Half derivative ln(x)

Рет қаралды 24,223

Күн бұрын

In this video, I calculate the half-derivative of ln(x) using the definition of the fractional derivative given in a previous video. Warning: The answer is NOT constant times 1/sqrt(x)

Пікірлер: 132

@granhermon2 6 жыл бұрын

Now half-differentiate again to check it's right :P

@albertodelaraza4475 6 жыл бұрын

Homework ;‑)

@skylardeslypere9909 5 жыл бұрын

This proof is trivial and left as an excercise to the readzr

@albertodelaraza4475 6 жыл бұрын

This problem requires a solid understanding of both Differential and Integral Single Variable Calculus to solve. Even if the definition of a half derivative is given, along with the gamma function value, any student that can complete the solution has definitely demonstrated mastery of the subject. Just go ahead and give them an A for the course.

@leonardromano1491 6 жыл бұрын

Do fractional derivatives also have things like the product rule or the chen lu? Also I'm triggered by ln(x)+ln(4) because ln(4x) is much more beautiful!

@steelviper7724 6 жыл бұрын

Leonard Romano A while back I played around with half derivatives on a heuristic non-rigorous level and I noticed that because the higher order derivatives of a product of functions followed the binomial theorem, you can use fractional power generalizations of the binomial theorem as well. I can't seem to find where I wrote anything about it down, but iirc it seemed like it converged to reasonable functions, but I was only using really simple products that I was able to take half derivatives of themselves, like f(x)=x^2=x*x or f(x)=sin(x)cos(x)=sin(2x)/2

@mathevengers1131 2 жыл бұрын

I am also triggered by that thing.

@Materialismodialecticohoy 6 жыл бұрын

The definition of the fractional derivative is very similar to the complex Cauchy theorem.

@drpeyam 6 жыл бұрын

Wow, it is! Didn’t realize that!

@112BALAGE112 6 жыл бұрын

Materialismo Dialéctico Hoy That can't be a coincidence.

@LilithLuz2 2 жыл бұрын

@@112BALAGE112 It isn't, the fractional derivative formula is derived from it

@MF-lg8mt 6 жыл бұрын

Riemann-Liouville derivative. Use now Caputo Derivative and compare results. Great video. Best regards from México!

@TZPlayer 6 жыл бұрын

Wait, what, literally yesterday I was thinking about this, I was just about to comment in Dr. Peyam's latest video to do this problem, I can't express how overjoyed I am! Ps: Hi from Brazil

@williamadams137 5 жыл бұрын

👍👍amazing work man!!! Best half derivative video

@kamilbizon8317 6 жыл бұрын

These half derivatives are amazing, i would love to see even more complex problems

@silasrodrigues1446 5 жыл бұрын

Man! That was really amazing! I lost my breath in some parts but I think I got it. I'll copy the initial problem and try to solve it on my own tomorrow. Then I'll start to play picking up other functions to half differentiate! Thank you very much Dr. P!

@eterty8335 Жыл бұрын

how'd you do

@cycklist 6 жыл бұрын

Absolutely superb. Love it!

@Rundas69420 6 жыл бұрын

The derivaive of a function is given by an integral? Feels like complex analysis though. :D

@Absilicon 6 жыл бұрын

Ok, you have a new a subscriber 😁

@marcelorogel9465 4 жыл бұрын

Greetings from Chile, you are the best mathTuber

@BabyXGlitz 5 жыл бұрын

بارك الله فيك يا دكتور پايام

@spockfan2000 6 жыл бұрын

Awesome channel. The one thing I kind of don't like is that you stay in front of the stuff you're writing. Can you position the camera more like they do in BlackPenRedPen, so that you're never in front of what you're writing? Anyway, awesome channel. I loved the episode with the i-th derivative, by the way.

@zathrasyes1287 6 жыл бұрын

Well Done. Could you make a video with an introduction into Fox H-funktion?

