It's a shame this video wasn't released as part of 3blue1brown's SoME series, you'd have blown everyone out of the water!
@imaginaryangle
10 ай бұрын
Thank you for saying that! I guess I needed more practice to iron out the **ahem** kinks 😁
@sleepycritical6950
10 ай бұрын
@@imaginaryangleclap…clap…clap
@williestroker3404
9 ай бұрын
@@hyperduality2838 yeah the number two shows up a lot...almost like it's the commonest nontrivial number.
@williestroker3404
9 ай бұрын
@@hyperduality2838 ok I guess it's a joke by now. But why can't you replace 'duality' with 'plurality'? Your enlightened BS is based on the most simply ignorant case it can be...why not obsess over trinity?
@ewthmatth
8 ай бұрын
@@hyperduality2838 sir, this is a Wendy's
@lumipakkanen3510
10 ай бұрын
I just love how the visualized elements jiggle when you mention them. "He's talking about me, yay!" ❤
@imaginaryangle
10 ай бұрын
Gotta show them affection 😉
@Procyon50
10 ай бұрын
When you revealed the "3d" structure of the cube root function by rotating (24:26), wow. I actually gasped.
@Verrisin
10 ай бұрын
whow, redefining plotting as `z = x + f(x)i` is such a brilliant idea!
@ILSCDF
10 ай бұрын
Wow, this channel is so underrated
@imaginaryangle
10 ай бұрын
Thank you! 🤗
@elilevio6948
10 ай бұрын
I completely agree Amazing
@wiltondewilligen3595
10 ай бұрын
Criminally so
@ILSCDF
10 ай бұрын
@@enantiodromia maybe, but my comment speaks 100% truth
@ILSCDF
10 ай бұрын
@@imaginaryanglebtw, congrats, this is now your most viewed video
@ER-uq9jw
10 ай бұрын
So, uh, thank you, for this. I haven’t enjoyed a math video this much since I discovered 3blue1brown a few years ago. This was amazing. I really loved that you opened this topic with a seemingly simple question and explored its complexity and beauty, while building a sense of intuition. Also, the pressed flowers metaphor was everything. So good, I’m obsessed. Chef’s kiss. Fingers crossed will get to see an explainer on the irrational powers someday.
@imaginaryangle
10 ай бұрын
Thank you so much for sharing those feelings in such detail! It really reaches me and gives me so much joy! I hope I get around to this topic, it's one of my favorites.
@AvanaVana
10 ай бұрын
This channel is one of my secret kinks. Thank you!
@tiagobeaulieu1745
10 ай бұрын
Never would I expect to find out the exponential function has a sort of focus, but its location does seem to line up with intuition. This was great work!
@imaginaryangle
10 ай бұрын
Thank you! I was also surprised!
@WhattheHectogon
10 ай бұрын
That was just wonderful....very enlightening, you should be quite proud of this work!
@imaginaryangle
10 ай бұрын
Thank you so much! A lot of love went into it!
@WhattheHectogon
10 ай бұрын
@@imaginaryangle I'm curious to know if you have found any kind of formula for the location of those "foci" for functions other than degree 0, 1, or 2 or the simple higher degree ones like x^n with no other terms
@imaginaryangle
10 ай бұрын
@@WhattheHectogon No, just the simple cases you've listed here (you can see it in the Desmos graphs given in the description). I also didn't come up with an elegant approach to look for it. Do you have an idea?
@lizzycoax
10 ай бұрын
yeah, this is awesome
@nicholassullivan6105
10 ай бұрын
@@imaginaryangle It seems like they happen where the derivative of x + f(x)i is 0, since the lines 'bunch up' around the kinks. So they would be the x where f'(x) = i, plugged back into x + f(x)i. Don't know about a general explicit formula though, and it doesn't explain why they would be like foci. Great video by the way!
