Imaginary derivative of x
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- čas přidán 13. 09. 2024
- This is the video you've all been waiting for!!! In this video, which is a sequel to my half-derivative of x video, I evaluate the imaginary derivative of x, that is the alpha-th derivative of x, where alpha = i. Although there is no formal definition of the imaginary derivative, I can still calculate it by analogy to what I did with the half-derivative video. Enjoy!
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 czcams.com/video/k2T0YilPrWw/video.html
Dr. Peyam's Show
Didn't you use the i in front of sinh(pi) to go from (1+i) to (i-1)?
you scared me, so I came to the comment section to see if I was right or wrong
We all make mistakes! Thanks for noticing & correcting.... but..... there is perhaps a deeper issue that shows just before 7:39. ....Does simply stating something to be true make it so? (I won’t drag in politics here, but there IS a real-life example or two in the current news)...You claim that the formula derived for real number derivatives is valid for complex numbers. In what way has this been demonstrated, shown or proven?!? [and I’m curious to know if any demonstrated results have important applications and uses). I haven’t finished watching due to other priority tasks, but this is in my « play » list ( = WORK!).
P.S. I love how you always thank us for watching first!!! That’s really Really nice of you!!
I paused at 17:06 to look for why the sinh(pi) vanished. Thanks for saving my day.
We're ready for quaternions, jth and kth derivatives, and Frobenius theorem
Throw some sparse matrices in there with some affine transformations...
@@skilz8098 Don't for get to add "Artificial Intelligence" into the title for good measure.
@@naterojas9272 but those two things are in fact relevant to this topic
Very interesting and well explained, but after about 15 minutes I couldn’t read all the scribbles.
Why not go straight to Geometric Algebra? Then you get imaginary, quaternions, vectors, and more automatically!
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!
Who tf is Peyman??
Thumbnail : *D i x*
Me : Looks *interesting*
nice
nice
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.
For every ex you've had you have to ask yourself: "Why?", so you can have a y for every x
I enjoyed this
all I got is a point at zero
I thought you were talking about ex girl friends
Very impressive, but can you do the derivative'th derivative of x
Hahaha, good one 😂
@@drpeyam Interestingly, raising to the power of a differential operator is possible: if D is the differential operator, then you can "formally" find its exponential via
e^D = 1 + D + D^2/2! + D^3/3! + ...
where D^n represents n-fold differentiation, and this acts as you'd expect on a function by
(e^D) f = (1 + D + D^2/2! + D^3/3! + ...)f = f + Df + (D^2 f)/2! + (D^3 f)/3! + ...
So you _could_ find that the Dth derivative of x should have x^(1-D) as power, which equals x x^(-D) = x e^(-ln(x) D) and the latter can be found using the above series expansion (only will have powers (-1)^n ln(x)^n D^n instead of just D^n in the numerators). Taking the gamma of D, on the other hand ... that I have no idea. But the Dth derivative will be an operator - a very weird one.
ADD: Actually, e^D has a nice interpretation as the unit translation operator - I just remember: [(e^D) f](x) = f(x + 1) for a suitable f. This has deep significance in quantum mechanics (in theoretical physics), too.
@@mike4ty4 what the FUCK
Now do a matrix-th derivative of x
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)
Plugging (i-1)! into Wolfram Alpha, we get that Gamma(i) is approximately -0.155 + 0.498i. So unfortunately, either something has gone wrong in your calculation, or we're dealing with a multivalued function for which your calculation gives a different branch.
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.
Peyam is a living legend.
Crazy Drummer he is our math lord..
alan tesla
and you are a dead legend
@@alex-cm9fd nice
This guy gets better and better.
Amazing ! This is new for me. Are these concepts of half-derivative and imaginary-derivative expandable to other functions that polynomial ones ?
Yep, see my playlist!
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
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.
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.
This is insane. I love it.
wow seems interesting , never seen this before !!!
6:53 "Proof by analogy"
Do fractional derivatives have any usefulness when analyzing physical systems?
I saw a video a week or two ago where it was used as an alternate way to solve the tautochrone problem.
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
You’re welcome! And probably just by taking limits, since every real number is a limit of rational numbers
Fascinating stuff! Also, love your style. Keep 'em coming!
