The Fractional Derivative, what is it? | Introduction to Fractional Calculus 

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 :)

This is an awesome video! I enjoyed it very much!

3blue1brown will be proud of you. Edit: 3blue1brown is proud of you. 😊

You’re explaining higher calculus and you are a person: no one would make fun of you. Fantastic video, by the way.

It seems that everything is beautiful with Manim

YES WE WANT THE VIDEO ON THE GAMMA FUNCTION. I'M GOING TO SEARCH YOUR PLAYLISTS NOW FINGERS CROSSED.

> 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 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!

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!! :-)

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).

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: ) 👍👍

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 thought that this was 3B1B until I heard the voice. Loved the video, loved the content, learned something! Good video.

Sometimes I love RU-vid recommendation

Oh dear god. I almost don't wanna know. Almost.

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

Me and my fellow student friends would definitely want a gamma function video!!

Wow, that elegant solution to the Tautochrone problem blew my mind. It's as if the solution just jumps out at you!

"I'M SORRY I CAN'T PRONOUNCE ANYTHING MB PLES DONT MAKE FUN OF ME THX".. we won't :)

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.

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.

What? I thought it was a recent idea, but Leibniz already thought of it!

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!!!

Just amazing, I was searching this for months, and got this recommended. *subscribed*

If first derivative represent the slope 2nd derivative : concavity Then what does 3rd, 4th and so on derivative represent. Please reply

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

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 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...

"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.

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!

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 RU-vid videos!

I started figuring out fractional calculus on my own during my Math minor. It's cool to finally learn about it officially.

This channel deserves more subscriptions.

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.

Brilliant. This is how higher Mathematics needs to be taught.

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!

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 !!

Man, this is awesome! I've been wanting some explanations about this for years. Thanks a lot!

This is great if you keep this up I'm sure this channel will be huge in no time

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.

This is the second video of yours that I’m watching and I already know that you’ve earned another subscriber

Cool, I've been interested in this for a few weeks.

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.

Very interesting video and well explained.

This is so beautiful. YOU U CAN HAVE ALL OF MY MONEY! COLLEGE CANNOT DO BETTER THAN THIS!

Wow bro this was nicely done. Good job!

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?

Keep up the good work Sir. No one usually touches these topics.

Your choice is so nice and smooth. Congratulations

wow - loved this! so glad i discovered your videos & thank you for making them!

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.

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.

Amazing video! I hope your channel blows up :)

I can't really say I understood how this works, but I hope I can get there someday. Thank you for explaining many complications.

This video reminds me of 3Blue1Brown. I like it.

I`m simply floored, awesome video!!

Absolutely great video :) I found it on Reddit, you should definitely have more subscribers

Brilliant video, you’ve earned a new subscriber! Keep up the amazing work!

it is very great first time to understand the meaning of fractional calculus thank you very much

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 .

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!)

Great video with excellent message at the end!

Alternatively, using integration by parts with u=t and dv=f(t) dt, the term -int(t*f(t),t,0,x) becomes -x*int(f(t),t,0,x)+int(int(f(t),0,s),s,0,x), and then the x*int(f(t),0,x) term cancels out, showing that g=I^(2)f. Also, by analytic continuation, it turns out that the only values of z at which Gamma(z) is not defined are the non-positive integers, while if Im(z)≠0, Gamma(z) is defined, even if Re(z) is a non-positive integer; additionally, 1/Gamma(z) has only removable singularities (again, at non-positive integers), meaning that it can be extended to an entire function; it still isn't helpful for defining multiple differentiation, because at non-positive integers, this extension of 1/Gamma(z) is 0.

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”*

Gamma function will work for me. Thanks. You guys on RU-vid are better than school. They need to teach Laws of Math furst b then show people how t I use math.

Great video! Keep on the great work!

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.

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.

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?

Wow... I can use fractional calculus to solve a group of variation calculus problems? That is intriguing!

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.

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

THIS IS A VERY GOOD STYLE

This is pure gold

A real-world application: Loudspeakers. To create a model of the speaker that fits measured data, fractional derivatives are needed both for the visco-elastic 'rubber'-suspension and the voice-coil. The coil becomes lossy when it is surrounded by ferromagnetic material and the equation describing the relationship between the voltage and current over the coil gets a fractional derivative typically in the range [0.5 0.8].

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?

Some fluid dynamicists use fractional calculus to better understand viscoelasticity. It also arises in the modeling of RL and RC circuits and the time evolution of electric charge within the circuit. Applications do exist.

Why the derivative in the animation of the evolution of the different fractional differintegrations when n equals -1 (the normal derivative) has like an overshoot near zero??

Thank you for making and posting this.

So cool that the fractional derivative of the sine function does a phase shift!

i hope see the man who made this wonderfull and amazing vedio and after that he asked from the people not make fun of him. you should ask from people saying thanks . really i liked you

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

I learned something new today. And thanks to you for explaining it so nicely!👍

Even though I am not a Mathematics Major , I liked the video ....I loved the presentation of part 1 and 3 a lot

The notation for I f(x) reminds me of the concept of a cumulative distribution function from statistics.

I knew you could solve that problem with variational calculus, but I didnt knew it relates to fractional derivates; by the way, there is a "logic" to solve the problem with fractional derivates? I just ask because they seems to be related, and dividing bt gamm function seems kind of random

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!

good video, I didnt know about that application.

If the function has a laplace transform, then I^n(f) equals L^-1(s^(-n)*L(f)).

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!

Yes a video on gamma function plz

thank you! I really enjoyed this video! hope you get back to making wonderful content.

Gamma function video !!!! :) Tnks for the video was really nice !

Excellent presentation.

That was awesome the animation of the Tautochrone Problem that you showed very Brilliantly describes why time period of Simple harmonic motion is independent of the displacement from the mean position . Like the time period of a simple pendulums is independent of the amplitude of vibration. The animation that you showed at tge end of the video reveals a mystery of transformation of a function into its derivatives or its integrals. Very very awesome visualizations. 💚💚💚 Even if I have studied calculus 3 but it still seems like I am a kindergarten student staring hopelessly at a teacher who is teaching university level students and I can't catch him even if I try my best. One question of mine is that can we move the integral sign inside a square root . If we want to to this then can it be accomplished with the help of fractional integral?

Great video, it was so good that i didn't notice any mistakes the first time i watched it :)

Awesome video! Thank you!

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?

Very interesting. It may have no relevance but I was curious that as the green line moves to match the orange line, there are a few little wobbles that look very like the transients you can see if you flick a switch and look at voltage against time on an oscilloscope.

Has anyone tried challenge 1? Im stuck. I tried using the same thing he does for n=2 but for n, not using induction as it is done in Wikipedia. I get to an expression which I have to relate to the n-th integral of f(x) but I cant see how