Support me on Patreon! patreon.com/vcubingx There are a lot of minor mistakes, like I said indefinite instead of definite and z instead of n. Sorry about that. Enjoy the video anyway :)
@@vcubingx indeed you have done a great job as i did read lot of material on FD and wanted to explain easily in my paper now i am able to understand it and able to put in my words. thanks
@@satwikchivukula8905 all I have is research papers as the FD textbooks are very expensive but there is one group on FB which provides freePDFs form there I downloaded one ebook on FD
Pedro Dumper explaining math concepts with minimalist animations is definitely not copyrightable lmao. And thank god its not, that would be an absolutely trash idea. I’m sure 3blue1brown would be really happy that this guy is competently explaining difficult concepts in a concise video
@Pedro Dumper ah I see what you mean. In my opinion that's different from saying a video is literally copyright infringement. Fair point though. Hell, if I made this video I'd take it as a huge compliment if someone said that it was that similar to 3b1b lol. He's basically the epitome of TH-cam math vids these days
@Pedro Dumper Look, the animation is from an open-source Python library called manim github.com/3b1b/manim. 3Blue1Brown wrote that and *everyone is allowed to use that*! This is not copyrighted or anything - this is completely legal. Research before making false accusations.
@@joulev Yes this is what I thought too. 3b1b does not own this software-like animation thing. I think the goal of 3b1b is clearly to popularize "the right way to think about maths" and moreover how to vizualise it to have an intuition. Just imagine if everyone get inspiration from 3b1b and it would become the common way to do maths, how far even humanity could get into complexity easily. I think 3b1b would be very happy of this and even do tutorials on how to use computers for this (maybe he did already i didn't check) Sorry for my english not my first language.
> The Gamma Function is not defined for n < 0 Actually, It isn’t defined for Re(z) in Negative integers. It is well defined for all complex numbers with negative real components, as long as that component is not an integer.
i mean, the graph clearly shows it. also, since Γ(n)=(n-1)!, we can see that it should be undefined for negative integers and 0, because the factorial of a negative integer contains division by 0, which is illegal.
It is defined for them, but not for the ones with Im(z) = 0. So for example -1+i is defined, whereas -1 or -2 isn't. You easily can evaluate -1 + i with the recursiv definition of the gamma function Gamma(z) = Gamm(z+1)/z.
It's well defined for all complex numbers with non-zero imaginary part, no matter what the real part is. So Γ(x) has points of where it is undefined, not lines
I am a returning student attempting a formal education in mathematics. It is horrible how many students just memorize, compute, and forget because they never get a chance to see the wounder and creativity of higher mathematics. your vids keep the alive!
There are some applications of fractional calculus in the design of PID-like controllers, but using fractional integral and fractional derivative instead of the simple integral and derivative used in PID. In some cases, those show some advantages in terms of robustness. A good survey about this topic is Dastjerdi, Ali Ahmadi, et al. "Linear fractional order controllers; A survey in the frequency domain." Annual Reviews in Control (2019).
Found this video from Showmakers interview of 3b1b. You have earned yourself another sub because of your quality content and to support the growth of manim. Cheers!! :-)
We can simply define fractional derivative using Fourier transformation. As the Fourrier transformation of a n-derived function is scaled by the n-power of the frequency, we can replace n by a real value and use inverse Fourrier to get the result. BTW this explain the oscillation that occurs at 9:23.
I was wondering if the Fourier transform would show up since the transition to the derivative in the animation showed some wave-like distortion around x=0 that I recognized as being reminiscent of what happens when taking the Fourier transform of a function with sharp jumps.
Yep thats one of the things I realized after making the video: if you don't give your full attention, its hard to understand everything. I think I'm gonna go a bit slower next time, but thanks anyway!!
Bro wrf ur literally the first guy I’ve seen to cover such complex topics like fractional calculus with beautiful animations. Subbed immediately keep it up vro and plz don’t worry bout ppl making fun of u, literally No One is thinking that: ) 👍👍
In 9:18 you can see oscillations of green function. I calculated that no oscillations should be there. Thank you for great and inspiring video. I am totally impressed.
You are right, there shouldn't be any oscillations. But the way I animated it relied on numerical integration, resulted in the little oscillations of greens function you mentioned. Thanks for watching the video 😊
I'd love to see a video about the gamma function + Pi function I really like your channel, keep up the good work I have two suggestion: 1- mention how advanced the math is before starting the video, and what I need to know to understand the content 2- Just watching a video is not going to be sufficient for understanding a concept, I hope that you put a link in the description for practice questions whether a pdf file or another video or a website
This was awesome!! I’m about to graduate in pure math as an undergrad and have been playing around with the Gamma Function! I feel like I just got a new toy!!!
