Typo: I forgot to put i sinh(pi) in the final answer. The answer should be: (i-1)/2pi Gamma(i) i sinh(pi) (x cos(ln(x)) + i x sin(ln(x))) Which can be written as: (i-1)/2pi Gamma(i) sinh(pi) (- x sin(ln(x)) + i x cos(ln(x))) Also, in case you’re wondering about e^x, cos, sin: Fractional derivatives of exponential and trigonometric functions kzfaq.info/get/bejne/oZiEY7ya0bbWh6s.html

We're ready for quaternions, jth and kth derivatives, and Frobenius theorem

Thumbnail : *D i x* Me : Looks *interesting*

For every ex you've had you have to ask yourself: "Why?", so you can have a y for every x

This is the right place to learn, to relax, to be amazed, to feel as you are sited in the front row of a master class of mathematics. Please Dr. Peyman, never stop to share with us your knowledge. Kind Regards!

Very impressive, but can you do the derivative'th derivative of x

Incredible. Pure maths at it's highest. Just wanted to mention that your presentation has improved remarkably. Whiteboard is clear and easy to read, audio is good and your dressed well for the camera.

Peyam is a living legend.

Hey, I found a way to think of Gamma(i), assuming I did it right. If you plug i into the integral and expand it with Euler's formula, you get two integrals: integral of 1/x*cos(lnx)e^-x and i*1/x*sin(lnx)e^-x. With the u sub: u = lnx, du = 1/x*dx, we get the integral from 0 to infinity of -1/u*cos(u) and -i/u*sin(u). The imaginary part is -pi/2, but the real part diverges. However, evidently the Gamma function integral does not converge absolutely for Re(z)

Thank you for your videos. Having only learned "vanilla" calculus and using it quite regularly in my day to day life, these videos have been inspiring to remember why I fell in love with mathematics when I was younger.

This guy gets better and better.

I think Dr Peyam is a great teacher in that his enthusiasm and positivity open the door to the student feeling that they too can learn this cool stuff.

When fractional derivative is not confusing enough

9 mins to come to the answer, then 13 minutes to rewrite a rewritten formula that you rewrote in order to rewrite it in a rewritten way.

Fascinating stuff! Also, love your style. Keep 'em coming!

This is the most fun math series ever----thanks so much!

When I saw you break out {tan x}, I got that feeling that only great, beautiful math can give you. Oh my lord that's some good stuff right there.

wow seems interesting , never seen this before !!!

Amazing ! This is new for me. Are these concepts of half-derivative and imaginary-derivative expandable to other functions that polynomial ones ?

No me canso de verlo, genial y gracias Dr. Tigre Peyam !

This is more valuable than a kg of GOLD to me.

This is insane. I love it.

To be convincing, this would need to work for functions that are not simple power laws.

My braines sanity: "Am I joke to you?"

6:53 "Proof by analogy"

Happy to watch informative video from a cheerful maths teacher :)

I WANT TO BELIEVE

Dr Peyam - a delight and pleasure as always! I must say, pulling that derivative out of thin air reminded me of a magician pulling a rabbit out of an 'empty' hat. then, of course, I remembered Oreo, and all was clear! Please would you do a sequence on transfinite numbers? I mean, well beyond 'countable and uncountable infinities', Hilert's hotel etc. Sam Sheppard's excellent book 'The Logic of Infinity', Cambridge U. Press, (no flakery here! ) - might give you some notions of the level to pitch your presentations on this. Not post-Postdoc, but past 1st yr undergrad. Thanks!

Nice video, thanks Payam jan! Keep the great work up!

Very enjoyable tutorial! Thank you for the video

Sometimes i just open your videos to listen the happiest "all right thanks for watching" ! Its so cool!!

Awesome, I love that passion! :D

"aye pi aye"... aye aye aye... :) Weird shit, but mind blowing. Never thought about a derivate this way. I always learn something new :)

Do fractional derivatives have any usefulness when analyzing physical systems?

