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1d
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.
1d
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
answered When this matrix is diagonalizable?
1d
revised Finding Possible Meromorphic functions on $\mathbb{C}$
added 10 characters in body
2d
answered Finding Possible Meromorphic functions on $\mathbb{C}$
2d
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.
2d
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.
2d
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 $\mathbb R$.
2d
answered Proof for the conversion of nanometers to inverse centimeters
2d
comment Show that determinant is equal to determinant of each variable
Sorry, I changed notation in the middle. Edited.
2d
revised Show that determinant is equal to determinant of each variable
edited body
2d
answered Show that determinant is equal to determinant of each variable
2d
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...
2d
comment Orthonormal basis for Hilbert space
Hint: $$\left\|\sum_{j=n+1}^m b_j/j\right\|^2 = \sum_{j=n+1}^m 1/j^2 < 1/n$$
2d
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.
2d
answered Orthonormal basis for Hilbert space
2d
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 $w \in \mathbb A$.
2d
answered Example of a bounded space which is not totally bounded
2d
answered Minimum and maximum with lagrange multiplier
2d
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 $z_n \to 1/(\pi/2 + k\pi)$.