uhh is $\int_{-\infty}^{\infty}\exp(-x^2)dx = \sqrt{\pi}$?

that's such a weird identity

As 3B1B says, if you see pi then there's always a circle lurking somewhere

In this case, the standard proof of that identity shows you where the circle lurks

I'm just learning about "arithmetic sequences of order $k$" and am hoping for some help on internalizing the definition. The $\Delta$ operator is defined as an endomorphism on $E^{\Bbb N}$ via $(\Delta f)_n = f_{n+1} - f_n$ for $f \in E^{\Bbb N}$ ($E$ is a vector space). Now I know that $(\Delta^kf)_n = \sum_{j=0}(-1)^{k-j}{k \choose j}f_{n+j}$ which appears relatively ugly, so I have very little inution for what $(\Delta^kf)_n$ constant would mean

Please feel free to ignore everything past the first sentence there, I am just sort of trying to parse that first sentence. What sort of sequence is an arithmetic sequence of order $k$?

@Obliv yes

@EE18 polynomial

$\Delta$ is like a discrete derivative

$\Delta^k f = 0$ is similar as to say that $k$th derivative of a function $f:\mathbb{R}\to\mathbb{R}$ vanishes

Got it. So there's nothing about repetition of values or something like that to be said?

e.g. after $k$ elements the cycle repeats

cycle?

(1,2,3,1,2,3,1,2,3,...)

Something like that

what about it

OK I see the definition has nothing to do with that, that was my first instinct

Thanks Jakobian

@EE18 In general those should be sequences of the form $f_n = \sum_{k=0}^N n^k v_k$ where $v_k\in E$

I think that's what the authors are driving at with the paragraph I gave but I'm still struggling to understand it

Will post back here in a little about it if that's ok

My basic set theory skills are a bit slow. Is there a typo in the yellow highlighted bit. Should $B$ (the second occurence) not be $Y$? This is from Folland's, but I haven't been able to find anything in his errata about this. The formula I have found about this is $(A \times B)^c=(X \times B^c) \cup (A^c \times Y)$.

Alternatively, one could write $$(A\times B)^c=(A^c\times B)\cup (A\times B^c)\cup (A^c\times B^c).$$

yes, it's a typo

ok, thanks, I know it's a small typo, but maybe an email wouldn't hurt, since it's in neither of his erratas, so he knows it's there

for the third edition, if it ever comes :D

OK I am still struggling with the second sentence in the picture I gave above. I'll label my questions with (x). What I have so far: given any $p \in K_k[X]$ and choice of $x_0,h$ I know (Remark 8.19(c)) that $p = N_k[p;x_0;h]$, where the RHS is the newton interpolation polynomial for that function $p$ ((1): how do $x_0,h$ factor into the discussion here? Shouldn't any choice of $x_0,h$ give the same $N_k$ because this is a polynomial, so that if $N_k$ agrees...

with $p$ at $k+1$ spots then they're equal?)

Agh

Trying to parse it still, don't even know how to ask the question about how the formula (12.15) gets used. Will keep at it

This is it FWIW. Will keep thinking on it