This video deserves more likes. Thank you for explaining something that a guy from my university madre seem so cryptic and difficult to understand in just 14 minutes
Omg this is really awesome! I love your explanation so much!! I'm really interested in fractional calculus. This channel needs to get noticed! You deserve more than that!
"The fractional calculus" by odham and spanier is a good book on the topic. Good work in the video! If you're interested in applications, "On the control and stability of variable-order mechanical systems" is a good paper that puts to use these concepts in control theory.
Your content is amazing. You are such a calm and patient teacher! What a great introduction into fractional calculus. So many complex pieces of mathematics in there.
Thank you for explaining this, I found it very interesting and I like how you do things! I would be interested in watching a video about the gamma function, as I don't really have any acces to other sources and I like how you make this accessible to people, like me, that don't really know a lot about maths except for other TH-cam videos!
Mine, too. I had to go back and single-step through it just so I could read it. FWIW, I couldn't find anything wrong with his pronunciation that couldn't be explained by his accent. Methinks the fellow is being too hard on himself.
It can be use in control theory for make controllers with greater degrees of freedom, it really different from the traditional calculus and the solution are non trivial, but it's a great subject
Everything is well explained and animated, great job on that! but underneath all that there's the same feeling which I used to hate in school: - Today we'll learn this new thing, let's deduce it by applying some arbitrary rules from last chapter. Did you understand each step? - Yes, but.. - - Now let's move on to some properties, are they clear? - Yes, but.. - - Now, let's solve this very specific problems which give a nice solution if we apply a sequence of its properties and that weird formula from chapter 3. Are there any questions? - Umm.. what do the in-betweens signify? are there analogies to different things? how does it connect to other concepts? what's the intuition behind it? - Look, that's not the focus of this class, you can look up Riemann-Liouville Integral and read these highly mathematical books if you wish to get a better sense. - Okay.. I know it's hard to find that simple intuition behind things and explain it in a relatable way which creates a sense of purpose, but that's the most precious thing I find in mathematics, I find it awful to reduce math to just heartless rules.
The fractional derivative has many applications! Every battery and every medical implant electrode has impedance that involves a fractional derivative. Every capacitor responds fractionally, from its current to its failure lifetime. Only most engineers do not know enough math...
I like the smoothness of this presentation but there is still a disadvantage: in all the cool looking animated transformation of the formulas I find it really hard to actually follow: for this I would usually stop the video and try to convince myself that the currant formula follows from the previous one(s). But then the previous formula isn’t on the screen anymore. So I end up having to write everything down so I can actually verify each step. When you do these animated changes of the formulas, which are appealing and have a bit of a sense of wizardry, then why don’t you keep the previous formula on the screen?
Nice intro to a topic I’d never heard of! In particle physics it is often useful to perform integrals over a non-integer number of spacetime dimensions. It offers a convenient way to regulate divergent integrals when the use of an explicit cutoff in the integral upper bound would break symmetries of the integrand like Lorentz invariance. I’m not sure how directly related to fractional integration this “dimensional regularization” is, but the gamma function certainly pops up a lot.
I think there is a tiny mistake in ur video. So at 7:14. The video is saying that the nth derivative of nth integral of a function is just the function itself, but it seems u wrote a extra "f" in the derivative. I don't if that's a mistake or not, i am just a 10th grader.
8:10 I was thinking that ceiling looks disgustingly horrendous but maybe it's just one of many branches and that's why the ceiling is there? Similar to how when you take a complex logarithm you chose the branch that gives you an angle between 0 and 2pi. 13:00 Why would that be true at all?
keep it up.. for me you hit the right note. I know enough calculus that your explanation was neatly added to what I know . BTW good teachers know this and your focus was good
Fractional integration/differentiation has many applications in time series analysis. A time series is often characterized by its mean reverting property. A series that keeps reverting back to its mean, and displays a similar variance over all periods, is called stationary. On the other hand, series that seem to follow a random walk are referred to as non-stationary and often are called integrated. The latter name makes sense because these series can be seen as an accumulation (i.e. integration) of random shocks. Now, in many economic time series, we observe that series seem te have some mean reversion, but very slowly over time. These series have very long memory and are kind of in between stationary and non-stationary series. Indeed, such series are "fractionally integrated" and the techniques in this video are very useful here.
