Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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41 views

Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove ...
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21 views

Convergence in probability or convergence in distribution?

Let $X$ be a random variable. let \begin{align*} Y=\alpha_1+\alpha_2 X \end{align*} where $\alpha_1$ and $\alpha_2$ are parameters. Now let \begin{align*} Z=\hat{\alpha}_1+\hat{\alpha}_2 X ...
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16 views

Big $O$ notation clarification

I've encountered something of the form: $$f(n)O(g(n))$$ I think this is equivalent to $O(g(n)f(n))$, but is this true?
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31 views

Asymptotic behavior of the function $e^{- \lambda t^2}$ when $\lambda$ is small

I wish to prove that when $\lambda$ is taken to be very small $$ \left| e^{- \lambda t^2} - \sum_{n=0}^N \frac{(- \lambda t^2)^n}{n!} \right| = O(e^{-\frac{a}{\lambda}})$$ for some constant $a \in ...
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21 views

Singularity of an oscillatory integral

Given $x\in\mathbb{R}^3\setminus \{0\}$, consider the following integral: $$I(x):=\int_{\mathbb{R}^3}\frac{e^{-i|x-y|^2}}{|y|} \, dy$$ Now $I(x)$ diverges as $x$ approaches to $0$, and it seems to me ...
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36 views

Fastest decaying Fourier transform of a function with compact support

Square-integrable complex-valued function $y(t)$ is defined on a finite domain $t \in [0, 1]$ and satisfies the following constraint: $$ \int_0^{1} |y(t)|^2 \, dt = 1 $$ I am seeking bounds on the ...
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49 views

Partitioning functions into equivalence classes based on running time?

I'm studying for my midterm and doing some practice problems, and I would be grateful if someone showed how to solve this. From my understanding you have to partition the functions into equivalence ...
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50 views

Asymptote produced from a composition of two polynomials and an arbitrary function.

I am interested in any references and/or information regarding this sort of asymptotic relationship. Given the composition: $$\frac{f(x)h(x)+g(x)}{f(x)}$$ With the following polynomial functions $\ ...
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46 views

How to prove the following conjecture.

Prove if $d(n)$ is $O(f(n))$ and $e(n)$ is $O(g(n))$, then $d(n) + e(n)$ is $O((f(n) + g(n)))$. What I tried. if $d(n)$ is $O(f(n))$ then $d(n) \leq c_1 \cdot f(n)$ if $e(n)$ is $O(g(n))$ then ...
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55 views

Asymptotic Approximation

After analyzing performance of a cooperative system, I get the following expression for the system outage probability: $P = 1 - \frac{{{e^{ - 2\mu /{\beta _M}}}}}{{\Gamma \left( {{\alpha _3}} ...
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146 views

On n! divided by a product of primes and related questions

We have the following Definition 1. For integers $n\geq 1$ we define $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\[2ex] \frac{n!}{\prod_{p\leq n}p}, & \text{if $n>1$} ...
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49 views

Find the angle between asymptotes

Sketch the locus of the centres of circles which touch two fixed and unequal circles. Find the angle between the asymptotes How shall I find the locus when the size of the circles are not ...
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77 views

Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...
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57 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 ...
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98 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
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35 views

asymptotics of Involutions recurrence relation

Consider the following recurrence relation where $t(n)$ is the number of involutions on $\{1,...,n\}$ \begin{equation} (n+1)t(n)+t(n+1)-t(n+2)=0 \end{equation} When $n \rightarrow \infty$, Wimp and ...
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53 views

Asymptotic conditional distribution of $\bar{Y}\mid\bar{X}=x$

I'm reviewing for my qualifying exam and I'm stuck on part of a problem. Setup Suppose that $(X,Y)$ are two random variables with joint distribution $ \begin{equation} ...
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37 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
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48 views

WKB leading order

I'm learning about the WKB method, and I'm applying it to an assignment. The assignment question asks to find the "leading order" WKB expansion for the particular equation. For WKB you make the ...
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32 views

Approximation of Hermite functions

I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials). The asymptotic expansion is $$ f_n(x) \sim_{n ...
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69 views

asymptotic expansion of hermite functions

Does anybody know how to proof the first asymptotic expansion of this page: http://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion ? (and how the physicist use this asymptotic ...
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23 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
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36 views

Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...
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37 views

The series may converge, but what about the series / n?

Let $a_i$ be a positive sequence such that $a_i \to 0$. I know that the series $\sum_{i=1}^\infty a_i$ may be divergent. But what about the series divided by $n$; does the following go to 0? ...
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51 views

Question related to the ballistic motion

A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the ...
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78 views

Asymptotic expansion of root of $\epsilon x \tan(x)=1$

Indicate a range of roots of $\epsilon x \tan(x)=1$ for which it is impossible to get an approximation using expansions. Since $\epsilon$ is small, I think for the equation to hold, we need ...
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52 views

Asymptotic analysis of coefficients of ordinary generating functions with radius of convergence $1$ seems to always predict polynomial growth rate

Wikipedia gives the following formula for obtaining asymptotic information about the coefficients of an ordinary generating function from information about the generating function itself: if the ...
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50 views

Multivariate Delta Method

If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose ...
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28 views

Biggest rate of growth of a sequence in $ℓ^2$

$ℓ^2$ is the space of complex sequences $u_n$ such that $\sum |u_n|^2$ converges. I'm wondering if there are asymptotic results known about such sequences. We have trivially $u_n=o(1)$. Are better ...
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28 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
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32 views

Leading behaviour of DE at infinity

This is taken from the book of Bender and Orszag, problem 3.44. Find the leading behavior as $x\rightarrow+\infty$ of the differential equation: $x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$ Explain ...
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85 views

Asymptotic expansion of integral of $e^{-t}/t^n$.