@AThagoras
10 ай бұрын
Wow! I studied complex analysis many years ago, and while I understood it well enough to get good marks in assignments and exams, I always felt that I didn't really understand it. It's like a jigsaw puzzle where I have all of the pieces and I know which pieces connect to which other pieces, but I can't see the whole picture. This gives me a new way to visualize and think about analytic functions and see the whole picture. I find, that I can't understand anything in mathematics unless I can find a way to visualize it. Thank you!
@AThagoras
9 ай бұрын
@@hyperduality2838 I do agree. I came to the same conclusion myself. "Being and non-being create each other. Difficult and easy support each other. Long and short define each other. High and low depend on each other. Before and after follow each other." -- Lao Tzu
@cheptan1996
10 ай бұрын
the title is a rollercoaster of emotions. secret (ok I like secrets) kinks (oooh sausy) of elementary (hol'up uhh) functions (ah ok)
@dolthhaven8564
10 ай бұрын
this is literally the best day of my life
@Sanchuniathon384
10 ай бұрын
This was awesome, I love seeing the Riemann sphere existing still hidden in this. Maybe you can do a video showing what's going on when you go into the extended complex plane with the Riemann sphere?
@imaginaryangle
10 ай бұрын
Thank you! That might come up at some point if I find a good story to tie it together.
@j.b.4090
9 ай бұрын
@@hyperduality2838but then why do we exist in three dimensions and not a multiple of two?
@samuelyigzaw
8 ай бұрын
@@imaginaryangle yes the Riemann Sphere was the only thing missing from this video. Adding it would've been perfect! An entirely geometric interpretation of all complex numbers, while also making your "complex infinity" (really just an unsigned infinity like 0) make intuitive sense. Better yet, graphing on the actual Riemann Sphere would show what physically happens when your graphs like 1/x shoot off to infinity. Hint: 1/x is just a Mobius transformation.
@SOBIESKI_freedom
10 ай бұрын
Marvelous and gorgeous! Please produce more like this. Truly enlightening and edifying. It would be fantastic to see more of the 3D renderings, though. All becomes clear when you add more dimensions. Keep up the good work. 👌👍👏
@imaginaryangle
10 ай бұрын
Thank you! As for 3D, my animation skills are not keeping up with everything I'd want to show. But I'm learning!
@EdTLive
9 ай бұрын
I have no idea what any of this means, but I like the smooth, soothing and synergistic voice of the narrator.
@sigmascrub
9 ай бұрын
I'll be honest, I read "Secret Kinks of Elementary" and I was _very_ concerned about how that was gonna end 😅
@dsdy1205
7 ай бұрын
💀
@douglasstrother6584
4 ай бұрын
"Complex Variables" by John W. Dettman (published by Dover) is a great read: the first part covers the geometry/topology of the complex plane 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. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation. The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.
@SandipChitale
10 ай бұрын
This is brilliant! Very brilliant! This channel belongs in the same league as Mathologer and 3Blue1Brown. This has echoes in three other videos: Mathologer - Times Tables, Mandelbrot and the Heart of Mathematics - where multiple foci of cardioids appear Welch Labs - Imaginary Numbers Are Real [Part 13: Riemann Surfaces] 3Blue1Brown - Taylor series | Chapter 11, Essence of calculus I wish imaginary numbers had more real name (like orthogonal numbers or some such - this idea was suggested by Riemann I think) so that they will not get a short shrift and thus allow development of more intuition about them. I know "imaginary" is just a word but sometime sociologically it has an effect of apathy. I think you are helping us develop that intuition which of course should be followed by mathematical rigor. But creativity starts with intuition.
@imaginaryangle
10 ай бұрын
Thank you so much, this means a lot! And you're right, my focus is more on assisting the building of intuition. It's awesome that there's a whole ecosystem of math educators and each of us can dive into our own approach without fear that something important won't be covered. And I might be biased (I definitely am), but I like the word "imaginary" 😉
@SandipChitale
10 ай бұрын
@@imaginaryangle I guess "imaginary" frees one to be more creative :) So in that sense I agree.
