Half Derivative Definition
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- Опубликовано: 25 авг 2024
- In this video, I give an actual definition of the half derivative and use it to calculate the half-derivative of x. Thankfully we get the same result.
In the next video, I use that definition to calculate the half-derivative of ln(x)
Some applications of fractional derivatives:
There are surprisingly many applications of this, because it turns out that some differential equations in physics are written in terms of fractional derivatives, see en.wikipedia.org/wiki/Fractional_calculus#Applications
There are three other ones I can think of:
1) In functional analysis, it's an important problem to find a square root of an operator (I don't really know why, maybe to decompose that operator?), and what we really did is to find a square root of the derivative operator, because if you apply D1/2 twice, you get D, so (D1/2)^2 = D, so D1/2 = sqrt(D) in some sense.
2) There is the nice formula in Fourier analysis that says that the Fourier transform of f' is integral of x e^(i something), and we have a similar formula for the fractional derivative, (I think, don't quote me on that) that the Fourier transform of D^(1/2) f is integral of x^(1/2) e^(i something).
3) Fractional derivatives allow us to define nice spaces of functions (for example, those whose fractional derivatives exist and are square integrable), and sometimes in differential equations you have a solution that is not defined in the classical sense (i.e. continuously differentiable), but might belong to this nice space, which allows us to study those equations.
Thank you
This helps! Thanks Dr. P!
ADDENDUM:
I now know that Fractional Quantum Mechanics is a thing!
Holy smokes!
Ultra Nice Dr. Tigre Peyam!!!
What about fraction integrals ? They can be defined as a linear transformation T(f) , such that T(T(f)) = Integral(f) +c (in case of half integral)
@@timurpryadilin8830 for the fractional integral I think you can integrate the function, and then perform a fractional derivative.
An intriguing idea. Before watching, I've tried to make sense of this. Here's what I got.
The plan is to first define half-derivative of a constant power of x, by using the general form of k'th derivative of xⁿ, and plugging k=½ into that.
Then it can be applied to any analytic function (defined as any fn that is equal to its Taylor Series), by applying it termwise to the fn's TS.
D(xⁿ) = nxⁿ⁻⁻¹
D²(xⁿ) = n(n-1)xⁿ⁻⁻²
:
Dᵏ(xⁿ) = [n!/(n-k)!] xⁿ⁻⁻ᵏ
For k=½, that becomes
D[½](xⁿ) = [n!/(n-½)!] x^(n-½)
And for n=1,
D[½](x) = [1/(½)!] √x = [2/√π]√x = 2√(x/π)
As a check, we can apply the same formula to this, where n=½, and get
D[½](√x) = ½√π
and putting the two together,
D[½](D[½](x)) = ½√π [2/√π] = 1 = D(x)
And more generally,
D[½](xⁿ) = [n!/(n-½)!] x^(n-½)
D[½] x^(n-½) = [(n-½)!/(n-1)!]xⁿ⁻⁻¹
so that:
D[½](D[½](xⁿ)) = [n!/(n-½)!] [(n-½)!/(n-1)!]xⁿ⁻⁻¹ = [n!/(n-1)!]xⁿ⁻⁻¹ = nxⁿ⁻⁻¹ = D(xⁿ)
showing that what we've developed is consistent with integer-order derivatives.
The second step of the plan, applying D[½] f(x) by first expanding f(x) in a Taylor Series, and then taking D[½] termwise, is clearly possible, but pretty complicated, and I leave that to later, or to Dr. Peyam, to see whether that's what he will do.
Watched.
That's neater in at least one aspect, because it eliminates the need for TS expansion when applying ½-derivative to a general f(x).
I'm pleased that our formulas agree, at least for α (= k) = ½. And I think that the way you've defined D[α] f(x) for α >1, insures that we will agree for general α.
And that this means that you could define fractional derivatives either way, using whichever form is more convenient to the problem at hand.
Separate question/issue: Do either/both of these break down for non-analytic functions? I'd kind of expect that both of them would...
What say you, Dr. Peyam?
PS, I haven't watched your earlier vid about ½-derivatives; I must remedy that!
Fred
im shocked that you did that half derivative of x lot more simpler. my mind=blown
Such a beautiful explanation. Thank you
@Cosmic Dissonance In my youth I had learned that a derivative is a limit: (F(x)-F(x0))/((x-x0) when x is going towards x0.