After taking real analysis in my undergrad, on the last day I went up to my professor and asked him about fractional derivatives because it was a random thought that came to mind. “What is a 3/4ths derivative?, what about the Pi th derivative? What about a derivative that changes, like x in certain intervals the 3rd derivative is taken, but x in other intervals the 2nd derivative is taken? Or what if it changes constantly based on another function?”. He didn’t answer my question and told me I would learn that in graduate level analysis. Well I took real analysis again (grad level) and I didn’t learn any of that. Thanks for the video !!
Damn, that's mind-blowing!!! I'm only going to enter university this year, but I do math for fun, and you've definitely motivated me to study further!)
It's a great video :-) really I like it very much. Within 15 mint u gave a small intro about Fractional calculus. Pls, upload more video related to Fractional derivative.
Gamma function will work for me. Thanks. You guys on TH-cam are better than school. They need to teach Laws of Math furst b then show people how t I use math.
Small remark: in LaTeX you can use the commath package, which defines \dif, yielding a non-italized d for differentials (e.g. \dif x). In addition, Leibniz notation is typed by e.g. \od[n]{f}{x} for the nth derivative of f.
4:00 Why would you use the Gamma Function? Wouldn't the Pi function be more convenient since it doesn't have the "n-1" and outputs the corresponding factorial for each value of n, without needing to add 1?
11:50 this is so freaky. Literally 2 days ago a video popped into my recommended about synchrous curves or something and I started thinking about what the derivative of that is. Started looking into cycloids and stuff but I haven’t really learned parametric graphing yet (Highschool junior) so there’s some prerequisites that I’m trying to get through at the same time. Now this video pops into my recommended that has the exact stuff that I’m looking for?
I’m glad you’re interested in this stuff, but be sure not to think of earlier math classes as just something you have to get through, they’re all important to understand :)
I would have got really mad if you didn’t mention 3b1b or at least the manim library. BUT YOU DID SO NOW YOU HAVE MY LIKE AND SUBSCRIPTION Great video, keep going!
great vid! there is also a way to introduce fractional derivatives using multipliers in the Fourier domain. If you also know things about this, i would really enjoy a video comparing those two concepts.
8:11, That sudden ghostly text you've been struggling to see what it is but it disappeared at light speed: *“I'M SORRY I CAN'T PRONOUNCE ANYTHING MB PLES DONT MAKE FUN OF ME THX”*
I'd like some extra info on the Gamma Fxn. Curious about its limitations. I love how concise you were in this video and the transition in n from a set of positive integers to n as the set of all positive real numbers to n as the set of all numbers. Your mistakes are forgivable since this is so well-presented on a conceptual level. Thank you so much for this work.
When the function x is being transformed by the Differintegral operator, at 9:16, it seems to me that the square root of x appears for a brief moment, at 9:17. Is the square root of x the Differintegral of x to some value between 0 and 1? Update: Yeah now I saw the rest of the video. The answer is Almost.
Nice video! I first time saw this defined through Fourier transformation. Since Fourier transformation is a isometry in L2 and by property of differentiation, it is easy to define for 1 dimensional functions. Then for n dimensional function, we defined the module of derivative operator (absolute value of differential operator) to proceed. Both end up losing the locality as mentioned. And the example at the end is very beautiful! Thank you for your effort!
There is one thing that i don't understand about fractional derivatives : based on the formula at 11:18, if you take the 3/2-derivative of th constant function 1, you get sonething that's nonzero. However, the 3/2-derivative is the 1/2-derivative of the 1-derivative, so the 3/2-derivative of 1 should be the 1/2 of 0, which is 0 by linearity of the D operator. So there is clearly a problem (it is zero and nonzero), what is the issue ? Is the "composition rule" (f^a * f^b = f^(a+b), a > 0, b > 0) true for I but not D ?
Nice. I'd like to actually make a video about the Gamma function eventually. Depending on what your idea for it is, let me know if you want to collaborate on that.
How does D^(1/2)(1) = 1/sqrt(pi * x) make intuitive sense? Shouldn't it be somewhere between D^1(1) = 0 and D^(0)(1) = 1, two terms that are both constant. Also, that initial spike during the transition at @9:20 between f(x) = x and f(x)=0 seems odd, but I guess must be somewhat related. In addition, it would be cool if some kind of generalized Taylor expansion existed where the error term would decay with a fractional exponential.
Nice explanation! By the way, I think there's a typo at 7:04. We'd want the nth derivative of the n-iterated integral of a function to be the function itself, not the nth derivative of the function evaluated at the nth integral evaluated at the input parameter.