@bensnodgrass6548 6 жыл бұрын

can you prove this alpha-derivative formula?? love your videos

@michalbotor 6 жыл бұрын

first we would have to ask ourselves what do we mean by proving this formula in the first place, since it's a definition after all snickily defined only for 0 < a < 1. i suppose, that (D^a)o(D^(1-a)) = d/dx for all 0 < a < 1 is our only constraint. there are many other *distinct* definitions of alpha-derivatives out there btw, similarly to how the (euler's) gamma function is *not* the only valid extension of the factorial function, see hadamard's gamma function. however if you're asking for the intuition for it, then it is cleverly deduced from the cauchy formula for the repeated integration. just find definition of f^(-n)(x) on its wiki page, base it at a=0, substitute gamma(n) for (n-1)!, change n to alpha: 0 < alpha < 1, notice, that (x-t)^(alpha-1) = 1/(x-t)^alpha, and finally take the derivative d/dx of this formula and *define* it to be the alpha-derivative.

@xCorvus7x 5 жыл бұрын

@@michalbotor Isn't 1/(x-t)^α equal to (x-t)^(-α), and not to (x-t)^(α-1)?

@epicmorphism2240 4 жыл бұрын

With cauchy formula for repeated integration, than use fundemental thrm of calc and logic ceiling function

@epicmorphism2240 4 жыл бұрын

Angel Mendez-Rivera it‘s not just the motivation, it‘s the derivitation of this firmula and at the same time also the proof.

@epicmorphism2240 4 жыл бұрын

Angel Mendez-Rivera 1) you can prove the CFFRI by induction and yes i know it‘s only defined for positiv integers, but if you expand a function to more values, it‘s an analytic continuation. And the whole topic of fractional or complex derivatives is an AC. You can derive the CF with pos intgers and proof it, so you are can also expand it. 2) the gamma function isn‘t just a defined function for complex arguments. Yes you can‘t evaluate -1/2! with the definition of the factorial for postiv integers. But the factorial has some specific properties. It‘s recursiv, G(1)=1 and super convex. With these properties you can expand the factorial for complex numbers.

@kharnakcrux2650 Жыл бұрын

24:45 i put ln(4x) I graph this stuff in GraffEQ imagine an animation, if alpha varies smoothly. you'll see that 0 point whip, up & down, while the Negative side makes weird spirals/irrational discontinuous. That's what i call a "Derivative spectrum". I would naiively let alpha be any Complex C

@Koisheep 6 жыл бұрын

Dr Peyam, may I ask if differentiability on an open interval is a sufficient condition to guarantee the existence of a half-derivative?

@drpeyam 6 жыл бұрын

We don’t even need differentiability, I feel just some sort of continuity is enough!

@michalbotor 6 жыл бұрын

.. so i was casually looking for this answer myself, and i came across the fractional derivative of the weierstrass function. and now i can't sleep. help?

@benjaminbrat3922 6 жыл бұрын

Not disappointed =)

@michalbotor 6 жыл бұрын

dr peyam, could you perhaps solve the equation D^a f = f, where 0 < a < 1 next? i believe, that it is a naturally coming to mind question to ask what is an analogue of an exponential function for this "fractional" case and quite possibly of great practical importance too. however as simple as it initially seemed to be to solve, it kind of quickly overgrown me, and i got stuck. yelp? ;p

@drpeyam 6 жыл бұрын

Wow, great question :) I feel it would be something like C e^bx, for some b in terms of a

@michalbotor 6 жыл бұрын

thank you. ;) whatever it is, i hope it will be as astounding, revealing and beautiful as the "fractional" derivatives of the trigonometic functions were.

@NAMEhzj 6 жыл бұрын

This is actually not that hard: As long as a is a rational number: a = n/m for n,m in Nat, iirc it follows from the definition of D^a , that (D^a)^m f = D^(a*m) f = D^(n/m *m) f = D^n f. So, if D^a f = f then iterating m times gives (D^a)^m f = f, so D^n f = f. but for this Differential equation we certainly know the unique solution f(x) = c*e^x for some constant c. So the awnser has to be this. (or no awnser exists, which would be sad). Then, if we assume that the D to the something operator is continuous with respect to the power (which should follow from the formula) , the real case is also clear, since rationals are dense in R. :)

@michalbotor 6 жыл бұрын

thank you kindly for giving it a go. however this cannot be the answer since it doesn't work for example for half-derivative of e^x. at least according to this big-headed genius: www.wolframalpha.com/input/?i=1%2Fgamma(1%2F2)+*+derivative+with+respect+to+x+of+integral+from+0+to+x+of+e%5Et%2F(x-t)%5E(1%2F2)+dt

@NAMEhzj 6 жыл бұрын

Well, it could just be that Wolframalpha cant compute that integral. (since its pretty complicated after all) Or maybe the equation doesnt have any solutions whatsoever. What i did was show that if there is any solution, it has to be e^x. That is assuming that these derivatives follow the rules Dr. Peyam explained here : kzfaq.info/get/bejne/ncdxm6aFqcman3U.html

@k.c.sunshine1934 4 жыл бұрын

The D1/2 ln(x) has a zero in addition to it's expected pole! Very interesting.