Hey Dr. Peyam Two questions Is there any Interpretation of imaginary differentiation? Would you like to do a video about fractional differential equations?

i love this!! you are so creative!

So how would this work for non-power functions, e.g. f(x)=ln(x)? One guess I have is, that you could use the Tailor expansion of f(x) and then get the i-th derivative for all terms. Not sure this would work though.

Dr peyam Thank you for fixing the angle of the camera with respect to the board from π/6 to π/4! :-) My question has to do with the generalization you made regarding the A. From integer to real and then to imaginary. How do you prove that this generalization is valid? And how this generalization is related to the original definition of a derivative which is a limit. Thank you George

what is the advantage of a fractional differential equation? why many of them converting their problems in integer order model to non-integer order model?

Does this satisfy D^a = e^(a log D), treating D as linear operator? Can you even take the log of D? It seems positive semidefinite but it's not index 0 and I can't recall the exact conditions.

Thanks Dr. Peyam, very interested.What is the interest to compute the imaginary derivative in our real Life ?

Is it possible to define a differential power derivative like D to the power of epsilon?

thank u, the illustration is realy down to every detail

Since your formula for the Ath derivative of x^N is proved by induction, it means it holds for all a in integers. I don't think you can generalise it just like that for complex numbers as well, because it's a different domain. Correct me if I'm wrong.

imaginary derivatives: the kind of derivatives year eleven students come up with on the exam after half a year of not doing their exercises?^^

Amazing!

Is there any equation expressible with elementary functions where the i-th derivative produces a result that is also expressible with elementary functions? Or any real function where the i-th derivative is also a real function?

Ok there is a simple formula for F(D)e^(ax) where D = d/dx operator of course ( *The D-Op Theorem* in fact used a lot in solving differential equations )so before I state the relevance here, I give quick simple example of the power of this theorem: Eg, solve y' ' -5y' +6y =e^(4x) then [D^2 -5D +6D]y = e^(4x) soln y = [1/(D-2)(D-3)] e^(4x) = F(D)e^(4x) where a = 4 y = [1/(4-2)(4-3)] e^(4x) = (1/2)e^(4x) the particular integral complete solution need to add homogeneous [D^2 -5D +6D]y_h = 0 the traditional method with y_h = Ae^(2x)+Be^(3x) of course. Now that out the way we need D^(i)x = D^(i)e^(lnx) = D^(i) e^(u) using u = ln x, unfortunately we need to redefine D for new variable u which I believe is D_x = {(u-1)e^(u)}D_u (this part I used d/dx = (d/du)(du/dx) chain rule = (xlnx - x)d/du but not 100% certain here) so D_x^(i) = (d/dx)^(i) x = {(u-1)e^(u)}^(i)}D_u^(i) e^(u) = {(u-1)e^(u)}^(i)}^(i)1^(i) = i(x)^(i+1)ln(x/e) which if correct should be equivalent to Dr Peyam's derivation. But who am i definitely not Pimi thats for sure.

Could you do all of this using the difference formula? It just seems like you can calculate any derivative of X using the difference formula so 1, 2, and 3 order are simply just re-applying the difference formala multiple times to X^5. So I ask, could you apply the difference formula half a time or i times to something? It has to be a natural number or something.

Didn't he loose a factor of sinh(pi) from the gamma function along the way?

Awesome presentation!!......Have you also done for quaternion order derivative?

You really have the gangsta way of doing calculus

Can someone please share with us some computer-rendered graphs (based on good numeric approximations) of the functions discussed in this series of videos?

It seems to me that you could check this definition by checking to see if D^-i(D^i{x^n)) = D^1(x^n). Though with how complicated the answer to one of those is, I'm not sure how well you could get everything to cancel out.

Can you use this continuous derivatives to find general solutions of families of differential equations? (math noob)

these are going to be 20 really good minutes :)

Thank you for detailed explain. But, i'm confusing that Gamma function is defined on "Re( z)>0". Gamma[z] when z=i --> Re(i)=0. I've been confused about that. Could you explain why gamma function is defined on "Re( z)>0".

