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18h
comment Complex analysis with $f(z)=\cot z$. Holomorphicity and residues
In principle, if $\Gamma$ is a simple closed positively oriented contour enclosing a region $R$, you can write (number of zeros in $R$) - (number of poles in $R$) as the integral of $2\pi i f'(z)/f(z)$ around $\Gamma$, but actually computing the integral will be harder than simply counting the zeros and poles.
18h
comment Continuity, algebraic and rational numbers
Well, it does get complicated trying to write out the details. But I'm quite confident the result is true. In the amended construction, the value at $a_j$ may be assigned before stage $j$, in which case it won't be reassigned. These values cannot be all different. For a particular $w \in \mathbb Q[i]$ and $R > 0$, the roots of $F_j(z) - w$ with $|z| < R$ will be fixed at some stage, and they can't move after that.
1d
comment What exactly is an “analytic function”?
Actually, power series with zero radius of convergence can be useful: see asymptotic series. Just not as nice as those that have positive radius of convergence.
1d
comment Proof for the conversion of nanometers to inverse centimeters
But that's not a conversion. The wave has two different quantities associated with it: wave number and wavelength. One is a function of the other, but they are different things. A conversion is when you express the same quantity in different units.
1d
comment Why do we study real numbers?
You might be surprised by how few mathematicians study $\mathbb R$. There are lots of areas of mathematics, and many of them don't have much to do with $\mathbb R$.
1d
comment Show that determinant is equal to determinant of each variable
Sorry, I changed notation in the middle. Edited.
1d
comment Continuity, algebraic and rational numbers
I think you misunderstand what I had in mind, but it'll be better to edit the answer rather than trying to clarify with further comments. I don't have time right now, but maybe later tonight...
1d
comment Orthonormal basis for Hilbert space
Hint: $$\left\|\sum_{j=n+1}^m b_j/j\right\|^2 = \sum_{j=n+1}^m 1/j^2 < 1/n$$
1d
comment How do I know the probability for me to be ranked in the 2nd place or last 2nd place?
You don't know them, and can't know them unless you assume a particular probability model for the scores.
1d
comment Continuity, algebraic and rational numbers
If $f(w) = z \in \mathbb Q[i]$ with multiplicity $m$, take $r > 0$ such that $w$ is the only zero of $f - z$ within distance $r$ of $w$. By Rouché's theorem there exists $N$ such that for $n > N$, $F_n - z$ has exactly $m$ zeros (counted by multiplicity) within distance $r$ of $w$. But by construction, such zeros (if $n$ is large enough) are in $\mathbb A$ and will also be zeros of $F_k - z$ for all $k > n$, and therefore of $f - z$: thus they must be $w$, and $w \in \mathbb A$.
1d
comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$
If you write $1/z_n = \pi/2 + k \pi + s$, $\cos(1/z_n) = (-1)^{k+1} \sin(s)$, and this will be $1/t$ if $s = (-1)^{k+1} \arcsin(1/t)$. As $n \to \infty$ you have $t = (n+1/2) \pi \to \infty$ so $1/t \to 0$ and $s \to 0$, i.e. $1/z_n \to \pi/2 + k \pi$ and $z_n \to 1/(\pi/2 + k\pi)$.
1d
comment Continuity, algebraic and rational numbers
I think we can still have $f^{-1}(\mathbb Q[i]) = \mathbb A$ by modifying the construction, ensuring at stage $j$ that any $w$ with $|w| \le j$ for which $F_j(w)$ is of the first $j$ members of $\mathbb Q[i]$ (in a fixed enumeration) is in $\mathbb A$, and making $f_k(w) = 0$ for $k > j$.
1d
comment Continuity, algebraic and rational numbers
Bijective on $\mathbb R$, I hope you mean. There aren't too many entire functions bijective on $\mathbb C$.
1d
comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$
$\sin(t) = \pm 1$ if $t$ is an odd multiple of $\pi/2$, say $(n+1/2) \pi$. Then you want $t = 1/\cos(1/z_n)$, with $1/z_n$ near $\pi/2 + k \pi$...
1d
comment Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?
The ratio test works well here.
1d
comment Multiplying binomials to come up with $ y^8 - 256 $
Right to left may be easiest.
2d
comment Proving a Weierstrass function is not differentiable anywhere
See e.g. this paper by Jon Johnsen.
2d
comment Bound of norm of the operator $T(f)=fg$ on $L^p$ space
Given $\epsilon > 0$, take such a set $A$ and let $f$ be the indicator function of $A$. Then $\|Tf\|_p \ge (\|g\|_\infty - \epsilon) \|f\|_p$.
2d
comment Why four roots to this equation: $(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$
More precisely, you have to take $(-t)^{1/3} = -t^{1/3}$ when $t > 0$, which is not the principal branch of the cube root. For each of the four values of $x$, one of the three cube roots will be of this type. If you use the principal branch of the cube root, there are no solutions.
2d
comment Nonlinear equations systems
OK, so choose some bounded subset of this domain containing the solution.