All Questions
Tagged with integration asymptotics
56 questions
2
votes
1
answer
430
views
Asymptotic expansion of a Laplace-type integral with a "manifold of maxima"
Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought.
Consider the integral
$$
I(\alpha)=\int_0^1 dx_1 \int_0^1 dy_1\int_{x_1}^1dx_2\...
- 131
0
votes
0
answers
94
views
Do they have the same limit?
Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
\frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
- 87
4
votes
1
answer
414
views
Convergence of the Double Integral of a Polynomial Reciprocal
Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...
- 3,142
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