11:00 This is saying that the semi-derivative of the constant function f(x)=1 with respect to x is 1/sqrt(pi*x) ? If I'm understanding that correctly, that is very odd indeed. Since the derivative of any constant is 0, which is another constant. So the transition between a full derivative and f(x) has some weird non-constant states in-between. I would love to see the animation of D^a(1) where a animates between 0 and 1
For what I understood, I^n seem to be more fundamental that D^n for fractional n, so I thought on a slightly different solution for the cycloid problem. We have sqrt(2g/pi) T= I^(1/2) Ds. Taking I^(1/2) on both sides: I^(1/2) sqrt(2g/pi) T = I Ds = s(y)-s(0), then s(y)-s(0)= sqrt(2g/pi) T 1/sqrt(pi) integral from 0 to y (1/sqrt(y-t)) dt = 2 sqrt(y) *sqrt(2g)/pi T . Differentiating both sides: ds/dy = T sqrt(2g/y)/pi .
Support me on Patreon! patreon.com/vcubingx
There are a lot of minor mistakes, like I said indefinite instead of definite and z instead of n. Sorry about that. Enjoy the video anyway :)
Also at 7min you have an extra 'f', otherwise a really nice introduction, well done mate.
@@bennettgardiner8936 you're right, thanks
@@vcubingx Would you consider doing a video on Fractional Brownian motion please, superb video btw :)
@@stephenv796 Thanks! I'll look into it and see if it's worth a video
@@vcubingx a video on fbm would be awesome
This is an awesome video! I enjoyed it very much!
Thank you so much Steve! I really appreciate it 😊
@blackpenredpen i love u🥺
@@emonph4463 x2
You’re explaining higher calculus and you are a person: no one would make fun of you. Fantastic video, by the way.
Thank you so much!!!
@@vcubingx indeed you have done a great job as i did read lot of material on FD and wanted to explain easily in my paper now i am able to understand it and able to put in my words. thanks
@@poonamdeshpande7832 can you please
suggest me any book that'll be great to learn this FDE topic that I can find online.
@@satwikchivukula8905 all I have is research papers as the FD textbooks are very expensive but there is one group on FB which provides freePDFs form there I downloaded one ebook on FD
@@poonamdeshpande7832 name of that page???
YES WE WANT THE VIDEO ON THE GAMMA FUNCTION. I'M GOING TO SEARCH YOUR PLAYLISTS NOW FINGERS CROSSED.
It seems that everything is beautiful with Manim
3blue1brown will be proud of you.
Edit: 3blue1brown is proud of you. 😊
Pedro Dumper What do you mean?
Pedro Dumper explaining math concepts with minimalist animations is definitely not copyrightable lmao. And thank god its not, that would be an absolutely trash idea. I’m sure 3blue1brown would be really happy that this guy is competently explaining difficult concepts in a concise video
@Pedro Dumper ah I see what you mean. In my opinion that's different from saying a video is literally copyright infringement. Fair point though. Hell, if I made this video I'd take it as a huge compliment if someone said that it was that similar to 3b1b lol. He's basically the epitome of TH-cam math vids these days
@Pedro Dumper Look, the animation is from an open-source Python library called manim github.com/3b1b/manim. 3Blue1Brown wrote that and *everyone is allowed to use that*! This is not copyrighted or anything - this is completely legal. Research before making false accusations.
@@joulev Yes this is what I thought too. 3b1b does not own this software-like animation thing. I think the goal of 3b1b is clearly to popularize "the right way to think about maths" and moreover how to vizualise it to have an intuition. Just imagine if everyone get inspiration from 3b1b and it would become the common way to do maths, how far even humanity could get into complexity easily. I think 3b1b would be very happy of this and even do tutorials on how to use computers for this (maybe he did already i didn't check) Sorry for my english not my first language.
> The Gamma Function is not defined for n < 0
Actually, It isn’t defined for Re(z) in Negative integers. It is well defined for all complex numbers with negative real components, as long as that component is not an integer.
you're right it is defined; my fault for the mistake
i mean, the graph clearly shows it. also, since Γ(n)=(n-1)!, we can see that it should be undefined for negative integers and 0, because the factorial of a negative integer contains division by 0, which is illegal.
It is defined for them, but not for the ones with Im(z) = 0.
So for example -1+i is defined, whereas -1 or -2 isn't.
You easily can evaluate -1 + i with the recursiv definition of the gamma function Gamma(z) = Gamm(z+1)/z.
@@epicmorphism2240 so do you want to take imagonary diferencition?
It's well defined for all complex numbers with non-zero imaginary part, no matter what the real part is. So Γ(x) has points of where it is undefined, not lines
I am a returning student attempting a formal education in mathematics. It is horrible how many students just memorize, compute, and forget because they never get a chance to see the wounder and creativity of higher mathematics. your vids keep the alive!