Is it possible to put the formula around 7:00 in it's generalized glory for complex a,b to D^b (x^a)=Gamma(b+1)/Gamma(b+-1-a)x^(b-a)?

Could we get the same result by using Fourier transform ? Given the fact that derivation is linear and that deriving sin(x) substracts pi/2 to the phase, I can guess that i-th derivative of sin(wt) is (w^i)*sin(wt-i*pi/2). And thus we should sum these sin functions to get de i-th derivative of any periodic function. Of course this doesn't work for x (aperiodic)

Love the idea of Dⁱ :) But I don't think the integral of Γ(i) converges. If I remember correctly, the integral representation of Γ(s) is only convergent for Re(s)>0.

Awesome nice way to explain this

Maybe using the series expansion of sine one could go on to define those

Best regards, I have a question, where can I find information or text to delve deeper into the fractional derivative of complex order, that is, when z has a real and imaginary part other than zero, it would also be good if you uploaded a video explaining this case. thank you

Just subscribed, you rock!

it's beautiful, love it :3 but i think you should put camera closer at the final answer, it's a little bit blurred

And here I was thinking that _real_-valued fractional derivatives were crazy.

After watching this I asked myself if there's exist g(x)-derivatives of f(x) ? Example what is d^(x)/dx of x ?

Uh, Dr. Peyam....I think you just broke calculus ;)

I have a question, your définition formula for the derivitive only works for α

Dr Peyam, what we can do with the fractional part of tanx or another fractional part? Its just and concept?Actually im studying Pure Mathematic but im starting, anyway, amazing video as always

Hi Drpeyam, may you please tell me what branch or research paper you got this from. If I can know more about this branch, I will be able to develop a formula that has the potential to solve the Riemann hypothesis

does the imaginary derivative mean the fractional integral ? since the integral is a derivative of the (-1) order or (inverse function)

amazing video! When I saw this, the I thought we can just take the square root of its integral as it’s the square root of its -1st derivative. How wrong I was....

I think the equation looks nicer using Gamma(i).

WOW WOW WOW this is so cool!! Never imagined that :) By the way: if you apply this i-derivative 2 times to x, since i*i = -1 , does this imply you get the -1-derivative of x, that is the integral of x?

Coool very nice

Can you put table of derivative for all basic function

What books do you use to prepare the information about fractional analysis?

Oooooohhh woww Dr. Tigre Peyam!!!

rip sinus hyperbolicus, it became as meaningless as 1 in multiplication

17:00 What's happend with this sinh???

wow... thats awesome :0 how would we integrate with respect to i now? :0 and how could we generalize that... Awesome concept and execution! Also... did you lose sinh(π) when simplifying or did I miss sth?

good lord! what is happening in there

What if f(x) = D^x(1), where D^x is the x-th derivative operator. So for example, f(0)=1, f(1/2)=2/√π, f(1)=0. Is there a nice way of representing f(x)?

sir,what is the derivative of x with respect to fractional part of x

Where does the "Fact" at 20:20 come from? I couldn't find anything like it. I tried to check numerically and it turned out to be false.

Does it also have a motivation?

Interesting. Don't you think that when you find the alpha derivative of x^5. There should be a condition that alpha must be less or equal to 5? Is that necessary?

your content is much more advanced and good as compared to *bprp* and fapable maths keep going :)

Cool video, got most of what you said, but what does sinh(x) mean?

To all those people asking where the sinh(π) went: Isn't it obvious that he was working in units where sinh(π)=1 in this certain part?

ما شاء الله دكتور پايام

After warching this video I'm completely convinced that you consumed some substances I named my channel after xD.

So does anyone know where to find at least a numerical approximation of “capital I”? Is it even finite?

Was hoping to find a way to find the integral by taking two imaginary derivatives... and of course it's complex. And would be the 2i'th not the i^2'th derivative.

i'd love to see a proof of the gamma(i) definition!

Do you recommend any books on Fractional Calculus? For the beginner

You discovered new Maths .. You are Euler in the present world

Fascinating

Very interesting.