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 🏆

@edwardzachary1426

2 жыл бұрын

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

@tanchienhao

Жыл бұрын

Agree with that once complex differentiable implies infinitely differentiable point!

@tens0r884

Жыл бұрын

If you're calc teacher NEVER taught that how bad were they lol isnt that a fundamental property

@nikitakipriyanov7260

Жыл бұрын

@@edwardzachary1426 Actually Mathologer explained that very well on a KZread 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.

@zetamath

Жыл бұрын

@@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.

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

These videos are criminally underwatched.

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. ⭐️

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.

What an excellent video! Definitely my favorite explanation of analytic continuation on KZread. 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.

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.

What a nerdy guy! On the other hand, what a fantastically clear explainer!

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!

@zetamath

2 жыл бұрын

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.

@baronvonbeandip

Жыл бұрын

Isn't that the point of choosing abs(x)?

@diribigal

Жыл бұрын

@@baronvonbeandip abs(x) isn't a "differentiable real function". So it doesn't show what's special about differentiability in the complex setting

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! 😄👏👏👏👏👏

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!

@zetamath

2 жыл бұрын

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!

@hyperduality2838

5 ай бұрын

@@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.

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.

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

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

@hyperduality2838

5 ай бұрын

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 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. =)

You guys are undoubtedly the best math channel on KZread!

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.

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.

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

this video is one of the best references to analytic continuation that is in about math yt. the goat!🐐

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

I’ve always wanted to understand analytic continuation and this video did it for me

Excellent video which is complementary to all the info you find on the web but of which you understand only a fraction

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!

@hyperduality2838

5 ай бұрын

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.

Great work. Congratulations. This is how math should be thought. Future of teaching is bright and this video is a great example.

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.

Why can't I Super Thanks this video...? You really deserve it for making this.

@zetamath

Жыл бұрын

We're glad you liked the video! I have no idea why you can't Super Thanks it, it should be enabled. Just KZread being weird I guess 🤷‍♂️ Maybe see if you can Super Thanks one of our other videos? Thank you regardless!

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.

this is by far the best video on complex analyses

Congratulations on 1.01K subscribers. You are hitting pay dirt!

@zetamath

2 жыл бұрын

Thanks! I'm super excited people like our content enough to get us up to 1k!

You're the kind of person who's voice makes me hungry Amazing video btw, well done! I learnt something today

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 is up there with the very best of maths videos on KZread, and all your others are up there with it. Great job!

I wish that such videos had been available when I was in Math grad school.

The best explaination of analytic continuation on KZread!

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

This video is incredibly well made and informative. It really helped me understand analytic continuation

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.

@hyperduality2838

5 ай бұрын

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 a gem! Initially wanted to say 'real' gem, then noticed that in literal truth it is 'complex'! (Pun intended, of course.) Well done!

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! :)

@hyperduality2838

5 ай бұрын

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 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.

@hyperduality2838

5 ай бұрын

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.

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.

As others have said, you did a great job putting these thorough videos together.

You have a real talent for explaining complicated math in a way, i believe, anyone can understand. Very interesting video! 😃

I knew what analytic continuation was trying to do, but never knew how it was actually computed Thank you for this

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

@zetamath

7 ай бұрын

It's always awesome to see people enjoying both sides of zetamath!

Absolutely love the format and detail, with simple visualizations. Keep it up!!

@zetamath

Жыл бұрын

Thank you so much, I'm glad this format works for you!

What I like most about your videos is the implicit presentation of topics as a design/engineering task. The mathematician is considering a problem, and DESIGNS and BUILDS a way to solve it. Very different from "here's a definition, here's a theorem, and here's a proof" that's unfortunately typical.

This is really outstanding work. I never did complex calculus in college so my only source of continued education on these topics is KZread 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.

Thanks , thanks, thanks a lot. It is really the best video for learning analytic continuation.

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!!

Math is so freaking cool! My high school could not conceive of all the crazy stuff I'd see starting down the math path.

Yes, wonderful videos. Extremely enlightening and well done.