There are some applications of fractional calculus in the design of PID-like controllers, but using fractional integral and fractional derivative instead of the simple integral and derivative used in PID. In some cases, those show some advantages in terms of robustness. A good survey about this topic is Dastjerdi, Ali Ahmadi, et al. "Linear fractional order controllers; A survey in the frequency domain." Annual Reviews in Control (2019).
Found this video from Showmakers interview of 3b1b. You have earned yourself another sub because of your quality content and to support the growth of manim. Cheers!! :-)
Thank you! Was I mentioned in the interview or something? I'm just curious as to how exactly you found my content
We can simply define fractional derivative using Fourier transformation. As the Fourrier transformation of a n-derived function is scaled by the n-power of the frequency, we can replace n by a real value and use inverse Fourrier to get the result. BTW this explain the oscillation that occurs at 9:23.
Yep. The Gibbs phenomenon.
That's exactly what I've thought the moment I read the title.
Interesting.
I was wondering if the Fourier transform would show up since the transition to the derivative in the animation showed some wave-like distortion around x=0 that I recognized as being reminiscent of what happens when taking the Fourier transform of a function with sharp jumps.
I thought that this was 3B1B until I heard the voice.
Loved the video, loved the content, learned something!
Good video.
A lot of this is over my head, though.
Yep thats one of the things I realized after making the video: if you don't give your full attention, its hard to understand everything. I think I'm gonna go a bit slower next time, but thanks anyway!!
I think that it's just over my head, I'm going into Calc this year.
@@benjamimapancake6429 ah, now it makes sense
Bro wrf ur literally the first guy I’ve seen to cover such complex topics like fractional calculus with beautiful animations. Subbed immediately keep it up vro and plz don’t worry bout ppl making fun of u, literally No One is thinking that: ) 👍👍
Ummm ... 3Blue1Brown ? You've _got_ to check out those videos. th-cam.com/channels/YO_jab_esuFRV4b17AJtAw.html
Yea it's quite similar to 3blue1brown
Thus video immediately reminded me of 3B1B!
@@flyingdonkey5488 he uses the same software (manim) to make the animation
It's an open source library made by 3b1b
In 9:18 you can see oscillations of green function. I calculated that no oscillations should be there. Thank you for great and inspiring video. I am totally impressed.
You are right, there shouldn't be any oscillations. But the way I animated it relied on numerical integration, resulted in the little oscillations of greens function you mentioned. Thanks for watching the video 😊
This artefact is well known in digital sound processing... causing disturbing audio effects...
I thought that actually takes place. So misleading. Thanks
@@pierrelacombe4757 Gibb's Phenomenon?
I'd love to see a video about the gamma function + Pi function
I really like your channel, keep up the good work
I have two suggestion:
1- mention how advanced the math is before starting the video, and what I need to know to understand the content
2- Just watching a video is not going to be sufficient for understanding a concept, I hope that you put a link in the description for practice questions whether a pdf file or another video or a website
Good points! I'll definitely be sure to include that in my next videos. Thanks for watching and thanks for the feedback 😊
Me and my fellow student friends would definitely want a gamma function video!!
Sometimes I love TH-cam recommendation
Wow, that elegant solution to the Tautochrone problem blew my mind. It's as if the solution just jumps out at you!
Today I learned that's the worst time to watch an amazing video on complex math is after a gigantic meal that has put you in a food coma.
Thats precisely what happened to me
Just amazing, I was searching this for months, and got this recommended. *subscribed*
This was awesome!! I’m about to graduate in pure math as an undergrad and have been playing around with the Gamma Function! I feel like I just got a new toy!!!
Oh dear god. I almost don't wanna know.
Almost.
dragon21516 lol me too
I want to know, but I can't understand this yet... Just missing too many pieces right now
Dwarf Fortress kills the cheap computer.
This video deserves more likes. Thank you for explaining something that a guy from my university madre seem so cryptic and difficult to understand in just 14 minutes
The video has a natural flow, the explanation is understandable. Thank you for doing this ✨🙂
This channel deserves more subscriptions.
Omg this is really awesome! I love your explanation so much!!
I'm really interested in fractional calculus. This channel needs to get noticed! You deserve more than that!
"I'M SORRY I CAN'T PRONOUNCE ANYTHING MB PLES DONT MAKE FUN OF ME THX".. we won't :)
Ahahahahaha
We should make fun of him saying that...
@engineer99 I did the same to look what just happened..
"The fractional calculus" by odham and spanier is a good book on the topic. Good work in the video! If you're interested in applications, "On the control and stability of variable-order mechanical systems" is a good paper that puts to use these concepts in control theory.