@mcmaho17
10 ай бұрын
This is excellent! Thank you for making it.
@imaginaryangle
10 ай бұрын
My pleasure! 😊
@oleksii7899
10 ай бұрын
Surprisingly low amount of subs for this channel, the vid is really insighful and clearly a lot of effort is put into it! Thanks!
9 ай бұрын
So... I was washing my dishes looking at KZitem videos because, you know, hardly anyone just enjoys washing dishes. And then I started looking at your video and actually stopped washing because my brain needs more processing power for your explanation. I stood there, hands wet, for the whole video. Great video sir. I liked it very much.
@phitsf5475
10 ай бұрын
You had me at secret kinks, you lost me at elementary functions. I love math but I can't do math. RIP
@pineapplequeen13
10 ай бұрын
This is the most intuitive way anyone has tried to explain to me the connections that each power graph has to any other power graph. Using complex number space and compressing it to fit on two and three axes really does show a lot of what's hidden on the real number line. And it was done in a way that retained the shape and certain key features of each power graph. Bravo!!!
@HypocriticalElitist
8 ай бұрын
If one of these days I ever study complex numbers, I'll rewatch this video and let it blow my mind for more than just the pretty visualizations.
@imaginaryangle
8 ай бұрын
Maybe check out my video about them: kzitem.info/news/bejne/z6KntYVmr3x3Y4Y
@Alexus00712
9 ай бұрын
I love how this title attracted two very different yet equally interesting people, some might fall in the venn diagram intersecting both groups.. Like me who finds maths interesting, and would also like to know about how secretly kinky elementary functions are..
@KarlWork-n3i
9 ай бұрын
AMAZING, BRILLIANT, INCREDIBLE, that's just FANTASTIC. Best mathematics video I've seen for complex numbers. Amazing beautiful video THANK YOU.
@shantilkhadatkar1195
10 ай бұрын
This is good stuff. Should put this in submission for Jjjackfilms.
@purple_sky
7 ай бұрын
Wowwww amazing visualisations! I love seeing functions' level curves and I don't think I've seen this used as a way of visualising complex functions before - I feel like I understand these functions better now, thank you so much for making this!!
@srandom3867
10 ай бұрын
Comment for the algorithm, this is pretty great and I love how you made the visuals work even under constraints
@fightocondria
10 ай бұрын
Im seriously looking forward to more deep dives from you
@twelvefootboy
10 ай бұрын
Captivating. I gorged on number theory and numberphile videos etc.. until I got burned out on the talking head, crude construction paper or blackboard approach. As much as I liked the presenters, I needed a snappier pace. This was the fastest 30 minute math vid I've seen in a while.
@flyingcroissant8555
10 ай бұрын
Reaching the end and watching everything come together was really cool to see
@ramziabbyad8816
10 ай бұрын
Polynomials and rational functions is all I ever needed
@thekutay25
10 ай бұрын
This is the best video I have watched in a long while, teaching me something elementary yet revealing about math. Looking forward to other videos!
@gebrehiwotewnetu358
9 ай бұрын
Fantastic. One of the best ways to visualise, theorise and conceptualise so many different parts of geometry, rays, graphs, and ellipses I have ever encountered.
@salsuginusrex5196
9 ай бұрын
Mind. Blown. Division by zero equals complex infinity, a circle of imaginary radius zero with a direction undetermined. Fantastic video, especially if I ever decide to do mushrooms.
@ghostchris519
10 ай бұрын
Actually one of my favorite videos on KZitem
@oooouaa
10 ай бұрын
nice work, bro rather than pumping out ai shitty content, you chose the true path of making quality content. this way, you will gain even more subscribers. wish you much very luck. thank you for making math more loveable. sincerely, freshman year cs major student. also i smell russian accent. are you, by any chance, russian? (i am though.)