Well done, congratulation.
@@VSP4591The limit definition can also be extended to the Grunwald-Letnikov fractional derivative from nesting limits n times (MVT and other stuff making it permissible) and then plugging in fractions
It would be fun to show that the limit as alpha approaches 1 D^alpha f(x) = f'(x)
John Baker you mean by using the definition of limits? Because if we just do it, we can just plug it in.
Definition of (-1)th derivative is how we call integration
You are right. The the negative one (-1) derivative suggests inversion to the derivative - it is integration. More interesting question would be about definition of the derivative for -0.5 (-1/2). In other words, the definition for the negative fraction derivative.
@@mireksoja9063 It is the fractional integral. It works both ways. Here you can see Dr Peyam using the notation D^a(f(x)), which is equivalent to d^a(f(x))/dt^a whereas for the indefinite integral (antiderivative) we can give it its own operator J^a(f(x))=D^(-a).
@@jerryflamholz8565 indefinite
That identity is to be fair one of the greatest formulas I've ever seen. Who came up with this??
Whoa, never knew this stuff existed... Cool!
You know its gonna be a hell of a formula when you start writing a parsec away from the equal line.
Hahahaha
These talks are ridiculously fun to watch. What a blast. I feel good about liking mathematics here.
i liked the other version more, but its always great to see other ways! thank you for the video! :D
by the way, is the half-derivative of ln(x) just the negative-half derivative of 1/x? then we could use the formula you've shown the other video
There will be a video on that :)
I think that the answer is no, because if you try to verify this doing two times the half derivative of ln(x) you should get 1/x and your answer doesn't respect this condition
Well, fractional derivatives only make sense within corresponding Hilbert or functional space. In that spaces you consider the eigenfunctions of D, so it becomes possible to introduce D^a. For example, if we consider functional space that is spanned by e^{i p x} for real p, then
D^a f(x) = \int_{-\infty}^\infty e^{i p x} (i p)^a f(p) dp/(2 pi)
Notice that (i p)^a is defined as a principle branch in case of non-integer a.
Non-locality of D^a can be the easiest seen at the differentiation of x^b at real positive x
because the answer will depend on how you extend x^b onto the negative values.
For example, for (x + i0)^b, (x - i0)^b and H(x) x^b, H(x) is the Heaviside step function,
the action of D^a is different at real positive x, even though all these three functions coincide at real positive x.
Is it possible to do a "Half Fourier Transformer?" When looking for patterns I think it would be useful.
Earthbjorn Nahkaimurrao He commented this above. Read his pinned comment.
And somebody dares to call category theory "abstract nonsense".
All hail Prof. Grothendieck
Wow! Thank you! I was not aware for such interesting construction as the fractional derivative. It's opening new concepts for differential equation of such type of derivative construction! Further, if we replace alpha with another functions of another variables such as g(x,y,z,t)... Interesting. Thank you again.
I wanted to take a moment to thank you for the excellent video on fractional derivatives. Your explanations were clear and concise, and I found the examples you provided to be very helpful in understanding this complex topic.
Your dedication to your students and your passion for teaching is evident in your teaching style and the effort you put into creating these valuable resources. Your video has helped me immensely in my studies, and I am grateful for the time and effort you have put into creating it.
Thank you again for your wonderful work and your commitment to teaching.
Sincerely,
Thank you so so much!!!
Ah I thought fractional derivatives were going to be harder, but this isn’t so bad!
sir excellent --please teach parabolic co ordinates -- u r topic r unique on internet
This seems like a hilbert transform or some such? does this come from the fact that the derivative in fourier space is a multiplication by (jw)^(n) where n is the degree of derivative.
Indeed!!! It is very closely related to the Fourier transform, and there will be a video on that next month or so!
Nice! I couldn't watch the videos for some time because of illness but I am happy to see interesting things here
That is a really interesting topic, make more videos about it!
Successive integration by parts will give you a series expansion for the fractional derivative too. It's an even easier method to get the half derivative of x than direct integration, in my opinion
Dear dr. Peyam, we all know that the first derivative is the tangent to the curve. What is the geometrical interpretation of half derivative???
Check out the video half derivative as a limit
I cannot tell you how much this helped me. I am so thankful for this. Subscribed and big like
Glad it helped!
3:02 is this commutative?
