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comment Help With Notation In Fermat's Last Theorem
If $S$ is a set, $S^4$ denotes the four-fold Cartesian product of $S$ with itself, $S\times S \times S \times S$, which is the set of (ordered) quadruples whose components are elements of $S$. Here, $S = \mathbb{Z}^+$.
1h
comment holomorphic function writen as a serie
You can get it into the sum as the case $n = 0$ of $$\Biggl(\prod_{k=1}^n\frac{\alpha+1-k}{k}\Biggr)\cdot z^n,$$ or, shorter, $\binom{\alpha}{n}z^n$.
1h
comment holomorphic function writen as a serie
Sure. Do you know that the sum function of a convergent power series is holomorphic on the disk of convergence?
1h
comment holomorphic function writen as a serie
The second term - the infinite sum - is only entire when $\alpha \in \mathbb{N}$. In general that series has finite radius of convergence (namely $1$).
2h
comment $\phi$, and the uses of an alternate formula
@Taylor $\frac{\sqrt{5}-1}{2} = \left(\frac{1+\sqrt{5}}{2}\right)^{-1}$. Apart from the unfortunate naming collision, this is exactly your relation.
2h
comment Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.
The norm on $\mathbb{Z}[\sqrt{2}]$ is not $a+b\sqrt{2}\mapsto a^2 + b^2$. It's $a+b\sqrt{2} \mapsto a^2 - 2b^2$.
2h
comment Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.
For $\mathbb{Z}[i]$, the norm works too. In cases where the norm doesn't work, I don't know of a general strategy, but I expect the algebraists have strategies that work in some families of cases. Just in case you misunderstood KCd's comment, that doesn't say that all rings $\mathbb{Z}[\sqrt{d}]$ are Euclidean (they aren't, e.g. $\mathbb{Z}[\sqrt{-5}]$ isn't a UFD), it asks whether you know the proof for some other example.
3h
comment Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book
Because for this function we have $\frac{1}{t^2} f\left(\frac{1}{t}\right) = t^3 f(t)$. Write down the left hand side, and try to transform it into the right hand side. Knowing where you want to end up, you should find the steps.
3h
comment Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book
For this specific $f$, the relation holds. It does not generally hold, but here it does.
3h
comment Need help finishing my proof about this complex recursive limit
You can do that (and then you avoid the "problematic" points of the Möbius transformation), but it's simpler if you include $\infty$ in the considerations.
3h
comment Need help finishing my proof about this complex recursive limit
@Anna When you look at the Riemann sphere, the Möbius transformations are continuous on the whole sphere, they are homeomorphisms of $\widehat{\mathbb{C}}$. To see the continuity at $\infty$ resp. at a pole, you look at $T(1/z)$ resp. $1/T(z)$ [and if $T(\infty) = \infty$, you look at $1/T(1/z)$]. If you try to restrict attention only to $\mathbb{C}$, you face the problem that $f$ has a pole at $0$, and zeros at $\pm ib$, and the set of points that are a pole of some iterate of $f$ is dense on the line $ib\cdot \mathbb{R}$, so you would have to leave that whole line out.
3h
comment Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.
The (absolute) norm is the first thing one usually tries. If it works, it's typically also the last. Have you looked at the norm here yet?
18h
comment Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$
@Did I wasn't chastising you, that would read differently. Just pointing out that the sequence of events may have been less unfavourable. Is it really so hard to believe that you pointing out a flaw in the question made the OP understand that flaw and enabled him/her to correct it?
18h
comment Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$
@Did It's plausible that futurebird edited his/her comment (at least, started editing it) before you posted yours. And I'm pretty confident that "I did not see your comment" means the reasonable "... before editing mine [that is, futurebird's] above it" here, not the ridiculous "... at all".
1d
revised computing an integration with a floor function
small LaTeX fix
1d
comment dual ball of L^1 w*-sequentially compact?
Could be that finiteness of the measure is not sufficient for $L^1$ to be separable. In that case, I have at the moment no idea for a direct proof of sequential compactness.
1d
comment $x$ is a left zero-divisor $\iff$ $x$ is a right zero-divisor.
You have received (good) answers to the question as it was originally asked. Do not edit the question to something different. If you missed an essential condition, ask a new question with that condition, don't edit the question so that the answers become invalid.
1d
revised $x$ is a left zero-divisor $\iff$ $x$ is a right zero-divisor.
rolled back to a previous revision
2d
comment Is my proof that $\frac{\pi}{4}=\sum\limits_{n\geq 0}(-1)^n \frac{1}{2n+1}$ correct?
@MarioCarneiro It has been reopened now.
2d
revised Values of the Herbrand quotient
More small fixes