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}.$$ I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now ...
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67 views

Picking codewords that are close

Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and number of minimum weight codewords $N_d$. How many ways can you select codewords $c_1,\dots,c_T$ (assume $T\ll q^k$) such ...
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41 views

Asymptotic behavior of $1/(a^2+\epsilon^2)$ as $\epsilon\to0$

A limit that often arises in physics is $$ \lim_{\epsilon \to 0} \frac{ \epsilon }{ a^2 + \epsilon^2 } = \pi \delta(a) ............ (1) $$ Is there a similar sort of limit for $$ \lim_{\epsilon \to 0} ...
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30 views

Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
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91 views

$f(x)\sim 1/x \implies (1+f(x))^x\to e$, but what family of functions maximizes the speed of convergence from below?

This problem is subordinate to finding out if $$\left(1+\frac{\log p_{n+1}}{p_n}\right)^{p_{n+1}/\log p_n},$$where $p_n$ is the $n$-th prime, never stabilizes above or below its limiting value, which ...
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87 views

Method of stationary phase for double integrals

I am looking for a reference for the leading term in the asymptotics of a double integral over a finite rectangle R of $K(x,y)\exp(i \,t\, h(x,y))$ as $t \to \infty$ in the following situation: the ...
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32 views

Asymptotic behaviour of $\sum_{k=0}^\infty \frac{n^k}{(k!)^\nu}$

Let $\nu>0$ be fixed. I am interested in the asymptotic behaviour of the series \begin{equation*}s(n,\nu)=\sum_{k=0}^\infty \frac{n^k}{(k!)^\nu} \end{equation*} in the limit $n\rightarrow\infty$. ...
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59 views

Asymptotic expansion of integral (Laguerre)

Consider $$L_n = \frac{1}{2\pi i } \oint_{C'} \frac{1}{(1-t)^{\alpha+1} t^{n+1}} e^{-\frac{xt}{1-t}} dt\,\,\,\,(1)$$ where $C'$ is an anticlockwise contour around zero. Now set $\alpha = n$ and I want ...
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Strategies for approximating fourier transform of $k$-th power of the $n$-th derivative of a function

For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I ...
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41 views

Asymptotics for the Alternating Mertens Function

Are there any tight bounds, or any nontrivial ones actually, known for, with the lack of a better name, the Alternating Mertens Function: $$ S(n) = \sum_{k=1}^{n} \left((-1)^k \mu\left(k ...
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67 views

$\epsilon y''+\sqrt{x}y'+y=0$, show there is no boundary layer at $x=1$ and a boundary layer of $\epsilon^{\frac{2}{3}}$ at $x=0$?

I'm so lost. If I use quadratic formula I obtain that: $$y(x) = ae^{-2\epsilon\sqrt{x}}+be^{-2x\sqrt{x}+2\epsilon\sqrt{x}}$$ with the boundary conditions $y(0)=0$ and $y(1)=1$ but how does this lead ...
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39 views

Asymptotic of a real double serie on $\mathbb{Z}$

I am interested by a real sequence $\{a_n\}_{n\in\mathbb{Z}}$ as $\sum_{n\in\mathbb{N}}\left(\vert a_n\vert + \vert a_{-n}\vert\right)$ converges. I want to find the asymptotic behavior of this ...
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45 views

Integral of product of Hermite functions with rescaled weights.

Let $$h_{k}(x)=c_{k}(-1)^k e^{\frac{x^2}{r^2}}\frac{d^k}{dx^k}e^{-\frac{x^2}{r^2}}$$ be the standard Hermite polynomials, rescaled with a given parameter $r>0$. The normalizing constant ...
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32 views

Number of ways to visit each cell of $n\times n$ board once

A piece lies on the upper-left corner of an $n\times n$ board. Let $f(n)$ denote the number of ways to move the pieces one step horizontally/vertically at a time, so that it visits each field of the ...
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71 views

Difficulty with Asymptotic Expansion of $\int_{0}^{1}\sqrt{t}e^{ixt}dt$

In the book Advanced Mathematical Methods for Scientists and Engineers by Bender and Orszag (question 6.50) we are asked to compute the asymptotic expansion of $\int_{0}^{1}\sqrt{t}e^{ixt}dt$ fully. ...
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36 views

Find $f(n)$ in $\binom {2^n} {n^4} = (f(n)+ o(1))^n$

Task is to find $f(n)$ in the following equation: $\binom {2^n} {n^4} = (f(n)+ o(1))^n$ I've found that the problem is a bit over my head. I'm attaching my partial solution below: With use of the ...
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59 views

Prove the big theta

I need to find a $n_0$ and $k$ for Big Oh and an $n_0$ and $k$ for Big Omega, to find a big theta bound for: $5n^2 - 9n = \theta(n^2)$ Can anyone help me and show me how to find these for this ...
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0answers
150 views

Comparing asymptotic growth of logarithmic functions by reasoning

As an exercise, we're sorting functions according to their asymptotical growth. When comparing these two functions, I'm getting stuck: $n^2/(\log_2 n)^3$ versus $n \log_2 n$. Using limits I am ...
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62 views

Find an asymptotic upper bound

Use the substistution method to find an asymptotic upper bound for the relation $$T(n)=3 T\left ( \frac{n}{3}+5 \right )+\frac{n}{2}$$ Try so that the bound is as accurate as possible. Consider that ...