great work. Congratulations. This is how math should be thought. Future of teaching is bright and this video is a great example.
What an excellent video! Definitely my favorite explanation of analytic continuation on RUclips. 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.
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.
Thank you so much for this incredible video. Everything is explained in a very comprehensible yet detailed manner. I am currently writing my final paper on the the Riemann Zeta Function and the connection between the values of its analytic continuation and those of (divergent) inifinte series, and this video has really helped me get started.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
Congratulations on 1.01K subscribers. You are hitting pay dirt!
Thanks! I'm super excited people like our content enough to get us up to 1k!
Sir, your videos -- while different in style than Grant's over at 3Blue1Brown -- are equal in clarity and educational quality. Bravo! You deserve several orders of magnitude of more subscribers.
This video is just amazing! It explains analytical continuation like no one has ever been able to do so. Not only that, it explains the nature of complex numbers and functions especially the illusive logarithm which i had a great difficulty understanding when i took complex analysis in the university! The explanation is so deep that after the 50 min presentation one is able to retrace the whole argument of analytic continuation from really scratch in ones own mind! I sometimes wonder why such easy concepts were taught to us in the university in such a convoluted and incomprehensible way!!! And I also wonder how much time was put to produce this beautiful video. The fact is that many people will appreciate this effort tremendously!!
As others have said, you did a great job putting these thorough videos together.
Thank you so much for this wonderful explanation !!
Really eye-opening for me. I have never thought of analytic continuation in this way. I thought I knew but I did not.
The best I have seen so far. Thanks again!!
Thanks , thanks, thanks a lot. It is really the best video for learning analytic continuation.
Excellent video which is complementary to all the info you find on the web but of which you understand only a fraction
Blown away how amazing that was. Bravo.
This is by far the best video on this topic I have found. In fact, this is the best presentation on this topic I have seen. I'm really looking forward to the next video! I wish I was able to watch this during my PhD, I would have benefited greatly!
Please, please keep up the incredible content!
Thank you so much for the kind words. I've really tried in these videos to explain things in the way I eventually came to think about them, in the hopes of giving people the scaffolding to wrap their minds around the more formal treatments of these topics. I'm glad it works for you, I'm really enjoying making these, and I can't wait to release the next one!
@@zetamath Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
Thank you for such a nice video on analytic continuation....
Best video on analytic continuation I've ever seen. Where was this 5 years ago when I was in Complex Analysis 😂
These videos are criminally underwatched.
There is a video on RUclips in which a Professor explains the gyroscopic effect to his students theoretically and exemplifies it experimentally after that.
One of the comments reads : " I was watching this when my mom entered my room and I had to switch to porn because it was easier to explain." This is one of the most explosive jokes I have ever heard.
You guys are undoubtedly the best math channel on RUclips!
FINALLY I understand how analytic continuation works!! Thanks loads, man!
What a great lesson! I learned a lot and am sure to learn more on subsequent viewings!
Wow! I have seen soooo many absolutely excellent videos related to this subject, I really didn’t think anybody could ever improve them any further. And yet here it is. This video has performed an analytic extension of my brain! 😄👏👏👏👏👏
Wow. Just wow. You lay out the concepts of analytic continuation clearly and succinctly. The "ah ha!" moment for me was pointing out how the absolute value function has a singular analytic continuation, but that continuation does NOT align with the evaluation of absolute value for negative numbers. This helps to clearly define what analytic continuation means: it's kind of a hypothetical, where we assume that the function has a valid definition on the complex plane, then there is only ONE valid definition, even if that definition is different from the function as defined for the reals, or even if the function is undefined on that part of the reals.
I am so interested in seeing the math behind analytical continuation of Zeta, but I am afraid I will get lost in the math without Zetamath to guide me.❤
WOW WOW WOW, incredible work man. super good stuff, perfect video.
Yes, wonderful videos. Extremely enlightening and well done.
Nothing better than a funky function ✌
Reverse engineers
This video is incredibly well made and informative.
It really helped me understand analytic continuation
Awesome, keep making these marvelous videos. Thank you.
I have never seen someone who rivals 3b1b in terms of clarity and introducing something intuitively. Thank you for such a clear explanation.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
I've been watching youtube for 20 years and I have never liked any video ever, until now! Thank you for putting in such effort, time, and sincerity in your explanation. Btw, if you don't mind me asking, what tools did you use to make this video?
We're really glad you liked it! These videos are a labor of love.
Apart from off-the-shelf video and audio editing software, our videos are made chiefly using a mathematical animation tool called manim. There's a link to their website in this video's description!
@@zetamath ty so much!
best channel ever
Muito bom este vídeo. Muito esclarecedor sobre alguns conceitos de difícil perceção em Análise Complexa.
Okay now imagine this plot twist:
This video series on the Zeta function eventually leads up to a proof of the Riemann hypothesis.
Excellent video. Thank you.
I don’t know why I haven’t subscribed to your channel already, but that glitch is now sorted. Brilliant content!
Wow, the square root of 4 is not only 2 but also -2.🤯
Great video!! I realy like to learn with examples. Thank you 😊
Absolutely outstanding!