@lalitverma5818 6 жыл бұрын

Very talented sir

@scottgoodson8295 6 жыл бұрын

But now you have to take the half derivative of this to check that you get 1/x 😉

@asmodeojung 5 жыл бұрын

Nah, just take a half-integral of 1/x. Piece of cake... probably.

@ashwinvishwakarma2531 6 жыл бұрын

Is the reason we differentiate because if alpha>=0 then the integral diverges (I'm not sure if it does, since it doesnt diverges if 1>alpha>0)?

@flowergirlkaomoji7361 6 жыл бұрын

tried to solve this in the spare time I had during my calc 2B final; no wonder it was so hard! (I ran into a wall :/ ) very cool :)

@radiotv624 6 жыл бұрын

Can you construct a proof of the formula for D^α f(x) please

@leonardromano1491 6 жыл бұрын

These fractional derivatives are linear operators, so it should be natural to ask about the maximum domain, and eigenfunctions + eigenvalues. Since the eigenfunctions of whole derivatives are exponentials/ plane waves and they are just functions of fractional derivatives they should be simultaneously diagonalizable on a sufficient domain. The quest is: Are the eigenfunctions of the fractional derivative known on their maximum domain, and if they are, are they just plane waves?

@dalek1099 6 жыл бұрын

Leonard Romano For a fractional derivative D(a/b) with eigenfunction f this implies that f is also an eigenfunction of D(a) by applying D(a/b) b times and the eigenfunctions of D(a) are just plane waves. However, this argument doesn't hold for irrational derivatives so it would be interesting if there are non plane wave eigenfunctions of irrational derivatives. I guess that would depend on whether eigenfunctions change in a limit of operators(approaching irrational using more and more accurate fractions)

@DrunkenUFOPilot 4 жыл бұрын

D^a D^n f(x) = D^n D^a f(x), and D^n = D D^(n-1), and so on... you find that given some eigen function g(x) of D, such that D g(x) = c g(x), then D D^a g(x) = D^a D g(x) = c D^a g(x), so the eigenfunctions of D^a are the same as for just D. Of course, the differential equation D g(x) = c g(x) is one of the simplest, with the result g(x) = exp(cx). It all boils down to D^a exp(cx) = c^a exp(cx). Let a be imaginary, and knowing D^a is linear, we can perform D^a on arbitrary functions using Fourier transforms. I once did that to a Cassini image :)

@szymon5830 6 жыл бұрын

How about i-th direvative?

@szymon5830 6 жыл бұрын

And how to do that? Is there any formula for that?

@drpeyam 6 жыл бұрын

There’s a video on that!

@christophem6373 6 жыл бұрын

Is there application for half derivative, as derivative for variation of function, andtangete equation, and second derivative for convexity .... ?

@drpeyam 6 жыл бұрын

Yeah, look at the pinned comment in my previous half derivative video

@t.diyarmath605 3 жыл бұрын

Hello Dr. this video it is so nice can you solve by Caputo fractional derivative

@Tomaplen 5 жыл бұрын

Does D^(alpha) conmutes with D^n when 0

@drpeyam 5 жыл бұрын

Yeah, that’s how you define D^alpha for alpha > 1 actually!

@Tomaplen 5 жыл бұрын

@@drpeyam Ooooh ok nice, thank you!!

@wankar0388 6 жыл бұрын

Tigre!!! Dr. P!!!!!!

@mattryan2006 4 жыл бұрын

if you switch variables to do it for ln(m) you get a factor of root Pi.m in the denominator

@digxx 4 жыл бұрын

What I always wonder, what the lower integration limit is supposed to be. You take x=0, but why not x=1? When integrating an integer number of times, this amounts to the successive integration constants coming in. When deriving an integer number of times they however vanish. But how many integration constants are there if you derive a half times = integrating a half times + deriving 1 time. When you choose x=1 as the lower bound since ln(t) vanishes there, you would instead get 2*\log(\sqrt{x}+\sqrt{x-1})/\sqrt{\pi x} which is somewhat different from \log(4x)/\sqrt{\pi x}. In fact, getting from the one to the other expression involves infinitely many integration constants c_n for the series \sum_{n=0}^\infty c_n x^n (this would only be finitely many terms when integrating an integer number of times).