@imaginaryangle
10 ай бұрын
Thank you! Math needs love ;) I'm not Russian, but my mother tongue is Slavic.
@LukePalmer
10 ай бұрын
Thank you, this is great! I get these glimmers of how beautiful complex analysis is, but it's too much to comprehend at once, I only get pieces. So thanks for helping me assemble a little bit more. I was dubious about the x.+ i f(x) trick at first but it's actually a pretty interesting tool for visualization. Thanks again!
@lexscarlet
9 ай бұрын
This is absolutely alien intelligence So so good man, excellent work. This is one of the best videos I've seen that translates math almost completely into art. You need to take these ideas and perspective shifts and put em up as AR assets because I would totally PAY to get to interact with knowledge like this. There's some that teach you foil, and there's some that incrementally bump up the understanding of everything you know anything about. This is totally core human curriculum
@enviroptic3342
9 ай бұрын
I think I would call this a pattern approach. Normally before we had the advent of computer graphics, we weren't able to see comparative variables without extensive, time-consuming drawing. This approach gives us a better understanding of effects and patterns that develop by creating multi-variable or parameterized. The parameterized graphs shows the interesting points in the equations by displaying the patterns. Keith Devlin, a mathematics professor of Stanford U. recently mentioned that we should start thinking in patterns rather than simply in equations since the patterns give us more information on the effect of equations on many processes.
@QP9237
5 ай бұрын
I tried playing with parametric representation on desmos to visualize space transformation when using complex variables, this really reminded me of that. In a similar vein I wanted to understand and play with raising non-unitary complex numbers (a+bi) to non-unitary complex powers which lead me to finding my favorite number: Gelfond's Constant (-1)^(-i)=e^pi.
@sock7896
10 ай бұрын
I always kinda viewed it in my head as a sort of 3 dimensional cylindrical system, with the x axis feeding in real numbers and then receiving magnitudes (y axis) and angles rotation (ø axis I guess, rotating around the x). there's some fun visualisations to be had in looking at the roots of unity or w/e for non integers, y'know of the form x^a=1. the way solutions pop in at and right after even integers and gradually repel each other as they spawn and split away is very pretty
@sock7896
10 ай бұрын
LMAO I should have just waited a minute further into the video
@JojiThomas7431
10 ай бұрын
Beautiful video. Nice presentation and ideas.
@Coolbluepinguen
10 ай бұрын
Just commenting to help promote this, fucking wild. I feel like many people have had the same question about the forms of graphs between powers. I got to the conclusion that its dirty and doesn't make to much sense because decimals (say x^1.7) turn into fractions (x^ 17/10) which turn into gross powers and roots (17th-root(x^10)), and that's it. Thank you for the new intuition. Most people here know 3blue1brown as the king of teaching like that, but man, this video is right up there with him. Edit: I recently rewatched Morphocular video: What Lies Between a Function and Its Derivative? | Fractional Calculus. I am VERY curious how someone smarter than me might relate these two concepts into the ultimate "graphing functions with powers and derivatives between whole numbers" video
@alex_creeper2752
10 ай бұрын
This video is simply amazing! Since i introduced myself to complex numbers through some youtube videos and wikipedia articles, i always wondered what were these rainbow-looking images, that on some resources were shown as "graphs". For a high schooler, that did not really learn anything complicated about calculus, (not even mentioning complex "world") this was rather distressing to read the information in overcomplicated and scientific way that is shown in almost all articles and pages. This video just united anything that i knew about essense of graphs and complex numbers and i am absolutely love it! Dear creator, you really deserve more views and i wish you it! IM IN FOR YOUR NEXT VIDEOS!
@mikemac5070
10 ай бұрын
This is the best video on complex numbers since welsh labs blew my mind w his series on them. This is just what ive been waiting for. Godspeed!
@imaginaryangle
10 ай бұрын
Thank you so much! Those videos were amazing!