To be convincing, this would need to work for functions that are not simple power laws.
I
WANT TO
BELIEVE
This is the exact beauty of math. No belief needed! Proof does it all. This is sweat of the intellectual brow - not divine revelation! go over the video carefully, write things down, puzzlements included, and don't take 'huh?'for an answer! Good luck!
david wright it was a reference to... nevermind
This is more valuable than a kg of GOLD to me.
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!
I was gonna do one on Hilbert’s Hotel, but there’s actually an excellent one around already, and I highly recommend you to watch it! czcams.com/video/Uj3_KqkI9Zo/video.html
Hey Dr. Peyam
Two questions
Is there any Interpretation of imaginary differentiation?
Would you like to do a video about fractional differential equations?
My braines sanity: "Am I joke to you?"
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.
Didn't he loose a factor of sinh(pi) from the gamma function along the way?
Yes, he forgot to write sinh(π).
I did, my bad!
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.
This is the most fun math series ever----thanks so much!
it's beautiful, love it :3 but i think you should put camera closer at the final answer, it's a little bit blurred
Sometimes i just open your videos to listen the happiest "all right thanks for watching" ! Its so cool!!
Awwwww ❤️
@@drpeyam I cant believe you just answered!! Best wishes from Brazil!! :))
17:00 What's happend with this sinh???
Ryba Plcaki My bad, it’s a typo
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?
Not quite I think you get the 2i derivative of x
@@drpeyam ahahahha yeah you are correct!!! Thanks for replaying i was a bit confused ;)
sir,what is the derivative of x with respect to fractional part of x
Wow, beautiful question! But it’s still the same answer but with alpha = {x}. I doubt that the gamma part can be simplified, but the x part becomes x^(1-{x})
I think the question wasn't (d/dx)^{x}*x but more of dx/d{x} which, I presume would be 1.
Except at integers where it's not differentiable (I think).
It is because you can make a continuation:
it's dx coming from left and from right, for every integer.
Angel Mendez-Rivera he meant (d/d{x}) x
"aye pi aye"... aye aye aye... :) Weird shit, but mind blowing. Never thought about a derivate this way. I always learn something new :)
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.
I think the equation looks nicer using Gamma(i).
Agreed :)
After watching this I asked myself if there's exist g(x)-derivatives of f(x) ? Example what is d^(x)/dx 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.
There’s a video about that coming on Monday. And it’s possible that the minus sign is a plus sign, that’s why numerically it might be false
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
Uh, Dr. Peyam....I think you just broke calculus ;)
imaginary derivatives: the kind of derivatives year eleven students come up with on the exam after half a year of not doing their exercises?^^
Nice video, thanks Payam jan! Keep the great work up!
I reinvented fractional derivatives myself in college, and I was always curious about D^i. Thank you!!!
Happy to watch informative video from a cheerful maths teacher :)
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.
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
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)
Thanks Dr. Peyam, very interested.What is the interest to compute the imaginary derivative in our real Life ?
I cannot read what is on the black /whiteboard ; is it possible to put the camera near the board?
Nice, but i have a question:
What is: D^i ! (x) = ?
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....
your content is much more advanced and good as compared to *bprp* and fapable maths keep going :)
Thanks so much!!! They do have some pretty advanced stuff too, though! :)
these are going to be 20 really good minutes :)
Non integer derivatives for non-polynomial functions?
Fractional derivatives of exponential and trigonometric functions: czcams.com/video/k2T0YilPrWw/video.html
You discovered new Maths ..
You are Euler in the present world
Awesome presentation!!......Have you also done for quaternion order derivative?
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?
Leonard Romano Hehehe, that’s a great way of putting it :)
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 someone please share with us some computer-rendered graphs (based on good numeric approximations) of the functions discussed in this series of videos?
No me canso de verlo, genial y gracias Dr. Tigre Peyam !
rip sinus hyperbolicus, it became as meaningless as 1 in multiplication
harisimer My bad!
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?
Is it possible to define a differential power derivative like D to the power of epsilon?
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
It’s okay to say that the fractional part of X its X - the greatest integer of X?