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.
This is really outstanding work. I never did complex calculus in college so my only source of continued education on these topics is RUclips and Wikipedia. I got a sense from other videos of the rigidity of complex valued functions but this explanation is intuitive and insightful. Really curious why some functions are single-valued or multi-valued and can’t wait for the next video. This is very cool.
Awesome explanation sir 👏 👌
This video is amazing. It's been 12 and a half years since I saw complex analysis in my university, and this video refreshed my memory. =)
beautiful clear n smooth explanations too much informations I love this video . thanks .
A really well made piece of education, it helped me make sense of a lot of concepts about complex analysis... Thanks!
Hi Zetamath - thought I'd drop by and check out your math videos - incredible stuff. Beautifully animated and brilliantly simple in explanation.
Thank you very much for the video.
Absolutely love the format and detail, with simple visualizations. Keep it up!!
Thank you so much, I'm glad this format works for you!
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.
This was very helpful for my thesis, which touches briefly on analytic continuation. I was very confused about the topic and couldn't find anything that explained it clearly until I stumbled upon your video. Thank you! :)
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
Your Riemann zeta math videos are amazing. I’ve watched every video at least twice. I’m looking forward to the next video about integrals…
So, just to confirm, analytic continuation was just using the polynomial version (Taylor series) of the function, and repeating that step?
Your videos are amazing I love them please keep making more!
I want to echo many of the comments I've seen below. Your video is spectacular in the sense that it approaches analytic continuation in far superior manner to (all) the other videos I have watched regarding the Riemann zeta function.
This channel is awesome
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.
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!
Are you telling me I can understand analytic continuation with very basic calculus?? Holy shit. This video is literally a game-changer.
Thank you so much for this fantastic work. I'm a nuclear engineer, but I have a love for math and physics. I casually want to learn more about the zeta function, not to win the millennium prize, but because analytic number theory is just really interesting. Please keep up the great content!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
Thanks, this is a fantastic series which finally helped me understand analytic continuation and the zeta function.
Amazing!
This is up there with the very best of maths videos on RUclips, and all your others are up there with it. Great job!
Great lecture!!!
PERFECT JUST AMAZING THANKSOMUCH
lmao the i into the bombelli transition
You have a real talent for explaining complicated math in a way, i believe, anyone can understand. Very interesting video! 😃
Huge thanks for what you are doing. It was a really nice video and I'm looking forward to see some new ones. Regards from Poland
Fabulous.😊 Thank you!
This is fantastic! I first learned about the concept of analytic continuation from 3b1b and you helped clarify what it actually. Absolutely fantastic work here!
Seeing Euler's Formula and (more generally) rotation being found using analytic continuation was soo beautiful!
I fololw the CTC channel almost everyday and have seen your (rather lovely) puzzles been featured. Today, I was curious about what zeta functions were and found your math channel and have absolutely loved another creation of yours today :D
Edit: Just wanted to comment more
It's always awesome to see people enjoying both sides of zetamath!
Looking for some help... Can anyone explain the answer to this question. "The local zoning ordinance requires that any property with impervious coverage that discharges more than 100,000 gallons of water in a 4-inch rainstorm must have a retention pond to handle the water. The proposed subject property will have 260,000 sf. of impervious surface. A cubic foot of water contains 7.48 gallons and retention ponds can hold no more that 500,000 gallons for safety reasons. There is no retention pond an the plan. What will this property discharge in a 4-inch rainstorm. Trying to use an HP C12 calculator... thank you.
You should do a series teaching from Davenport’s book on analytic number theory. I recommend the edition from 1967.
Beautiful job. I subscribed.
Incredible video. I have a correction to make though.
Around 27:00, you said that "every value of a complex function carries information about other points". However the correct statement should be that **analytic** complex functions have this property.
This sentiment is again repeated at 39:00, but is seemingly contrasted with the real function |x|. You claim that real functions can be modified in one part without affecting other parts. However, |x| is a piecewise-defined function (it's equal to x if x is positive and -x if x is negative). If you similarly defined a complex function piecewise, e.g. f(z) = z for Re(z) ≥ 0 and f(z) = -z for Re(z) < 0, then you could again draw the exact same conclusion.
In reality, the problem is that none of these functions are complex analytic, as |x| is really just √zz* (z* being conjugate) and it is easy to directly show that conjugation doesn't satisfy CR equations and is hence non-analytic. The main reason I stress this point is because there is a very tangible class of real functions that have the property of varying everywhere as a result of a change to any single point: polynomials! Every polynomial of any degree can be extrapolated by simply knowing a continuous segment and it's degree! And that continuous segment, strangely enough, can be as tiny as you want. This is exactly as you said for complex analytic functions, because polynomials are trivially analytic as they are just truncated Taylor series.
If you know you have a quadratic polynomial and you have a small snippet of the curve around one of the roots, you can easily extrapolate this out to infinity. This actually works for any size segment, located anywhere. In fact, instead of needing a whole continuous infinitum of points from a segment, a polynomial of degree k can be fully recreated by k discrete points. We've dropped a whole ℵ class down! This set is generally chosen to be the zeros of the polynomial, but any k distinct points are sufficient.