This is the second video of yours that I’m watching and I already know that you’ve earned another subscriber
Thanks!
What? I thought it was a recent idea, but Leibniz already thought of it!
Leibniz and Euler, between the two of them, have thought of _everything._ 😁
This is great if you keep this up I'm sure this channel will be huge in no time
Thanks!!
Your content is amazing. You are such a calm and patient teacher! What a great introduction into fractional calculus. So many complex pieces of mathematics in there.
Thank you so much!
Keep up the good work Sir. No one usually touches these topics.
Thank you for explaining this, I found it very interesting and I like how you do things! I would be interested in watching a video about the gamma function, as I don't really have any acces to other sources and I like how you make this accessible to people, like me, that don't really know a lot about maths except for other TH-cam videos!
Man, this is awesome! I've been wanting some explanations about this for years.
Thanks a lot!
8:11 says "I'm sorry i can't prounounce anything mb plez dont make fun of me thx"
barely a frame.. my ocd nearly killed me
Mine, too. I had to go back and single-step through it just so I could read it. FWIW, I couldn't find anything wrong with his pronunciation that couldn't be explained by his accent. Methinks the fellow is being too hard on himself.
Got it on my first try xd... Guess I was lucky
. = +1 frame; , = -1 frame.
I started figuring out fractional calculus on my own during my Math minor. It's cool to finally learn about it officially.
😂
Wow bro this was nicely done. Good job!
wow - loved this! so glad i discovered your videos & thank you for making them!
I've always wondered if there was something along the lines of fractional derivatives. I didn't know there would be actual applications for it!
BROTHER WHAT ARE YOU DOING HERE WTF
I'm a math nerd, what do you expect? :)
It can be use in control theory for make controllers with greater degrees of freedom, it really different from the traditional calculus and the solution are non trivial, but it's a great subject
Jafet Ríos Durán that sounds very interesting. I just learned control theory and viscosity solution recently. Is there any recommended references?
@@gunhasirac of course you can start with fractional calculos of Igor Podlubny and search about the Mittag Leffler function
Brilliant. This is how higher Mathematics needs to be taught.
Everything is well explained and animated, great job on that! but underneath all that there's the same feeling which I used to hate in school:
- Today we'll learn this new thing, let's deduce it by applying some arbitrary rules from last chapter. Did you understand each step?
- Yes, but.. -
- Now let's move on to some properties, are they clear?
- Yes, but.. -
- Now, let's solve this very specific problems which give a nice solution if we apply a sequence of its properties and that weird formula from chapter 3. Are there any questions?
- Umm.. what do the in-betweens signify? are there analogies to different things? how does it connect to other concepts? what's the intuition behind it?
- Look, that's not the focus of this class, you can look up Riemann-Liouville Integral and read these highly mathematical books if you wish to get a better sense.
- Okay..
I know it's hard to find that simple intuition behind things and explain it in a relatable way which creates a sense of purpose, but that's the most precious thing I find in mathematics, I find it awful to reduce math to just heartless rules.
Your choice is so nice and smooth. Congratulations
Brilliant video, you’ve earned a new subscriber! Keep up the amazing work!
Absolutely great video :)
I found it on Reddit, you should definitely have more subscribers
Thanks!
Cool, I've been interested in this for a few weeks.
The fractional derivative has many applications! Every battery and every medical implant electrode has impedance that involves a fractional derivative. Every capacitor responds fractionally, from its current to its failure lifetime. Only most engineers do not know enough math...
Amazing video! I hope your channel blows up :)
I like the smoothness of this presentation but there is still a disadvantage: in all the cool looking animated transformation of the formulas I find it really hard to actually follow: for this I would usually stop the video and try to convince myself that the currant formula follows from the previous one(s). But then the previous formula isn’t on the screen anymore. So I end up having to write everything down so I can actually verify each step. When you do these animated changes of the formulas, which are appealing and have a bit of a sense of wizardry, then why don’t you keep the previous formula on the screen?
3:45 you can do a simple induction by applying the formula to I(f) instead of f, by making use of integration by parts
Nice intro to a topic I’d never heard of!
In particle physics it is often useful to perform integrals over a non-integer number of spacetime dimensions. It offers a convenient way to regulate divergent integrals when the use of an explicit cutoff in the integral upper bound would break symmetries of the integrand like Lorentz invariance. I’m not sure how directly related to fractional integration this “dimensional regularization” is, but the gamma function certainly pops up a lot.
Interesting! That's a fascinating application. Thanks for sharing!