@magicorigamipiano21
10 ай бұрын
This was absolutely brilliant the build up to the “flower pressing” was jaw dropping. Immediately liked and subscribed! Keep up the great work!!!
@imaginaryangle
10 ай бұрын
Thank you! Will do!
@AlanKey86
10 ай бұрын
Great visualisations!
@AlanKey86
10 ай бұрын
Hi @imaginaryangle - if you're ever looking for music for your videos, let me know!
@imaginaryangle
10 ай бұрын
Thank you! I subscribed to your channel, I will keep it in mind 💙🎼
@MarkTimeMiles
10 ай бұрын
Beautiful, thank you. 🙏
@0FAS1
10 ай бұрын
Amazing video! Mind blowing stuff - love your initial explanations of imaginary numbers too, somehow it feels intuitive - we gain freedom (of rotation in this case) when negation comes into the picture. Thanks!
@OfficialFoliLucker
10 ай бұрын
This is an excellent representation of how graphs transform into another function in the complex plane and how would they behave. I might have to look deeper into this to understand what really is going on!! (Congrats in advance when reaching 5k subs btw)
@imaginaryangle
10 ай бұрын
Thank you!
@juanpablo2097
10 ай бұрын
Extremely value content, this remains me that everything in nature are dicted by mathematical laws, simply amazing
@wellscampbell9858
10 ай бұрын
@ 28:43 Imaginary Angle--"Keep that in mind as we travel down to the negative one power" My Cranium--"Kablooey!"
@BMXaster
10 ай бұрын
Oh my god, this is absolutely beautiful
@newwaveinfantry8362
10 ай бұрын
14:45 - That's why you take the principal branch of theta being in [0,2п). Otherwise x send to x^1.1 is not a function. The same goes for any power. 27:15 - We're approaching progective geometry now. lol
@morgan0
10 ай бұрын
also notable with the negative powers is folding the rings inside out. it's something i explored a bit as i tried to come up with some way to get the complex conjugate (with the purpose of flipping the phase shift of a filter) a while back (never got anything that works, and moved on to other things before trying to make an approximation), i think because it also changed the phase of the complex number, or it in combination with something else got me close. it was like most of a year ago so my memory is a bit hazy, but it was nice to see something about complex plotting again, very cool topic.
@BoutiqueLaTrice
10 ай бұрын
This was AMAZING! Thank you for sharing! I’ll be watching this on the big screen next time!!
@jelenahegser445
10 ай бұрын
what a beautiful video! the "spirals" from the negative exponents reminded me a lot of the graphs of trig functions in polar coordinates ( r(theta)=cos(a*theta) ).
@imaginaryangle
10 ай бұрын
That's no accident 😉
@GameplayAndRelaxation
9 ай бұрын
Absolutely sublime work.
@RichConnerGMN
10 ай бұрын
what i learned from this video: e^x likes feet wait, wrong kind of kink
@johnmcdonough955
10 ай бұрын
'A somewhat unorthodox approach' = A ballerina with a crew cut. Agreed.
@MisakaMikotoDesu
10 ай бұрын
Commenting to make sure the algorithm picks this up.
@ImaginaryMdA
10 ай бұрын
This is a pretty demonstration! Not sure what it demonstrates, but very nice regardless!
@Jerry-u3v
10 ай бұрын
This title had me going until I read the last word 😂
@Nuovoswiss
10 ай бұрын
I'd love to see more 3-d-ization of the real component with the complex components. I never knew that the two complex components are identical, but "compressed" into the 2d complex plane. Would love to see more than just the cube-root case.
@imaginaryangle
10 ай бұрын
They're identical in the sense that they are just collections of step lengths that can be taken in some unit, and then they apply those identical steps to different units: 1, -1, i, -i. I steered a bit away from too much 3D because lines hanging loose in 3D space are difficult to correctly visually interpret. Is it going towards you, away from you, how far... Maybe when I get better at 3D shading and environment building.