Correct, the frac part of x is x minus the integer part (floor) of x. So it’s basically as important as the floor of x, except what’s nice is that it’s always between 0 and 1.
if complex numbers aren’t well ordered, how does it make sense to have a z factorial
But the gamma function is convegent when Re(alpha)is positive .
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 have a question, your définition formula for the derivitive only works for α
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.
Very enjoyable tutorial! Thank you for the video
Awesome, I love that passion! :D
thank u, the illustration is realy down to every detail
what is the justin beiber th derivative of x
Does it also have a motivation?
i'd love to see a proof of the gamma(i) definition!
does the imaginary derivative mean the fractional integral ? since the integral is a derivative of the (-1) order or (inverse function)
I have to ask: Did you make this up or is this something that has been done before?
I made it up :)
@@drpeyam I think Dr Peyam you must release these results to AMS ..
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)?
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".
Wouldn't it be cleaner to use the Pi function? It's exactly like the Gamma function, except that Pi(n)=n! when n is a natural number.
Gamma is more common
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.
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?
After warching this video I'm completely convinced that you consumed some substances I named my channel after xD.
Beautiful, thank you for your work!
One small nitpick about Gamma: I've read, in a great book by the great author and expositor H. M. Eswards "Riemann's Zeta Function", that the whole Gamma(x) = (x-1)! (i.e. that *shift by 1* ) was a blunder by the great Legendre. Gauss used Pi(x) which is *not* shifted by 1. So for obscure reasons we keep needlessly adding "+1" or "-1" to our formulas :-) Same with Pi that sould have been 2Pi btw.
Cool video, got most of what you said, but what does sinh(x) mean?
Linus Schwan hyperbolic sine
You can find more about it here :D : en.m.wikipedia.org/wiki/Hyperbolic_function
great !
Do we have D[1](D[i] (D[i](x)) ) = x?
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)?
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?
Integrating with respect to i is differentiation with respect to -i, so just use the formulas with -i :) And yep, I forgot about that factor
Dr. Peyam's Show oh wow! thank you!!!
I don't understand, at about the minute 17 of the video, how do you simplify (1 + i) i sinh(pi) to achieve i - 1. According to my calculator, sinh(pi) is approximately 11,548 739 357 3...
(1+i)i = i + i^2 = i-1
I think this is what they call Umbral techniques. shady dark techniques that can still give some insight.
This is what i tinkered with years ago... arriving at exactly this.
I saw the complex order as just being "rotations".
So, i apply it to X^p in general.... something strange happens when p=integer..... VS non-integer powers.
what i call a "Derivative spectrum", just collapses to Zero.
But non-integer: the Derivatives go on & on to lower and lower degrees.
-- Integrals become VERY ugly. I had an odd hunch...... that the subtle reason why integrals are such beasts... has to do with the poles Gamma has on the negative side, while it's a nice curve on the right.
it's beyond my why. But i had another hunch that i could test whether any function's integral is non-elementary.....
i applied fractional derivatives to X^X. couldn't get far with that ;0
You really have the gangsta way of doing calculus
Very nice, but I wonder where you would find a complex derivative? I always think of j as being a rotation operator (sorry i'm an engineer). So "rate of rotation" in a complex (Hilbert) space ?? Doesn't this come up in quantum mechanics ?? Forgive my guess work I'm only a humble electrical engineer, but I sometimes worship at the church of mathematics.
Super interesting questions!!! I’m not really sure, but there should be applications somewhere
@@drpeyam In vector calculus we have the gradient, divergence and curl operators, in a normal 3D vector space. These operators use integer derivatives that we are used to. But what about in higher dimensional complex spaces. What are the equivalent operators? Is the j derivative a curl in a complex space ?
And here I was thinking that _real_-valued fractional derivatives were crazy.
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 mean that the square of the imaginary derivative = the -1th derivative, aka the integral?
Yep :)
So, derivative degrades the function by a power of 1(x^n becomes x^n-1], half derivative degrades it by a power of half (x becomes square root of x), and now imaginary derivative comes up with ln(x) (which is far more degradation than square root of x)... no idea what to do with this one, I’ll just step back and admire it from afar.