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May
17
comment Concerning Hurwitz Zeta function, how to prove the following identity?
Done: math.stackexchange.com/questions/1287008/…
May
17
asked Can these integrals be represented in closed form?
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
accepted Concerning Hurwitz Zeta function, how to prove the following identity?
May
17
revised Concerning Hurwitz Zeta function, how to prove the following identity?
added 132 characters in body
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
17
asked Concerning Hurwitz Zeta function, how to prove the following identity?
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
7
revised Finding two sequences with a limsup value
not related to euler gamma
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
answered Comparing infinite numbers
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.
May
6
asked Compare density of rationals to the density of integers