# Tagged Questions

**4**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**3**answers

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

**3**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

298 views

### Quotient of two Laplace integrals (2)

In one attempt to prove a probability theorem (of K.L. Chung and P. ErdÅ‘s, 1951) using analytic argument, I try to prove the following
Let $\varphi(x)$ and $\psi(x)$ be two
complex-valued continuous ...

**6**

votes

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**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
$$ ...

**5**

votes

**2**answers

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

**1**answer

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

**3**answers

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 ...