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 😅

If they charge us a phi, we Peyam.

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!

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

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

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!

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

@ 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.

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

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

is there something that this property can be useful for

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?

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!

Great T Shirt Peyam - how do I get one?

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

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

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. :/

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

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.

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...

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

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

9:10 Dr. Peyam is sans undertale confirmed

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

First

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

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