I think there is a tiny mistake in ur video. So at 7:14. The video is saying that the nth derivative of nth integral of a function is just the function itself, but it seems u wrote a extra "f" in the derivative. I don't if that's a mistake or not, i am just a 10th grader.
Great video with excellent message at the end!
Just watch this impressive Math channel th-cam.com/channels/ZDkxpcvd-T1uR65Feuj5Yg.html
I`m simply floored, awesome video!!
8:10 I was thinking that ceiling looks disgustingly horrendous but maybe it's just one of many branches and that's why the ceiling is there? Similar to how when you take a complex logarithm you chose the branch that gives you an angle between 0 and 2pi.
13:00 Why would that be true at all?
This is so beautiful. YOU U CAN HAVE ALL OF MY MONEY! COLLEGE CANNOT DO BETTER THAN THIS!
keep it up.. for me you hit the right note. I know enough calculus that your explanation was neatly added to what I know . BTW
good teachers know this and your focus was good
Fractional integration/differentiation has many applications in time series analysis. A time series is often characterized by its mean reverting property. A series that keeps reverting back to its mean, and displays a similar variance over all periods, is called stationary. On the other hand, series that seem to follow a random walk are referred to as non-stationary and often are called integrated. The latter name makes sense because these series can be seen as an accumulation (i.e. integration) of random shocks. Now, in many economic time series, we observe that series seem te have some mean reversion, but very slowly over time. These series have very long memory and are kind of in between stationary and non-stationary series. Indeed, such series are "fractionally integrated" and the techniques in this video are very useful here.
A particular formal model of this type is "fractional Brownian motion".
Great video, it was so good that i didn't notice any mistakes the first time i watched it :)
After taking real analysis in my undergrad, on the last day I went up to my professor and asked him about fractional derivatives because it was a random thought that came to mind. “What is a 3/4ths derivative?, what about the Pi th derivative? What about a derivative that changes, like x in certain intervals the 3rd derivative is taken, but x in other intervals the 2nd derivative is taken? Or what if it changes constantly based on another function?”. He didn’t answer my question and told me I would learn that in graduate level analysis. Well I took real analysis again (grad level) and I didn’t learn any of that. Thanks for the video !!
Great video! Keep on the great work!
Damn, that's mind-blowing!!! I'm only going to enter university this year, but I do math for fun, and you've definitely motivated me to study further!)
It's a great video :-)
really I like it very much. Within 15 mint u gave a small intro about Fractional calculus.
Pls, upload more video related to Fractional derivative.
Very interesting video and well explained.
Glad you liked it!
Gamma function will work for me. Thanks. You guys on TH-cam are better than school. They need to teach Laws of Math furst b then show people how t I use math.
Small remark: in LaTeX you can use the commath package, which defines \dif, yielding a non-italized d for differentials (e.g. \dif x). In addition, Leibniz notation is typed by e.g. \od[n]{f}{x} for the nth derivative of f.
got it, thanks!
I can't really say I understood how this works, but I hope I can get there someday. Thank you for explaining many complications.
4:00 Why would you use the Gamma Function? Wouldn't the Pi function be more convenient since it doesn't have the "n-1" and outputs the corresponding factorial for each value of n, without needing to add 1?
11:50 this is so freaky. Literally 2 days ago a video popped into my recommended about synchrous curves or something and I started thinking about what the derivative of that is. Started looking into cycloids and stuff but I haven’t really learned parametric graphing yet (Highschool junior) so there’s some prerequisites that I’m trying to get through at the same time. Now this video pops into my recommended that has the exact stuff that I’m looking for?
I’m glad you’re interested in this stuff, but be sure not to think of earlier math classes as just something you have to get through, they’re all important to understand :)
it is very great first time to understand the meaning of fractional calculus thank you very much
I would have got really mad if you didn’t mention 3b1b or at least the manim library.
BUT YOU DID SO NOW YOU HAVE MY LIKE AND SUBSCRIPTION
Great video, keep going!
I learned something new today. And thanks to you for explaining it so nicely!👍
Yes a video on gamma function plz
great vid! there is also a way to introduce fractional derivatives using multipliers in the Fourier domain. If you also know things about this, i would really enjoy a video comparing those two concepts.
Even though I am not a Mathematics Major , I liked the video ....I loved the presentation of part 1 and 3 a lot
Wow... I can use fractional calculus to solve a group of variation calculus problems? That is intriguing!
Thank you for making and posting this.
