All Questions
12 questions
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
0
votes
0
answers
182
views
Why is the sign of the integration negative?
Let
\begin{aligned}
I=\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
\end{aligned}
...
- 9
2
votes
1
answer
102
views
Approximation of $\Phi (p)$
I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
- 23
2
votes
2
answers
321
views
Asymptotic of an improper integral
I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is:
Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...
- 43
2
votes
2
answers
399
views
Asymptotic decay rate of an oscillatory integral
Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...
- 2,712
4
votes
1
answer
351
views
Asymptotic behaviour of function using Fox $H$-function representation
In equation (9) of this paper, it is claimed that the limiting behaviour
$$
\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk
\sim
\frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
- 85
5
votes
1
answer
294
views
Asymptotic of integral $\int_{1}^{e^n}(1-\frac{\ln(x)}{n})^n\,dx$
How could we find the large-$n$ asymptotic of $$\int_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx.$$
I have a suspicion that this is $\sqrt{n}$.
6
votes
1
answer
560
views
Asymptotic Expansion of Bessel Function Integral
I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...
- 275
10
votes
1
answer
328
views
Asymptotic behavior of an integral depending on an integer
A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \...
- 1,991
3
votes
3
answers
480
views
Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?
What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...
- 1,256
3
votes
1
answer
2k
views
Approximating a multiple sum with an integral
Hi,
I want to approximate a multiple sum of the form
$$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$
where each $x_i$ is an integer between $0$ and $n$,
by an integral
$$\int_{x_1+x_2+\...
- 491
8
votes
4
answers
2k
views
An integral that somehow equals pi^2/6 and involves dilogarithms?
I am attempting to show that
$$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$
as $x$ approaches 1 from below. The sum can be approximated by the integral
$$ \...
- 14k