Imaginary derivative of x
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- Опубликовано: 8 мар 2018
- 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!
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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 ruclips.net/video/k2T0YilPrWw/видео.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.
Thumbnail : *D i x*
Me : Looks *interesting*
nice
nice
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??
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
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
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.
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.
Fascinating stuff! Also, love your style. Keep 'em coming!
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.
This is insane. I love it.
This is the most fun math series ever----thanks so much!
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!
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.
wow seems interesting , never seen this before !!!
No me canso de verlo, genial y gracias Dr. Tigre Peyam !
Nice video, thanks Payam jan! Keep the great work up!
6:53 "Proof by analogy"
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! ruclips.net/video/Uj3_KqkI9Zo/видео.html
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 more valuable than a kg of GOLD to me.
Happy to watch informative video from a cheerful maths teacher :)
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
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!!
Awwwww ❤️
@@drpeyam I cant believe you just answered!! Best wishes from Brazil!! :))
Hey Dr. Peyam
Two questions
Is there any Interpretation of imaginary differentiation?
Would you like to do a video about fractional differential equations?
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.
Amazing!
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.
i love this!! you are so creative!
Awesome, I love that passion! :D
My braines sanity: "Am I joke to you?"
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.
thank u, the illustration is realy down to every detail
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.
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.
Is it possible to define a differential power derivative like D to the power of epsilon?
it's beautiful, love it :3 but i think you should put camera closer at the final answer, it's a little bit blurred
Thanks Dr. Peyam, very interested.What is the interest to compute the imaginary derivative in our real Life ?
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.
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.
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)
To be convincing, this would need to work for functions that are not simple power laws.
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?
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?
Awesome presentation!!......Have you also done for quaternion order derivative?
Just subscribed, you rock!
Thank you!!! :D
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 ;)
"aye pi aye"... aye aye aye... :) Weird shit, but mind blowing. Never thought about a derivate this way. I always learn something new :)
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
I have a question, your définition formula for the derivitive only works for α
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
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
Awesome nice way to explain this
Good!!
Fascinating
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)?
What books do you use to prepare the information about fractional analysis?
Brezis functional analysis
Oooooohhh woww Dr. Tigre Peyam!!!
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".
these are going to be 20 really good minutes :)
Coool very nice
Maybe using the series expansion of sine one could go on to define those
does the imaginary derivative mean the fractional integral ? since the integral is a derivative of the (-1) order or (inverse function)
Can you use this continuous derivatives to find general solutions of families of differential equations? (math noob)
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.
You really have the gangsta way of doing calculus
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!
ما شاء الله دكتور پايام
After watching this I asked myself if there's exist g(x)-derivatives of f(x) ? Example what is d^(x)/dx of x ?
Does it also have a motivation?
Do you recommend any books on Fractional Calculus?
For the beginner
My videos
Awesome
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)?
Very interesting.
what a god
Intresting
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!!!
it was stunning af
Is the fractional of the tangent function riemann integrable?
Probably not; the upper sums are 1 but the lower sums are 0, and to be integrable we need the two to be the same in the limit
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?
17:00 What's happend with this sinh???
Ryba Plcaki My bad, it’s a typo
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 ..
Wow so cool
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
Wow! Just... wow!
Can you put table of derivative for all basic function
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
I think the equation looks nicer using Gamma(i).
Agreed :)
I cannot read what is on the black /whiteboard ; is it possible to put the camera near the board?
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?^^
if complex numbers aren’t well ordered, how does it make sense to have a z factorial
i'd love to see a proof of the gamma(i) definition!
Is it possible to solve something like D^αX²(f)+D^βX(f)+D^γ(f)=g, when f is a function you can derivate at least αX² times?
Actually yes, using the Fourier transform :) What is X(f), though?