I don't think I've ever commented on a KZread video. This explanation on analytic continuation is absolutely superb and it almost brought tears to my eyes as I remembered the amazing math (calculus 101) lessons from my high school teacher. He had a very similar style as yours (without the animations of course as it was 30 years ago) and has been the most influential teacher in my life! Watching your video I felt like Anton Egon tasting Ratatouille!

Clearest explanation of analytic continuation I've ever seen

Just adding to the pile of praise. This was masterful education. Thanks.

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…

lmao the i into the bombelli transition

Nice video need more math channels like this❤️❤️

Deep and humble!!! All teachers should be like this

A really well made piece of education, it helped me make sense of a lot of concepts about complex analysis... Thanks!

Your videos are amazing I love them please keep making more!

I never really understood why textbooks on complex analysis emphasized the radius of convergence, all the time, until I saw your video. Really good material.

@hyperduality2838

5 ай бұрын

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.

These videos are incredible. As someone who has a degree in math but never took complex analysis, these videos are super accessible and actually make these ideas intuitive, yet also have a level of rigor that makes me feel like you're not just glossing over the messy details.

@hyperduality2838

5 ай бұрын

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!

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

Blown away how amazing that was. Bravo.

This is the best explanation of analytic continuation I have seen that has all of the details but is easy to understand.

Nothing better than a funky function ✌

@brendawilliams8062

Жыл бұрын

Reverse engineers

Hi Zetamath - thought I'd drop by and check out your math videos - incredible stuff. Beautifully animated and brilliantly simple in explanation.

Thank-you for such a detailed and straight-forward expose on analytic continuation!

Thank you for this. I’m excited to see the next video on integration in the complex domain.

Thank you for such a nice video on analytic continuation....

This is one of the best math videos I've watched in a while, thank you!

Very good. I didn't understand the idea of analytic continuation before. I do now. Thanks.

Thank you for explaining the term analytic continuation. Every time I came across the term, I got at best a line of explanation. It really takes a 1-hour video, which amounts to > 20 written pages. 🙂🙂🙂

this is the greatest video ever

ur videos and explanations are awesome - very clear and logical

This is a great video, looking forward to checking out the rest of your channel!!

This was extremely well done! Looking forward to more of your videos:)

Have seen many math videos on analytic continuations. This is easily one of the best explanations. Thank you for this video. Hoping to see and learn more.

amazing videos, amazing maths, amazing popularization

This is an absolutely superb exposition - thanks so much

Watching again - you explanations are so very good - I can use some of these words to explain these concepts to my son. It fascinating how one needs to “steer around” the singularities in the neighborhood to keep the results meaningful.

So clear!! Much better than Roger Penrose’s “Road to Reality” explanation that took 8 chapters to explain it.

Wonderfully clear and articulate explanation of analytic continuation. What a thrilling, albeit totally serendipitous, discovery!! I have now subscribed to your channel. One minor point: At one point in your presentation, you refer to a number as being complex when it has no real component. My understanding, (admittedly from my high school teacher), was a complex number with no real component was referred to as an imaginary number, and that a complex number has both a real and an imaginary component. I can’t wait to watch your other videos!

Best analytic continuation lesson yet

What a great lesson! I learned a lot and am sure to learn more on subsequent viewings!

Amazing video!! You managed to explain complex (pun intended) topics so clearly! Many of these visuals clicked with me as I have tried reading complex analysis text previously and struggled to develop an intuition :) keep up the great work! Subscribed!

FINALLY I understand how analytic continuation works!! Thanks loads, man!

You talks are very clear. I have enjoyed them a lot and hope you produce more :)

So happy this video came out, last two parts have been great! Now i'm gona enjoy this one :D

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 :)

Thank you sir for making these videos. Keep it up!

This was lovely, and I'm sure very time consuming to make, so thank you!

Very well made, illustrative, well explained, complementary to other channels that I now need to re-watch, such as the series by Petra Bonfert-Taylor

@hyperduality2838

5 ай бұрын

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.

Wow. This was extremely informative and thorough. Many thanks!

WOW WOW WOW, incredible work man. super good stuff, perfect video.