@yoavshati 6 жыл бұрын

Can you use the alpha differentiation to solve integrals by doing D(-1)?

@justinji1595 4 ай бұрын

Hi, does this work for alpha = a complex number? Such as i?

@damiandassen7763 6 жыл бұрын

I have a question: do these half derivatives have any "real" meaning; i.e are they usefull in physics. I can't think of what they might mean. And I have a request: can you do the half derivative again to see if it truly works. Btw I don't doubt your expertise. :)

@drpeyam 6 жыл бұрын

Yes, see the pinned comment on a previous video

@damiandassen7763 6 жыл бұрын

Dr. Peyam's Show btw I forgot to mention. This was a great video. Keep up the good work.

@aee220phmunirabad 3 жыл бұрын

Sir, Is it possible to find general real derivative for ln(x). Eg: D^r(ln(x)) Where r is real number

@txikitofandango 4 жыл бұрын

So, if we let f(x) = ln(4x)/sqrt(pi*x), then f(f(x)) should just equal 1/x, right? Well, that's not the case when I graph it, so where am I wrong?

@txikitofandango 4 жыл бұрын

In Desmos the graphs of f(f(x)) as defined above and 1/x look different.

@francescocapacci9941 4 жыл бұрын

I loved this calculus, but my question now is : does really exist a function which the half derivative is actually 1/x, like the log in the usual derivate??? I wonder what 'd be.

@drpeyam 4 жыл бұрын

Yeah, that would be interesting! I wonder if 1/x^1/2 works!

@christophem6373 6 жыл бұрын

In which theorical context this half derivative appear ? Same answer to my first question: Fractional Analysis. en.wikipedia.org/wiki/Fractional_calculus

@pedrocusinato02 6 жыл бұрын

Calculate the half derivative of a constant

@drpeyam 6 жыл бұрын

Coming on Monday ;)

@drdca8263 4 жыл бұрын

Now I’m curious as to what the half integral of x^(-1/2) is. Edit: it appears to be x*sqrt(pi)/2 ? That is based on just plugging the formula into Wolfram alpha though, and may be mistaken.

@drpeyam 4 жыл бұрын

That doesn’t seem right, are you sure you didn’t do the half integral of x^1/2 or the integral of x^-1/2?

@drdca8263 4 жыл бұрын

Dr Peyam I did (1/sqrt(pi)) * integral from 0 to t of (t-x)^(1/2) * x^(-1/2) dt Maybe I should have used (t-x)^(-3/2) ? Using that, Wolfram alpha tells me that the integral does not converge. Is this what you would expect?

@drpeyam 4 жыл бұрын

Ah, I know where the mistake is: The formula is only valid for alpha between 0 and 1, so to get the half integral, you take the integral and then the half derivative (using the formula above)

@pickleyt6432 Жыл бұрын

So I just calculated it and you should get zero as the result for D^(1/2)(1/sqrt(x)) via the formula D^n(x^k)=x^(k-n)Gamma(k+1)/Gamma(k+1-n), and plugging in k=-1/2. This is because the term in the bottom goes to Gamma(1/2-n), which approaches infinity as n approaches 1/2, thus making the whole expression tend towards zero

@MrRyanroberson1 6 жыл бұрын

I think you could prove that this is the half derivative by taking the half derivative of the answer to get 1/x. You could also prove your prediction of 1/sqrt(x) wrong by half differentiating that as well

@thomaswilliams5320 6 жыл бұрын

Can anyone explain why 2t*ln(t) = 0 when t = 0, it seems like this implies that 0^0 = 1

@jordancole4004 6 жыл бұрын

Thomas Williams wanna go park???

@BabyXGlitz 2 жыл бұрын

2t*ln(t) = 2*ln (t^t) lim t^t as t goes to 0 is 1 and 2ln(1) =0

@tricky778 2 жыл бұрын

Rational derivatives are all very well, but what about irrational derivatives?