AndDiracisHisProphet incorrect. That property of semi-group is valid only under some conditions. For fractional integrals, there is no problem, however, for derivatives the situation is differente.
What do you mean, by "incorrect"? I asked IF it is commutative, and you can either answer "yes" or "no", but not "incorrect". That has zero semantic value.
But thanks anyway, I extracted the information I wanted out of your "answer".
@@MF-lg8mt I think he meant if it is the same if you first take de 1/2 derivative and after that the normal derivative
Very good. Thanks a lot! You are coolest guy Dr Peyam!
I think you explained it, but why Leibniz rule can’t be use to evaluate the derivative of the integral?
Thank you
Does this and the other definition that you showed quite a while ago always give the same results?
Interestingly, it does for powers of x, and it doesn’t for 1 (there will be a video about that), and apparently it gives slightly different answers for sin and cos
Thanks, your videos are great and remind me of school
Does the half-derivative have an application in physics or is it just some abstract continuous extension of the usual derivative?
I'm not an expert, but I seem to recall that Feynman enjoyed a bit of fractional calculus. I also know of some books and talks which focus on applying the ideas to physics, like this one: www.worldscientific.com/worldscibooks/10.1142/8072. How much mileage it gets in practice, I don't know. Doesn't seem too popular yet.
Thanks for the info.
See the pinned comment
What if you use the Cauchy integral formula instead? I assume that would work as well.
This video defines D(n)f when n is a suitable real number.
Which leads me to wonder about generalizing....
* Suppose we let n vary on some (not necessarily real) interval I. Or path P.
Under what conditions is D(P)f continuous? That is,
If a --> b, then D(a)f --> D(b)f
Is there some resource on this ?
Just curious.
Thank you.
I didn’t understand what a fractional derivative actually is. If a normal derivative of a function is a linear projection at a particular point, what is a half derivative then?
Very nice.
from which text book you have taken this definition sir it would be helpful for me
Once we have defined a fractional derivative operator, could you define a new operator for the derivative of the derivative; using the same limit expression used in the conventional derivative definition?
Hi!!! Welcome to Fractional Calculus!
At 3:13 shouldnt we first take the half derivative, then the first derivative? Because if we first differentiate then take the half derivative our result will have some extra term from the lower bound of the integral. Like how integral of e^y dy from x to b is e^x-e^b, where instead of C we have e^b. If we first differentiate and then take the half derivative this constant disappears.
I know I've asked about this before, but do such things as a product rule and composition rule exist for fractional derivatives? I.e., is D^a[f(t)g(t)] = f(t)g^(a)(t) + f^(a)(t)g(t), 0
How is it generated? Could you please share with me some understandable sources?
so very helpful!!! thank you so much :)
It's amazing :D, I really like your channel (I'm engineer) but, can you record your videos with higher volume? Please... Greetings from Colombia
Why do the limits of 0 to x always work? Is there a general proof that these singularities you mentioned balance out? I can't find one anywhere for the life of me 🤣
How about if i want alpha to be some imaginery number?
Szymon Misiewicz Dont think it could work since there is no order to complex numbers, thus the interval of 0
OneInfiniti MathBoi So i am asking how to find a derivative where we can use complex number as a alpha. Is there a formula for this?
@@restitutororbis964 he is asking the ith derivative and is not fractional, so he is asking if it is possible to have formula for the ith derivative
why we use integration in this formula please give me information
Well, i tried to use the half-derivative again but got stuck on a bunch of annoying integral (integrals with those sin-1 tan-1 log answers), but i was able to get the x out and be left with a pure annoying integral (2/pi * integral sqrt(u)/(u+1)^2) from 0 to inf- which had to overall equal one since D(0.5)^2 x=1
can you use the half derivative (and other fractional derivative) to find formulas for these type of integrals by using half derivative twice, using substitutions to extract x out of the integral and comparing with normal derivation?
Hey Dr. Peyam..... I've a question..... For 1/2 integration how many constant should be there in solution 🤔
Taking the half integral means integrating once and differentiating 1/2 times afterwards, so there will be 1 constant.
Hey Dr Peyam I recently tried to find the Taylor series of the inverse function of x^x but I failed. Could you do a video on that?
Because you take the (regular) derivative in the definition, it looks like to take the "half derivative" you take the derivative of the "half integral" isn't it?
is there something like half gradient of a function?