@wellscampbell9858
10 ай бұрын
@@imaginaryangle I feel that motion in 3D graphs can go a long way towards conveying information and cleaning up what's being displayed. I was just about to write a separate post when I saw this one; I was going to ask if the rainbow lines (z-values?) could be plotted in the environment introduced at 24:02. I'm guessing they would form a complicated structure but I'd imagine it would be interesting...
@imaginaryangle
10 ай бұрын
@@wellscampbell9858 They could, and I thought about that, but it didn't make the final cut. I think I'll make a part 2 of this some time next year.
@wellscampbell9858
10 ай бұрын
@@imaginaryangle I think you made a good choice, restricting most of the concepts to a 2D representation I believe forced them to be more understandable (and kept the time under control). Plus, now I have yet another reason to keep checking back :)
@moocatmeow
10 ай бұрын
28:32 the unsigned infinity is finally being appreciated, even if it is in the form of the true unsigned infinity
@user-rx5dh4le5x
10 ай бұрын
I’ve always thought that it’s counterintuitive to identify points on the complex plane by the sum of their respective coordinates, although this might be simpler and it can work alright, it just makes more sense to identify it by the length of the segment from the origin to the point times i to the power of twice the adjacent angle (in radians) over pi.
@aidarosullivan5269
10 ай бұрын
I, too, have a secret kink for a good math.
@sossololpipi9633
10 ай бұрын
that is one hell of a title
@Taskarnin
10 ай бұрын
If math could be ballet this would be it. I’m going to have to wrap my head around this. I have a masters degree in stress analysis and vibrations (engineering) and this is melting my brain. Would be cool to see how this interacts with diff equations for vibrational systems. The resulting Fourier series could look wild I’m sure. I’ll let my brain chew on this more tomorrow, it’s either trivial or wild and I can’t figure it out lol.
@FredericoKlein
9 ай бұрын
"why are you so negative? why is everything with you so complex?" well, this is where the interesting things happen
@tylerboulware6510
10 ай бұрын
Very cool, and amazing visuals! I learned some things. I think I'll have to watch at least one more time to understand it better.
@Seiffouri
8 ай бұрын
Such an amazing journey. Thank you so much for your hard work.
@imaginaryangle
8 ай бұрын
Much appreciated, I'm happy you enjoyed it!
@craftycurate
10 ай бұрын
Beautifully presented and highly engaging, and was I able to follow it.
@MrSilverSerf
9 ай бұрын
Wow! That was amazing journey! Thank you!
@serdarakalin2209
6 ай бұрын
ITS beautiful, to See that there ist only one Infinity in complex world, Not a plus and minus Infinity.
@serdarakalin2209
6 ай бұрын
What Software ist used for visualisation, May bei too dumb to ask?
@imaginaryangle
6 ай бұрын
The animations are generated using Python and manim, a Python module for mathematical animations.
@mia-ul4ou
10 ай бұрын
don't forget about x³ being into cuffs and ropes
@adammonteleone7748
10 ай бұрын
Something like this but for visualizing elliptic curves as we vary the j-invariant would be amazing and afaik hasn't been done yet by anyone on KZitem (This can also be done for elliptic curves over finite fields).
@carloscampello8406
10 ай бұрын
Very good video. Much interesting the graphics.
@maythefaie
10 ай бұрын
not the video i was expecting :( but still nice!! ♥️
@TheGluemess
9 ай бұрын
Mathologer tier video, please make more
@imaginaryangle
9 ай бұрын
Thank you! I will!
@z-file9321
10 ай бұрын
Amazing video. it was very fun & intriguing to watch. (might have to rewatch a couple times to digest it tho lol) I got like 40% of what was bein said (I barely know enough mafs to get thru highschool). The visual representations were top-tier. I don't think i would've understood anything without them.
