# Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

888
questions

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### Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term?

Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate
\begin{equation}
\lvert f(x) \rvert \leq \lvert x \rvert.
\end{equation}
Also, let $d\mu$ ...

6
votes

0
answers

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### You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?
How to ...

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votes

2
answers

142
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### How fast does this summation grow?

$n,i\in\mathbb N$.
The summation in question is
$$\sum_{k=1}^n\prod_{l=1}^k\binom{2^n}{2^l}^i.$$
How fast does this grow? I am specifically looking at $i=1,2$.

0
votes

0
answers

58
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### Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed

The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...

1
vote

0
answers

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### Asymptotic location of zeros of of a sequence of analytic functions

Assume we have a sequence of functions $f_n$ analytic in a bounded domain $\Omega \subset \{ |z|\ge 1 \}$ of the complex plane, such that the sequence
$$
g_n(z) = f_n(z) - z^n
$$
converges to an ...

2
votes

0
answers

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### Ratio of the number of solutions to unit fraction equations with shifted prime and natural denominators

In a 2018 question posed by Zhi-Wei Sun, he conjectures that for any rational number $r>0$, there are finite sets $P_r^-$ and $P_r^+$ of primes such that
$$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in ...

1
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0
answers

129
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### Asymptotics for $\mathrm{Zi}(x)$ and comparison to $\mathrm{Li}(x)$

Consider a function that attempts to count primes
$$\mathrm{Zi}(x)=\frac{1}{e}\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\phi(k)}$$
where
$$ \phi(k)= \sum_{n=1}^\infty e^{-n^k} $$
Based on some preliminary ...

6
votes

1
answer

385
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### On an asymptotic integral decay

Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...

2
votes

1
answer

122
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### The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$

Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...

5
votes

0
answers

157
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### When does the Fourier transform of a measure decay?

Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...

2
votes

2
answers

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### Asymptotics of an integral requested

Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...

0
votes

1
answer

117
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### Asymptotic approximation of a convolution of binomial coefficients

I would like to find the following limit which is somewhat similar to the usual Vandermonde's convolution for binomial coefficients. Define $L$ as follows.
$$ L \triangleq \lim_{N\to\infty} \frac{1}{2^...

11
votes

2
answers

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### Mertens-like theorem

Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...

2
votes

1
answer

96
views

### Proof of Szegö asymptotic theorem

Consider the truncated exponential series
$$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$
The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...

2
votes

0
answers

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### Asymptotic expansion of Jacobi function

For $\alpha,\beta \in \mathbb{C},\, \alpha$ a non-negative integer, we define $$A_{\alpha,\beta}(t)=(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1} $$
and $$ \mathcal{L}_{\alpha,\beta}=\frac{d^2}{dt^2}+\...

4
votes

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### Asymptotic analysis for a double integral related to Airy functions

Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...

0
votes

1
answer

125
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### The asymptotic behaviour of a singular integral

Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...

1
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0
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### Asymptotic behavior of sum of regularly varying function

In the proof of Lemma 4.9 of Beran et al (2013), the authors consider a strictly stationary time series $X = \{X_t, t \in \mathbb{N} \}$ with regularly varying autocovariance function $\gamma_X(k) \...

1
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0
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### CLT of the left singular vectors of i.i.d. data matrix

Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...

5
votes

2
answers

257
views

### Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?

The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ...

0
votes

2
answers

159
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### Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions

Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by
$$
F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}.
$$
It is clear that $F$ is strictly ...

6
votes

1
answer

178
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### Density of extended Mersenne numbers?

Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in ...

4
votes

0
answers

102
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### Permutations avoiding a family of consecutive patterns

Let $B=\{1324,14325,154326,1654327,\ldots\}$ be the set of permutation patterns of the form $1(m-1)(m-2)\cdots 2m$ for $m\geq 4$. I'm interested in the set $\mathcal P$ of permutations that avoid all ...

1
vote

1
answer

76
views

### Asymptotic expansion on the following integral of exponential function

I wish to obtain the asymptotic for the following integral
$$
\int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr,
$$
where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...