@mokouf3 3 жыл бұрын

That integral is best for high school students, for scaring them!

@zwz.zdenek 6 жыл бұрын

What is the intuitive meaning of a non-whole-number derivative? Where is it useful?

@drpeyam 6 жыл бұрын

See the pinned comment on the previous video

@zwz.zdenek 6 жыл бұрын

Thanks for replying, but I'm not at all familiar with fluid dynamics; I was hoping for getting the relationship between a graph of a function and its half derivative. I can see how these could be useful for fitting a model, but any model fitting is a virtue out of necessity and far from the exactness of mathematics.

@DrunkenUFOPilot 4 жыл бұрын

I've been fascinated by this topic for many years. I still don't have a good intuitive understanding of semi-derivatives and semi-integrals, but I do know that integrating white noise makes brown noise. Half-integrating white noise makes pink noise, useful in audio testing. Also, the isochrone problem. Then there are uses in electrochemistry and fluid dynamics, and I think I once saw something concerning antenna design. But an elegant, "aha!" getting of the concept intuitively, not yet. A semi-integral is a mild low pass filter, in a way, but that just isn't good enough for a sense of "getting" it.

@thomasblackwell9507 5 жыл бұрын

Who survived? In the words of the Great Bugs Bunny “I should have turned left at Albuquerque!”

@adonaythegreat8426 2 жыл бұрын

What is the significance of half derivative

@drpeyam 2 жыл бұрын

See pinned comment on the original half derivative video

@cachamuertos 6 жыл бұрын

why your profile photo is a rabbit? it is funyy XD

@drpeyam 6 жыл бұрын

It’s my bunny Oreo!

@cachamuertos 6 жыл бұрын

Dr. Peyam's Show wow its cute. Ohh i just remembered something. Its possible to express cbrt(a+bi) in terms of p+qi?

@xy9439 5 жыл бұрын

Cesar Cruz Yes of course

@insouciantFox 4 жыл бұрын

The way you write your math gives me flashbacks to ‘Nam - Calc I. My Calc I teacher was very persnickety about what form the answers could be in: only in “simplified, factored, reduced, elementary form” This meant a few things: 1) No decimals of any kind for any reason. 1.4236 had to be 3559/2500. 2) Any fraction must be extended to the entire expression, e.g., 1/2 x + x/ln(x) must be (ln(x)+2x)/(2lnx) 3) Denominators and logs must be rationalized/ in elementary form: ln16/sqrt(136) must be ln(2)sqrt(34)/34 4) If possible, all nat logs should be written as one function, e.g. .5ln(x)-ln(sinx) must be ln(x/(sinx)^2)/2 5) Any polynomial that can be factored through any known algebraic method must be so 6) No negative exponents. Ever. 7) All logs must be natural or common. No log_2(x), only ln(x)/ln(2) 8) Euler’s notation for derivatives (i.e. the “D” functional operator). Prime notation was only for f(x) and the like. 9) Trig must be the most compact/ reduced form possible. No x/sinx, only xcscx etc. This may not seem annoying, but for really complex exp, it can get tedious. Liked him as a teacher and person. Scarred me as a mathematician.

@drpeyam 4 жыл бұрын

Good, I like your teacher haha

@user-dv4eq7tv3c 3 жыл бұрын

Что на 2:51 произошло с -1/2 ?

@user-dv4eq7tv3c 3 жыл бұрын

И если что я дальше смотреть не стал. Потому-что мне уже то непонятно.

@alberteinstein7571 6 жыл бұрын

👍 :D

@insouciantFox 3 жыл бұрын

So it’s a derivative of a convolution. Interesting.

@GhostyOcean 6 жыл бұрын

Why not use int(db/(a^2-b^2))=(1/a)arctanh(b/a)+c at 9:20, then use the definition of the arctanh function to find it in algebraic terms? You'll get the same answer with less work (assuming you didn't want to do that partial fraction) Edit: fixed some typos Edit 2: I guess you just didn't want to use it? To each their own 🤷‍♂️

@drpeyam 6 жыл бұрын

Yeah, but I don’t like obscure integration formulas ;)

@GhostyOcean 6 жыл бұрын

@@drpeyam ahhh I see. Seeing the integration in full is more useful to the casual watcher since they get to learn how to do it if they didn't already.