7:50 Just a tiny nitpick, but the left dots in your graph should be open circles, no? (otherwise you have a one-to-many function)
8:11, That sudden ghostly text you've been struggling to see what it is but it disappeared at light speed: *“I'M SORRY I CAN'T PRONOUNCE ANYTHING MB PLES DONT MAKE FUN OF ME THX”*
I'd like some extra info on the Gamma Fxn. Curious about its limitations. I love how concise you were in this video and the transition in n from a set of positive integers to n as the set of all positive real numbers to n as the set of all numbers. Your mistakes are forgivable since this is so well-presented on a conceptual level. Thank you so much for this work.
Gamma function video !!!! :) Tnks for the video was really nice !
Glad you liked it!
Lovely one. One of the best videos. Please make a video on gamma function as well. Waiting to see.
When the function x is being transformed by the Differintegral operator, at 9:16, it seems to me that the square root of x appears for a brief moment, at 9:17.
Is the square root of x the Differintegral of x to some value between 0 and 1?
Update: Yeah now I saw the rest of the video. The answer is Almost.
Nice video! I first time saw this defined through Fourier transformation. Since Fourier transformation is a isometry in L2 and by property of differentiation, it is easy to define for 1 dimensional functions. Then for n dimensional function, we defined the module of derivative operator (absolute value of differential operator) to proceed. Both end up losing the locality as mentioned. And the example at the end is very beautiful! Thank you for your effort!
Thank you so much! I haven't heard of this definition through the Fourier transform but I'll be sure to read up about it. Thanks for watching 😊
I was wondering why the (first, full) fractional derivative of the function f(x)=x looked like it was the Fourier transform of 1. This helps, thanks!
(oh, it appears that might just be an animation artefact, and not actually related....)
So cool that the fractional derivative of the sine function does a phase shift!
THIS IS A VERY GOOD STYLE
fascinating ... and great production quality!!
thank you! I really enjoyed this video! hope you get back to making wonderful content.
I was sure I was the first to invent this in the 9th grade, but now I'm not sure my formula is correct
This is really well done. You should be proud! I’m sure the math teachers and profs you’ve had in your life are proud too.
There is one thing that i don't understand about fractional derivatives : based on the formula at 11:18, if you take the 3/2-derivative of th constant function 1, you get sonething that's nonzero. However, the 3/2-derivative is the 1/2-derivative of the 1-derivative, so the 3/2-derivative of 1 should be the 1/2 of 0, which is 0 by linearity of the D operator.
So there is clearly a problem (it is zero and nonzero), what is the issue ?
Is the "composition rule" (f^a * f^b = f^(a+b), a > 0, b > 0) true for I but not D ?
That was pretty hard stuff, I love it!
Amazing quality :D. In here before you blow up (19k subs)
Nice. I'd like to actually make a video about the Gamma function eventually.
Depending on what your idea for it is, let me know if you want to collaborate on that.
How does D^(1/2)(1) = 1/sqrt(pi * x) make intuitive sense?
Shouldn't it be somewhere between D^1(1) = 0 and D^(0)(1) = 1, two terms that are both constant.
Also, that initial spike during the transition at @9:20 between f(x) = x and f(x)=0 seems odd, but I guess must be somewhat related.
In addition, it would be cool if some kind of generalized Taylor expansion existed where the error term would decay with a fractional exponential.
11:04 Why would D^(1/2) evaluated at 1 equal a function of x? Did you mean D^(1/2) evaluated at x=1 equals 1/sqrt(pi)?
nope what's on the screen is what I meant. The half derivative of a constant equals a function in terms of x. Quite intriguing!
8:11 for integer values , ceil(n)=n and gamma 0 is not defined .
Nice explanation! By the way, I think there's a typo at 7:04. We'd want the nth derivative of the n-iterated integral of a function to be the function itself, not the nth derivative of the function evaluated at the nth integral evaluated at the input parameter.
11:00 This is saying that the semi-derivative of the constant function f(x)=1 with respect to x is 1/sqrt(pi*x) ?
If I'm understanding that correctly, that is very odd indeed. Since the derivative of any constant is 0, which is another constant. So the transition between a full derivative and f(x) has some weird non-constant states in-between. I would love to see the animation of D^a(1) where a animates between 0 and 1
For what I understood, I^n seem to be more fundamental that D^n for fractional n, so I thought on a slightly different solution for the cycloid problem. We have sqrt(2g/pi) T= I^(1/2) Ds. Taking I^(1/2) on both sides: I^(1/2) sqrt(2g/pi) T = I Ds = s(y)-s(0), then s(y)-s(0)= sqrt(2g/pi) T 1/sqrt(pi) integral from 0 to y (1/sqrt(y-t)) dt = 2 sqrt(y) *sqrt(2g)/pi T . Differentiating both sides: ds/dy = T sqrt(2g/y)/pi .