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That was one of the most entertaining things I've ever watched. Bravo, subscribed.

This video changed my (math) life. I can't think of anything else anymore.Thanks

Closed forms for all kinds of nasty discrete infinite sums could be derived by using integration and phi and phi^-1 transformations?!

It is so sad that the real good content creators don't get enough attention and need to stop. And we get stuck with so many overrated sh1tty fake content creators.

my god, ϕ(eᵃˣ)=(a+1)ˣ finally explains why Δ(2ˣ)=2ˣ

I am now upset that they didnt teach us operational calculus upfront when i was learning quantum mechanics. Wtf, this clicked immediately

“give bro the evil calculus”

Your presentation is outstanding. I wonder if you can direct me to the software you use that prints the equations on the screen in such an amazing way.

5:22 Where did the minus disappear?

Thank you so much!! 🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻

I don't understand what's happening at 13:30. Firstly, D^n f(0) is essentially a constant, and the operator phi cannot act on it. And if it can, then phi is not multiplicative, and therefore it cannot act on both x^n and D^n f(0). The final answer is correct, of course, but the approach is very strange

Using the first principles of differentiation you can right D in terms of T, h, and the "limit as h approaches 0" operator, D=L_(h -> 0)h^-1(T^h-1). Rearranging and replacing T with e^D, you can get a formula for this limit operator, L_(h -> 0) = hD(e^(hD)-1)^-1. Let h = 1 and replace e^D-1 with Delta to get L_(1 -> 0)=D(Delta)^-1 so the Bernoulli operator is the same as taking the limit as 1 approaches 0. The inverse of the forward difference is the sum so L_(1 -> 0)=D*Sigma is a cleaner form. This operator converts discrete problems into continuous. If you want to calculate the sum you can instead take the integral of the limit as 1->0 of the function. of if you want the forward difference you can instead take the limit as 1->0 of the derivative.

Really good video, although you made leaps way too big, I think you kinda skimmed over how linearity mattered, so I didn't trust you

The diagram at 7:45 looks suspiciously like the commuting diagram for a naturality square where phi is the natural transformation between two functors. This is why you can't 'divide by phi', phi is actually a family of arrows with components at each continuous function up top. I'd wager the two functors are the ones that map onto 'continuous' and 'discrete' functions, though I'm not sure from which category they'd map from; that much is far above my head.

newton series is such a nice analogue to taylor series and a special case of the interpolation polynomials when written in newton form. Tried to use this on a test and the teacher just said "there is no such thing as a 0 in the indexing set" and didn't even bother looking at the rest. like just change the index if you don't like it? it's much uglier with index 1 than 0 because it doesn't ressemble classical calculus anymore. it'd be quite unnatural though still technically correct if taylor series started at 1 instead of 0.

Just superb

this channel is underrated

I wish math videos would stop trying to be like 3b1b. 3b1b is popular because it's watered down 'pop-math' that still somehow does a better job than institutionalized education (at least in America). Then again, the diploma mills select for youtube channels that grift.

Wonderful!

🤯

Great explanation of this concept! Congrats and thank you

Does anyone have anymore reading or good books on this subject? I cant seem to find information about the phi function

(T-1) is the forward difference, -(1+T+T^2+T^3+...) is the taylor series of its inverse which just so happens to be the negative sum from 0 to infinity of f(x+k), since one is the inverse of the other this is what you're supposed to get: -(1+T+T^2+T^3+...) * (T-1) = 1 On one hand it makes sense for everything to cancel out because of the properties of infinity, on the other someone might read this as -T^inf+1 = 1 or T^inf=0 which seems like nonsense

Excellent Excellent Excellent👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏

hi

Umbral calculus is truly a shadow of school calculus. I played around with umbral calculus and discovered that the sequence 2^n is its own difference. Therefore, 2^n is a shadow of e^x. Really cool. EDIT: If you apply newton's forward difference formula to 2^n, you get something that is disturbingly similar to the maclaurin series for e^x

Oooh more stuff from umbral calculus

I just found that x sub n is the same as nPr(x,n), which is quite useful

I wonder if there’s any way to use normal or umbral calculus to find an exact functional way to do that, I believe you could very much just use factorials or something

Nice video but how is that phi operator defined formally, I mean does such an operator exist in terms of functions or integral transforms or even limits?

Umbral Calculus is just the best when you're deep in some special functions, like Bessel, Laguerre, and so on.

i used something similar to this to derive the binet formula when i was just trying new things without concern for rigor usually you derive the binet formula using a generating function, but I actually imagined the (naturally-indexed) fibonacci numbers as the components of a vector in an infinite-dimensional vector space (ie f = 1e1+1e2+2e3+3e4+5e5+8e6+...) and then, i kind-of defined into existence a linear transformation that brought every basis vector to the next-indexed one, ie. s(e_i)=e_(i+1), pretended i had an inverse for this even though obviously one doesn't exist for e1, and it led to a polynomial in s applying to the fibonacci vector equaling the RHS, so the next problem was to find the inverse of this polynomial in s i got stuck there, until i realized i could factor the polynomial in s into two monomials and then just apply the inverse to each monomial separately, eventually bringing me to the Binet Formula as well as some very cool identities involving power series of the golden ratio i was unaware of its a very fun thing to work through i highly encourage, because ive never seen anyone else fiddle with a "generating vector" but essentially my approach seems to just be 'operational calculus' but translated to the language of linear algebrs

There's a interesting relation between \Delta and D \Delta = 1 - e^D and S = 1/\Delta = 1/D (D/(1-e^D)) = 1/D + B_0 + B_1/1! D + B_2/2! D^2 + B_3/3! D^3 ... which is Euler-Maclaurin formula. The relation mentioned in this video is also interesting. thanks.

Last time I watched this I was very confused, is thinking of these things as linear operators on a vector space of functions valid?

Aren't these just linear operators on a vector space of functions? Oh wait I think the shift operator isn't linear 😢. And I thought I had figured out why this worked.

Really nice presentation .

Hrnnnnnggg isomorphisms

Looking forward to more videos like this one.

Cool

ooh this looks fun! no hesitation to wishlist on this one :3 btw, have you heard of Wang Tiles? you might find them interesting to look into, iirc the original aperiodic tiling

I was here for the calculus but I'll follow this project also

Really exciting! Already wishlisted ;)

what is this is this a game?

what is this is this a game?

You'll never know how i got so many likes.

finnaly

This is the missing motivation from the semesters of calculus i took.

I'm speechless :D

B l ack browser / st /u/ dy