Anixx
Reputation
2,274
Top tag
Next privilege 2,500 Rep.
Create tag synonyms
 May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? Can the F.3.6 integrals be represented in closed form? May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? I've got the following: $$\int_0^{\infty } \frac{1}{4} (\coth (\pi t)-1) \left(a^2+t^2\right)^{-\frac{s}{2}} \left(2 \tan ^{-1}\left(\frac{t}{a}\right) \cos \left(s \tan ^{-1}\left(\frac{t}{a}\right)\right)-\log \left(a^2+t^2\right) \sin \left(s \tan ^{-1}\left(\frac{t}{a}\right)\right)\right) \, dt$$ using Mathematica. May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? After I differentiate the expression under the integral I get something different. How do you get this? May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? Great! Is there a way to obtain a more general formula, that is with a variable instead of 0? May 17 comment Concerning Hurwitz Zeta function, how to prove the following identity? @tired I do not see how the formula is connected here. May 7 comment Comparing infinite numbers @Ben Crowell I always was wondering how one can reconcile diverging series with ordinals. I think I know a sketch. May 7 comment Comparing infinite numbers @Asaf Karagila see here: en.wikipedia.org/wiki/… Cardinality of $\omega$ is $\aleph_0$ se they are usually considered equal. May 6 comment Comparing infinite numbers @Asaf Karagila they do not. But the ordinal $\omega$ is usually identified with the cardinal $\aleph_0$ yet with different operations, see here: en.wikipedia.org/wiki/Ordinal_number . May 6 comment Comparing infinite numbers @Asaf Karagila exactly my point. The operations on surreals correspond not to the classical operations on ordinals but to the so-called "natural operations" (follow the second link). May 6 comment Compare density of rationals to the density of integers @Travis Can we compare the density of rationals to the density of a uniform dense countable set at all? For instance the density of rationals to the density of its subset - rationals with, say, even numerator or denomenator? May 6 comment Compare density of rationals to the density of integers OK, infinite but quantify it. Use ordinals, cardinals, complex numbers, whatever else. But in such a way that the desities came different. May 6 comment Compare density of rationals to the density of integers No, I want a measure where the desnsity of integers is not zero. And greater than the density of, say, evens. Apr 25 comment What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument? No Wikipedia entry? Apr 25 comment What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument? @mercio you are correct. Apr 25 comment What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument? @Lee Mosher yes, typo Apr 25 comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always? @Alamos the first symbol definitely represents $\lim_{n\to\infty}\sum_{k=-n}^0f(k)$ Apr 25 comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always? @Alamos divergent series depend on the order Apr 25 comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always? It this true for all summation techniques used for divergent series? Apr 25 comment Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always? I made a typo in the question, sorry.