1
vote

1
answer

53
views

### Asymptotic property of the left singular vectors of i.i.d. data matrix

Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...

1
vote

1
answer

101
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### Hypothesis to guarantee Lindeberg's condition

Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....

6
votes

3
answers

597
views

### How do I solve the following definite integral (preferably by an asymptotic method)?

$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...

7
votes

2
answers

244
views

### Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...

4
votes

1
answer

394
views

### Is there a "convolution" of asymptotic growth?

Suppose that I have two asymptotic counts given by
$$
\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)
$$
and also
$$
\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).
$$
From these two ...

3
votes

2
answers

140
views

### Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...

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votes

1
answer

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### applying the watson lemma to an integral [closed]

So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral
$$
I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt,
$$
as $x \rightarrow \infty$.
I think ...

4
votes

1
answer

159
views

### How to compute the asymptotics of this oscillatory integral?

I posted this on Stackexchange but got no responses or comments.
Consider the following integral, for $\epsilon\ne 0:$
$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...

1
vote

1
answer

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### Convergent condition of the high-dimensional submatrix of some orthogonal matrix

Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...

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votes

2
answers

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### Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?

The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...

0
votes

1
answer

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### Hamiltonian particle system and its frequency domain

I am interested in the following question.
So let suppose we have finite number of point particles on plane $\mathbb{R}^2$.
We can assume that every $j$ point is represented by Dirac delta function $\...

8
votes

1
answer

171
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### Asymptotic number of permutation representations of a given group

Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S_n$.
Equivalently this is ...

2
votes

1
answer

134
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### Asymptotic analysis of an expression involving a Fox's H function

One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...

1
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0
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### Necessary and sufficient conditions so that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...

2
votes

1
answer

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### When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...

12
votes

2
answers

456
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### Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$

I'm interested in the asymptotics of
$$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$
as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...

2
votes

2
answers

304
views

### Asymptotic behavior of a hypergeometric function

Can anybody see how to deduce an asymptotic formula for the hypergeometric function
$$ _3F_2\left(\frac{1}{2},x,x;x+\frac{1}{2},x+\frac{1}{2} \hskip2pt\bigg|\hskip2pt 1\right), \quad\mbox{ as } x\to\...

0
votes

0
answers

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### Asymptotic bound of some number theoretic function

I asked this in stack exchange but did not get anything so I am posting it here.
I am self-studying asymptotic behavior of some number theoretic function and the following question comes up.
Let $n$ ...

10
votes

2
answers

426
views

### How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$

I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...

1
vote

0
answers

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### Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?

I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...

3
votes

0
answers

140
views

### The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$

Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...

3
votes

0
answers

134
views

### Spectrum of large Hilbert matrices

Let $x_k>0$ be a increasing sequence of real numbers, such that
$$\sum_0^\infty\frac1{x_k}<+\infty.$$
Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with
$$a_{...

0
votes

0
answers

66
views

### Stability of a special singular perturbation problem

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a lower bounded smooth function, i.e., $\inf_{x\in\mathbb{R}^n} f(x)>-\infty$. Consider the following singular perturbation problem:
$$\begin{cases}\dot{...

5
votes

1
answer

98
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### Asymptotic expansion for the number of self-avoiding random walks

This question is cross-posted from https://math.stackexchange.com/questions/4580314/asymptotic-expansion-for-the-number-of-self-avoiding-random-walks.
Let $c_n$ be the number of self-avoiding random ...

3
votes

1
answer

118
views

### What is the optimal asymptotic behavior of this integral over the sphere?

Let $k_{1},\dots, k_{d}>1$ be integers and consider the integral
$$J_{\lambda }=\int_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k_{1}}_{1}+\dots+ x^{2k_{d}}_{d}\right)} d\sigma(x)$$
where $d\sigma$ ...

2
votes

1
answer

90
views

### Asymptotic analysis of a peculiar sum of squares sequence

Let $a,b$ be two positive integers. Let the sequence $\{s_n\}_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in ascending order
\begin{align*}
& n=1 & s_1=1^2+1^2=...