This but in higher dimensions
How would you apply the definition of limit to find the half derivative?
You can, I’ve done a video on that, it’s on my playlist
@@drpeyam Ok thanks. Let's see if I can find it. ^^
I'm having difficulty to find the video. Can you link it to me please?
Thank you.
"Partial differential equations are hard for you? ha! don't make me laugh, my boy here be solving them partial fractional differential equations, and let me tell you, linear is the last word you'd want to describe them as..."
So, that was kind of a "removable" singularity?
Could you explain the motivation for this definition (if there exists) ?
What is the meaning of this derivative? Does It have geometrical meaning?
Does it only work for fractions? Is there a sqrt(2) derivative or a pi derivative? Or even an i-derivative?
Works for any real and even complex number! Check out my fractional derivatives playlist
Is there an intuitive meaning of the half derivative result e.g. in terms of ideas like slope of the line or area under the curve?
Yes, check out my playlist (I think it’s called half derivative as a limit or something)
is it me, or should the integration limits have been substituted too since u=x-t?
I am looking forward to the half derivative of ln, I got it but it's a mess.
Apparently the full derivative and partial derivative do not commute, but is there a rationale for taking first the full derivatives and then only the partial derivative?
There will be a video on that :)
Dr. I wanna know "Is there any half integral?"I think the integration of 1/2 order derivative will give us the 1/2 order integral.But,l am trying to relate it with area formed by the curve.Is it possible?
It’s the -1/2 th derivative
Is it possible to define half-derivative for holomorphic functions? What properties do they have in complex plane? What happens on poles and with cauchy integrals?
Great insight! I’m pretty sure the definition stays the same, except you do a contour integral instead. And not sure what happens to poles there, it would be interesting to explore this further
@@drpeyam It is! I am asking that because if I prove that some peoperties remain in half derivatives than I have a way to prove for example that there is about x/sqrt(lnx) numbers that are sum of two squares and less than x.
Wow, that would be so cool! But I really don’t know what the answer is, since it’s mainly used for real functions
En qué contexto surgió esto?
Realmente no me hago idea de sus aplicaciones pero sí me recuerda el método de operadores para resolver ED's.
Could you please prove that formula? I have never seen it. Also, does it have geometrical meaning? Thanks
It’s a definition, it can’t be proven. And yeah, there’s another video on the geometric meaning
But how does that look on a graph?
Beyond of maths for fun, does half derivative has an actual meaning or use in applications?
See the pinned comment
Not in hollidays ? Thanks !
What about complex derivative? D^i f?
There’s a video on that, I think it’s called imaginary derivative
@@drpeyam Oh thanks!
I call the gamma function the hangman function
Just a bunch of thought now: how about fractional integral? alpha=a complex number? maybe fractional differentiable function? fractional integrable function? fractional partial differential equation? maybe FTC can apply for the half derivative of a half integral? maybe our knowledge can be extended to "fractional-dimention" world? people can talk about things in 2.5-dimention perhaps? or more fansy, functional derivative and integral?
The alpha fractional integral is the -alpha derivative.
Alpha being a complex number is completely ok, and there’s a related video somewhere.
Fractional PDE absolutely exist, in fact that’s what motivates fractional derivatives.
And there are fractional dimensions, google Hausdorff dimension :)
never knew Arnold Schwarzenegger was a mathematician
why extend the definition of the derivitive in this way ? as far as I see there isn't much of a connection between this and the actual derivative
it satisfies the properties that:
1) it is linear
2) and D^(1/2) of D^(1/2) of f(x) = D of f(x)
interesting , and has it been proven that it is the only application that satisfies this property ?
Hi! I want to ask about the geometric interpretation of the fractional dervatives
There’s a video on that, check out my playlist!
clearly (D^0.5(X))(D^0.5(X)) is not equal to D(x),in this case, how could you say D^3/2F(X) is equal to DF(X)*D^0.5F(X)
can you make a proof video?
?
Dr. Peyam's Show
what I meant to say is that it will be beautiful to learn the proof of this interesting formula...
There is no proof, it’s a definition!
Dr. Peyam's Show yes but how can I show that apply this operation a times(if alpha is 1/a) will give me the first derivative?
Then could u explain us how the formula means....?
7:22 2×sqrt(u)=2x^(3/2) ? What did i miss?
Its would be 2Xsqrt(X) which is 2X^(3/2). It was an X not a multiplication symbol.
If u do half derivative of 2 u get half derivative of x too... Why?
Wo hast du eigentlich studiert? :D
In Berkeley, tolle Uni :)
It also Works for alpha=0
And if alpha is some imaginary number?
Works too
f (x+h)
Hi ,professor how are you
ahhh god damn it this does not make any geometric sense to my head!! It never stops. The funny thing is that I legit thought about that in my calculous class the other day... Like, I thought "hey, we now know what's a derivative and an integral... but does something like half an integral exists or is properly defined? " or so. I did not ask my teacher because I thought it would upset my class about it (I always ask out-of-my-knowledge-set kind of questions, and so I often get big stares, like I should just shut up lol).
What about D(i) ( i, the complex number, th derivative). What about D(i/2)? Although I guess you can simply apply D(i) before or after D(1/2) to do the operation.
Each time I encountet a concept in math that seem to utilize natural numbers only, there is always somebody somewhere who destroys my comfort zone and show me that no, in fact, it can be used with real numbers and sometimes even complex numbers!! Ah!
Like factorials.
Does half integration exists? Is there a formula for this?
Also, has anybody heard of continuous operations? Like something between an addition and a multiplication (operation 3/2)? or like between a multiplication and an exponentiation (operation 5/2)? Or... a complex operation? Like operation i lol
Is there some concept that seem to only work with integers in math that cannot be "realyfied"?
Check out my playlist
Check out medium.com/@olydis/fractional-derivative-playground-74e61c28721f if you want to play with fractional derivatives interactively :)
That's cool that's cool. How about imaginary derivates? :)
There’s a video on that! Check out the fractional derivative playlist
A derivative is a derivative, you can't say it's half.
Someone here doesn't understand mathematics, and it's not Dr Peyam.
Edit: en.wikipedia.org/wiki/Fractional_calculus
In the spirit of this video ruclips.net/video/VI6ZlnkuxuA/видео.html , couldn't one take the Maclaurin series for $x^{\alpha} and plug in $\mathbf{D}$, then have that linear operator act on f(x)?
Dr Peyam=Dr Banner
Hello, I was wondering if it is possible to show an equivalence between this integral formula and to decompose the function in a Taylor series (for a nice fonction) and diferanciate each monome using the extension of the derivative of such monome ?
Also if it is possible just to go in Fourier's world, dericate the exponential easy peasy and go back.
Thanks
At least the exponential part were 1-1/2
How are you,
Not a cynical question: When is this ever useful?
search for the broader term 'fractional calculus'
crowd modelling, since most humans dont follow smooth differentiable paths
also sometimes physics, like statisitical mechanics, is done in 3-epsilon dimensions with lim[epsilon -> 0], so you work with fractional differential equations
112BALAGE112 that question is forbidden!!!!!
Czeckie Where is fractional calculus used?
Cartoon dekhna jada sahi rahega
great video but i still don't understand how this is useful
See the pinned comment
Big deal--you haven't proven the formula; you just plugged & chugged---develop the rich history of it's development going back to Leibniz & L'Hospital !
Is the half derivative of ln(x) just 2/√(x) ?
No, if you try to verify this doing two times the half derivative of ln(x) you should get 1/x and your answer doesn't respect this condition
Gino Giotto it depends on what the half derivative of 2/√x=
You lol the half derivative of 2/sqrt(x) is exact ln(x)
Hears the explanation......
D^(1/2)(2/(x)^(1/2)) is similar to differentiate 2/x^0.5 wrt x^0.5 so asuming x^0.5=y d/dy of 2/y is 2ln(y)
=2ln(x^0.5)=2×0.5ln(x)=ln(x)
Philipp Hoehn If you use the first definition you get that the half derivative of 2/sqrt(x) is equal to (-1/2)!/((-1)!*x) but (-1)! (which is Gamma(0)) doesn't exist so this formula doesn't work
And if you use the definition in this video you get an answer that has no sense, because if you solve this integral at some point you have to divide by zero (and also using the limit doesn't work)
So since the answer doesn't work in both cases, it is clearly incorrect
Debraj Banerjee Your calculations have no sense, we are talking about if 2/√x is or not the fractional derivative of ln(x) and you try to demonstrate that the half derivative of 2/√x is ln(x)
:O