With this minimal set of information we use to describe a polynomial, it should be clear that modifying a single root will produce a different polynomial of the same degree.
Back to the discussion of |x|, I believe the overall idea you are trying to convey is that you are trying to show the distinction between real analytic functions and complex analytic functions, but this example does not show that. |x| is not even real analytic.
All that being said, incredible video. Extremely keen to watch this whole series, and especially excited to see a discussion of branch points and branch cuts at some point.
When I made this part, I definitely thought "This is going to upset the experts." But part of providing condensed explanations is allowing some degree of imprecision. Talking about "nice" functions does well enough for most purposes.
Were I to talk about this to an analysis class, I would say : The difference between real and complex functions is that for complex functions C1 implies Analytic where as for real functions that is very much not the case.
By the way, I think it is great when people put some of the technical details I omit in the comments, so well said!
@@zetamath I agree, and if I were in your position I would've said something like "nice complex functions cannot be modified in one place..." with a snippet explaining what "nice" means shown on screen for those who are curious that fades away after a brief period. The only reason it even struck to me comment is that I didn't recall any sort of asterisk being highlighted, so I just wanted to bring it up in the comments instead.
Bro, I'm just a 14 year old boy from central Europe and I'm very grateful for these youtube tutorials, nice work I think this video deserves to have more views than it really has. I have a question, where do you animate these videos, is it just python or some special program, I ask because I want to do some math videos also. Thanks for the answer.
Of course you can define a complex valued function to take on whatever values you please on the complex plane. It is only analytical functions that possess these rigid properties. Yes, it's semantics, but it's an important distinction
Nice video need more math channels like this❤️❤️
Why people use complex numbers. Because they relate to waves like sine wave cosine waves and merge etc. They can be represented as numbers as convenience. So number series was discovered. But waves are so many different types and so description is difficult. Some may have solution depending on sine wave or other harmonica is used. Like discrete waves pulse waves Etc. Zeta is specific to number line.
I love this approach. One learns a significant slab of real math with great clarity, and why it's important. Complementary to the equally excellent 3blue1brown style, I think.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Real is dual to imaginary -- complex numbers are dual.
The integers are self dual as they are their own conjugates.
Injective is dual to surjective synthesizes bijective or isomorphism.
Elliptic curves are dual to modular forms.
Subgroups are dual to subfields -- the Galois correspondence.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Homology (syntropy, convergence) is dual to co-homology or dual homology (entropy, divergence).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics!
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
Points are dual to lines -- the principle of duality in geometry.
Perpendicularity or orthogonality = duality in mathematics.
"Always two there are" -- Yoda.
It is very well explained. Thanks!
I am familiar with the concepts here. What isn't so easy, is actually how to use numerical methods to achieve analytic continuation itself. And, even if you can approximate it numerically, how can you prove that a particular zero lies exactly on the critical line? That's what I hope to learn from this series.
Good news, this is exactly the intention of this three video series. The second one is out, and the third is still being animated.
Great videos, thoroughly enjoyable and a lot of fun - probably the nicest introduction to complex analysis I have found. Why wasn't the series continued? What happened?
Why does the radius of convergence always follow the singularities like this?
This is an excellent question, and ultimately not one I'm intending to get into in this series I'm afraid. To even state as a theorem what is being claimed here in a way that isn't a tautology requires a lot of nuance that would really derail the series.
But it should not derail you from learning it! Any book on complex analysis will discuss this, but I personally am very fond of Gamelin as far as introductions to complex analysis go.
45:45 e^(2*n*pi)=1 for integral n, so the analytic continuation might have some "meaning". Is this just a coincidence?
By pleasant coincidence, we talk about this in the next video!
It looks like the cube root problem like relative of factoring that maybe good for quantum computer to solve?
Can you make video of 'irreducibles'?
Props for explaining analytic continuation but all in all the video seems too long and over explained.
These videos are top-notch. Great graphics, explanations and story-telling. To get more subs, you should put out some "short stories" that are around 15 minutes long. Little math tidbits and interesting things. You can use a lot of your existing material. It's really a shame you only got 6.36K subs! (I'm a non-mathematician interested in the subject.)
I'm overall very happy with how the channel has been growing, but I certainly have been considering what sorts of things I could do as short videos that people would find meaningful. I'm always open to suggestions!
Why can't I Super Thanks this video...? You really deserve it for making this.
We're glad you liked the video! I have no idea why you can't Super Thanks it, it should be enabled. Just RUclips being weird I guess 🤷♂️ Maybe see if you can Super Thanks one of our other videos? Thank you regardless!
This is one of the best math videos I've watched in a while, thank you!
ur videos and explanations are awesome - very clear and logical
this is the best video I've seen on the subject !
very engaging, and all information was presented in a manner which really gave a good understanding and intuition of the basics :)
This is a wonderful video with very clear and intuitive explanantions of stuff and pulling in someone who has detatched for math a long time ! Thanks a lot for all the effort in this content creation.