All Questions
10 questions
2
votes
1
answer
215
views
An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
- 21
10
votes
7
answers
875
views
$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
Let $y>0$, $L>0$. Has the (special?) function given by
$$f(y,L) = \int_{L}^\infty e^{- t - y/t} \, dt$$
been studied? Are there precise, simple bounds?
Let me try to attempt to reinvent the ...
- 20.2k
3
votes
0
answers
186
views
Is factorial the restriction of some elementary function?
Hölder's theorem says that Gamma function is very non-elementary, but it does not exclude the possibility that factorial is the restriction of some elementary function to natural numbers. The answer ...
- 623
11
votes
2
answers
882
views
Do infinitely nested radicals have any applications?
There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
- 4,943
5
votes
1
answer
728
views
Linear independence of exponentials
Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...
- 3,873
6
votes
2
answers
1k
views
A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
- 128k
42
votes
7
answers
5k
views
How should an analytic number theorist look at Bessel functions?
(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
- 7,337
3
votes
0
answers
867
views
Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta
The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation.
In the discrete calculus there is ...
- 10.1k
7
votes
2
answers
464
views
Asymptotics of the $q$-harmonic series as $q\to1$
The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
h(q):=(1-q)\sum_{n=1}^\infty\...
- 13.5k
21
votes
1
answer
2k
views
Trigonometry related to Rogers–Ramanujan identities
For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric numbers:...
- 13.5k