4
votes
1answer
112 views

Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in ...
-1
votes
1answer
87 views

Method of steepest descents for $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt\sim\frac{i}{z}$

Through some calculations I ended up with the integral $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt$. I would like to obtain the result that the behaviour of the integral is $\sim\frac{i}{z}$ as ...
3
votes
1answer
107 views

Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...
10
votes
3answers
274 views

The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
3
votes
1answer
173 views

Asymptotic behavior for the solution of a nonlinear ODE

In a nutshell, if $u$ is a solution to $$ \partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\ \lim_{r \to \infty} u(r) = 0, \quad \text{and} ...
5
votes
1answer
134 views

Asymptotic solution of the integral equation

What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$z(s)=1+\gamma\int\limits_{-\infty}^s ds_1\int\limits_{-\infty}^{s_1}ds_2 \cos{(s_1^2-s_2^2)}z(s_2)\;?$$ In fact I ...
0
votes
0answers
90 views

Bound on a sum of Laguerre polynomials

I am trying to find an asymptotic behavior, for large real $t$, of the following sum \begin{align} Q(t)=\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{1+n}L_n^{(1)}(t-n) \end{align} where $L_n^{(\alpha)}$ is ...
6
votes
1answer
228 views

Certain asymptotics involving double infinite sum

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow ...
11
votes
1answer
270 views

Is there something wrong with Hörmander's theorem on stationary phase method

It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - ...
1
vote
1answer
90 views

Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series $$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$ where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, ...
1
vote
0answers
56 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question. Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose, ...
4
votes
2answers
214 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose $$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$ Is $a(x,0)=0$, $\forall x\in[0,1]$? ...
1
vote
3answers
245 views

Asymptotic estimates for the Exponential [closed]

Consider the function $z(n) = (1-f(n))^{g(n)}$. For $f(n) = \frac 1n, g(n) = n$ we have that $\lim z(n) = e^{-1}$; more generally, when $f(n) = \frac cn$ for any constant $c$, we have $\lim z(n) = ...
3
votes
3answers
280 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? ...
7
votes
2answers
463 views

Asymptotic question about time ordered exponentials

Let $A(t)$ be a smooth function from $[-1,1]$ to the $n \times n$ complex matrices. Define the time ordered exponential $$\prod_{-1}^1 \exp(A(t) dt)$$ as in this question, as the limit of Riemann ...
7
votes
2answers
341 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 $$ ...
0
votes
2answers
732 views

Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too! The original motivation for this is the ...
3
votes
0answers
489 views

Asymptotic form of $L^1$-norm of Hermite functions

Background Working on a quantum mechanics problem, I've stumbled on the problem of maximising the functional $$\int_A \varphi_m \varphi_n$$ in the limit of large $m$ and $n$, given that $n \gg m$. ...
6
votes
0answers
396 views

Phase perturbations in oscillatory integrals

I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...
3
votes
0answers
298 views

Quotient of two Laplace integrals (2)

In one attempt to prove a probability theorem (of K.L. Chung and P. Erdős, 1951) using analytic argument, I try to prove the following Let $\varphi(x)$ and $\psi(x)$ be two complex-valued continuous ...
6
votes
1answer
710 views

Quotient of two Laplace integrals

Let $\varphi(x)$ and $\psi(x)$ be two complex-valued continuous functions on $[a,b]$, and let $f(x)$ be a complex-valued continuously differentiable function on $[a,b]$. Suppose that $|f(x)|$ has an ...
8
votes
3answers
767 views

Big O notation and the maximal set of comparable functions

One can easily find a set of functions that are comparable with respect to the big O notation that is, $$f \leq g \Leftrightarrow \exists c \exists x_0 \forall x\geq x_0: |f(x)| \leq c|g(x)|,$$ for ...
1
vote
2answers
485 views

Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$

Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the ...
1
vote
1answer
548 views

Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral $\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ ...
2
votes
1answer
573 views

Partial sums of the Chu--Vandermonde identity

I am interested in finding a lower bound of the sum: $$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right) \left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$ when $d < k$ (and assuming both $n\geq k$, ...
3
votes
0answers
340 views

Asymptotics related to the Erdos--Moser diophantine equation

I share the authorship of this question with Pieter Moree. In our recent joint work with Y. Gallot (arXiv:0907.1356 [math.NT]) we attack the Erdős--Moser diophantine equation $$ ...
3
votes
1answer
1k 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 ...
7
votes
2answers
2k views

Estimate for tail of power series of exponential function?

I would like to have an estimate for the infinite series $$ \sum_{k=B}^\infty \frac{A^k}{k!}, $$ where $A$ is a large positive quantity and $B$ is just a little bit bigger than $A$, namely, $B = A + C ...
8
votes
4answers
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 $$ ...
5
votes
2answers
351 views

Asymptotics of iterated polynomials

Let the sequence $u_1, u_2, \ldots$ satisfy $u_{n+1} = u_n - u_n^2 + O(u_n^3)$. Then it can be shown that if $u_n \to 0$ as $n \to \infty$, then $u_n = n^{-1} + O(n^{-2} \log n)$. (See N. G. de ...
2
votes
1answer
319 views

How to do asymptotics for integrals?

What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example. Look at 1/z. If I want to find out how fast ∫0a ...
4
votes
3answers
976 views

Asymptotics of a hypergeometric series/Taylor series coefficient.

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes: I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh ...