Robert Israel
Reputation
397/400 score
 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 $z_n \to 1/(\pi/2 + k\pi)$.