Robert Israel
Reputation
397/400 score
 1d comment Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge? Comparison test. Apr 28 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. Apr 28 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. Apr 28 answered When this matrix is diagonalizable? Apr 28 revised Finding Possible Meromorphic functions on $\mathbb{C}$ added 10 characters in body Apr 28 answered Finding Possible Meromorphic functions on $\mathbb{C}$ Apr 28 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. Apr 28 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. Apr 28 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 $\mathbb R$. Apr 28 answered Proof for the conversion of nanometers to inverse centimeters Apr 28 comment Show that determinant is equal to determinant of each variable Sorry, I changed notation in the middle. Edited. Apr 28 revised Show that determinant is equal to determinant of each variable edited body Apr 28 answered Show that determinant is equal to determinant of each variable Apr 28 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... Apr 27 comment Orthonormal basis for Hilbert space Hint: $$\left\|\sum_{j=n+1}^m b_j/j\right\|^2 = \sum_{j=n+1}^m 1/j^2 < 1/n$$ Apr 27 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. Apr 27 answered Orthonormal basis for Hilbert space Apr 27 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 $w \in \mathbb A$. Apr 27 answered Example of a bounded space which is not totally bounded Apr 27 answered Minimum and maximum with lagrange multiplier