@jaysmooveV2
6 ай бұрын
Can you do a video when you explain where sin cosine tangent come from/are and where the unit cirlce comes from and why it is what it is and how it is important in nature architechture etcccc LOVE THE CHANNEL MATE NICE WORK !
@imaginaryangle
6 ай бұрын
Thanks! There's a video I'm working on further down the line that may be the closest to what you're referring to, about roundness and continuity. But it's probably not going to be out before summer.
@empireempire3545
10 ай бұрын
Absolutely stunning and very clever. You should make a paper out of this. Also, Gaussian is such a mess as always xDDD
@TheLyndontp
10 ай бұрын
Loved this video, super kinky
@DeJay7
5 ай бұрын
Absolutely fascinating!
@imaginaryangle
5 ай бұрын
Thank you!
@KineHjeldnes
10 ай бұрын
Woooow this was amazing! I learned a lot, and suspect I will rewatch many times
@sertacatac0
10 ай бұрын
This is the channel that I was looking for, great content!
@imaginaryangle
10 ай бұрын
Welcome aboard!
@artmaiq
9 ай бұрын
Finally, mathematical acid Great video!
@HuxleysShaggyDog
10 ай бұрын
Subbed. Breathtaking. Play with the Riemann zeta this way? Please. You’re on to something. Branches, too!
@brandonchelstrom2260
10 ай бұрын
Absolutely Beautiful
@bradleygaddis5155
10 ай бұрын
Sir Roger Penrose made several videos here on yt about how the universe expands to such a point that time becomes meaningless and mixes, starting over. He didn't explain how this happens. This video at time 28+ gives a viable explanation. Excellent video 👍
@Avokadik13
10 ай бұрын
Just wow You really found something new
@awefan
10 ай бұрын
You destroyed the image of complex numbers for me Somehow, you've broke mathematics, but in a stylish way. I like you!
@hp2594
9 ай бұрын
You should make a similar video on why integral of 1/x is log(x) whereas for all other powers of x it is x^(n+1)/(n+1)
@imaginaryangle
9 ай бұрын
That's an interesting suggestion, thank you! This should get covered in one of my future videos about Euler's number, but I'm still knee deep in research on it, so I have no idea when it will be ready. I'm guessing you'd like to see the gradual approach to the log, like this video does with the powers.
@KarlWork-n3i
9 ай бұрын
You can actually do complex integration without using fundamental theorem of calculus f(b)-f(a) where f is some integrand for your function you are integrating. You can use the Div & Curl operators from vector analysis(physics) If you are integrating F(z) = u(x,y) + v(x,y) where u,v are your usual harmonic functions. You use the POYLA VECTOR FIELD (u, -v) and integrate along path using tangent and normal for Work and Flux. For closed path which is Curl and Div free = 0 the original function F(z) will be analytic everywhere within the closed contour. It offers a physical way of understanding complex integration and can be used to integrate any complex function even tricky ones like z conjugate. It also allows you to visualize how one method of integration relates to another without the use of integrands and the fundamental theorem. Google Polya vector field.
@KarlWork-n3i
9 ай бұрын
It's something you never see in undergrad mathematics. R Penrose and one of his students wrote a book about visualizing complex analysis, if you search around you should easily find it.
@MNanme1z4xs
10 ай бұрын
This is too interesting it start to become scary. Those are things told in high school, you picked a bit into the reality behind and suddenly discover a Lovecraftian dimension. The way you visualized these carry implications in biology. If add constrain with finite and entropy, math may be able to construct life form. I hope next time you can do these transitions in Real and Imagery 3D coordinate. Lets hope you don't need to overclock GPU to do that :P
@ObamaTron
9 ай бұрын
We need a name for this graphing/plotting technique!
@margaritashcheglova8670
10 ай бұрын
Thank you! And I thought that discovering that hyperbola (and parabola) is a fancy circle was ming-boggling enough...
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