nth derivative equals inverse

This video is a sequel to my video on the derivative equals to the inverse, which you can find here: • Derivative equals inverse
In this video, I find a function f with the property that the n-th derivative of f equals to the inverse of f. In other words, for which f do we have f^(n) = f^(-1)
The cool thing is that the result features the metallic numbers, which is a neat generalization of the golden ratio. There will be many videos on the metallic numbers in the weeks to come. Enjoy!

[@drpeyam]

Typo: Starting from 10:20 it should be Gamma(phi-n+1)/Gamma(phi+1), so the roles of Gamma are reversed. So a should be (Gamma(phi-n+1)/Gamma(phi+1))^1/(phi-n+1) I got confused because what’s on the whiteboard didn’t match what I wrote in my notes, haha 😅

[@GreenMeansGOF]

There is no -N in the x exponent, right?

[@carlosmonte9597]

You need a PhD if you want to work with any function not a particular function? XD

[@mehdisi9194]

I was also surprised that when I saw this, I guessed you did not realize this mistake but I was sure you would point it to the comments. Thank you so much for this video

[@koenth2359]

@@GreenMeansGOF yes, f(x)=((Γ(b-n+1)/ Γ(b+1))^(1/(b-n+1)) x^b where b= ½(n±√(n²+4) )

[@TheWhitequark]

At that point I exclaimed, "no this is not correct" and my wife was like as if he could listen to you :)

[@johnkampmeyer5296]

Great video as always, Dr. Peyam; I feel so honored that my solution was featured on your channel! ! You had mentioned towards the end of the video that if we consider the solution we get for b = -1/phi, we could end up with a complex number for the coefficient a. I used Mathematica to check this for the first 100 values of N, and it turns out that this is indeed the case, but only for the odd values of N! (that's excitement, not a factorial) I think this is a most interesting pattern indeed, and one that I intend to investigate to see if it holds in general. I'll let you know about the results of my research!

[@notspaso6644]

While watching this video my mom came up to me and said: "He looks so enthusiastic!" I was like, "Indeed, it's Dr. Peyam!"

[@drpeyam]

😄😄😄

[@blackpenredpen]

SpikePowerHD yup!!!!

[@rogerkearns8094]

If they charge us a phi, we Peyam.

[@drpeyam]

Hahaha

[@rbradhill]

me playing w math, going down rabbit holes for hours and finding something cool. P-dog drops some knowledge and lays out an elegant and succinct distillation of the whole thing. this happens all. the. time. beautiful work as ever- keep it coming!

[@ozzyfromspace]

When he got the golden ratio, I knew there was no going back. Awesome video, Dr. Peyam!

[@atrumluminarium]

I think for higher values of N, (b-N+1) tends to 1, and (b+1) tends to N so there could be some interesting approximations done to this for large N like a(N) tending to N! Unfortunately b-N tends to zero so it's kinda hard to approximate that down without losing too much information and turning the function into a constant but I think f(x) ~ N! x^(b-N) is still quite an improvement

[@Arycke]

I also forgot to say DR PEYAM IN THE HOUSE BOI. Always love your videos

[@panagiotisapostolidis6424]

is there something that this property can be useful for

[@Arycke]

@ 3:52 Dr. Peyam, I thought the Gamma Function was only defined for non-positive integers or for the analytic continuation for all complex numbers z sans integers less than or equal to zero? I may have misunderstood what you said sir. Excluding incomplete gamma function, digamma, Legendre, etc. as well, I just mean the regular old gamma function.

[@Arycke]

@@angelmendez-rivera351 yeah that's what I said. It wasn't explicit in the video and was just wondering if it was a slip up or just a gloss over. Thanks.

[@mrmeowtv6248]

Is there such a thing as a nth derivative of a function as n approaches infinity or is that just a silly thing to ask?

[@drpeyam]

You can always define it pointwise as f^infty (x) = lim n goes to infty of f^n (x) at every x. Doesn’t mean that the limit exists, though!

[@MrNerst]

If N>b, then the derivative is zero; you should include that in the analysis.... In addition, since N is a natural number, the formula you found in terms of the gamma function, should be proven using Mathematical Induction in order to be complete the demonstration.

[@PontusLarsson1]

b is some real number. Not just a positive integer.

[@MrNerst]

@@PontusLarsson1 the induction is on N, not on b

[@PontusLarsson1]

@@MrNerst I was referring to the statement "If N > b, then the derivative is zero". This is not necessarily true. A simple counterexample is f(x) = sqrt(x).

[@TheNachoesuncapo]

I wish they taught me this at uni....

[@rajendramisir3530]

Few thoughts: Wouldn't integrating the derivative and the inverse of the power function separately result in the original power function? And, finding the inverse of the inverse of the power function should result in the original power function. Stimulating video!

[@dhunt6618]

I can't gammaigin a more amazing mathematical phi(t)! :)

[@andygregory2390]

Great T Shirt Peyam - how do I get one?

[@drpeyam]

At the blackpenredpen shop :)

[@M0rph1sm55]

Are you familiar with the fractional laplacian? Maybe it would make a nice topic for a video. :)

[@alimghazzawi3700]

I liked the video and the mistake was miner so don’t worry most of us noticed and liked the idea in general

[@kokainum]

Gamma function is defined only for positive arguments, right? That's why it works for phi with a plus. For a minus you just can't simplify formula with gamma, but still you have 2 possibilities for phi. And also remember that functions f that we got are also defined for positive x. In fact it's easily to prove for example for N=1 that there is no continuous (and ofc differential) function from R to R that would satisfy f'(x)=f^-1(x). So I'd say problem is a lil bit badly stated. We didn't say at first anything about our domain. Also we don't know how many other functions satisfy it. So after few hours I gave up and watched the video only to find out we didn't really fully answer the problem. :/

[@kokainum]

@@angelmendez-rivera351 I know it's enough for real part to positive. This is he suggested it has to be greater than 1. Just listen to video.

[@elbilongon4826]

9:10 Dr. Peyam is sans undertale confirmed

[@gabef9538]

I wish that the gamma function was shifted so that gamma_new(x) = gamma(x-1). It would clean up a lot of equations.

[@drpeyam]

I agree

[@gabef9538]

@@drpeyam I think the shift is used to clean up the conditions in which the gamma function is infinite. Feals unnecisary.

[@WildAnimalChannel]

I got confused at first because I thought f^(n)(x) meant f(f(f(f...(x)...)))

[@drpeyam]

That’s an interesting question as well, but that would be equivalent to f^n+1 = x

[@radiotv624]

Is there a way to make this hold for its half derivative and also its nth derivative where n isn’t an integer?

[@drpeyam]

Yeah, exact same process! We never really used the fact that n is an integer here!

[@radiotv624]

Dr. Peyam's Show So cool!

[@ahmadkalaoun3473]

Very cool.. But u commuted a mistake at 10:20.... U didn't reverse the gamma functions... I don't if u see wut i mean but u have to chek it...

[@martincaputo2361]

Also noticed that

[@drpeyam]

Ahmad Kalaoun Indeed! I’ve been wondering why it’s not the same in my notes!

[@ahmadkalaoun3473]

@@drpeyam i noticed u was wondering lol..

[@Dionisi0]

THIS IS NOT Nth Derivate equals to the inverse, you were just finding solutions to that affirmation

[@drpeyam]

?

[@shandyverdyo7688]

[@martincaputo2361]

First

[@maxsungwd]

I used to bully kids like that back in my school days