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

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37
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0answers
844 views

Why are asymptotically one half of the integer compositions gap-free?

This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ...
9
votes
0answers
335 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
8
votes
0answers
163 views

How to estimate $Pr[vr_i=ur_i]$ in the presence of rotations

Suppose we want to compute the probability that for two different random vectors (with elements that are $0$ or $1$), denoted by $v$ and $u$, multiplying them with the rotations of a random vector $r$ ...
8
votes
0answers
149 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by ...
8
votes
0answers
157 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega ...
7
votes
0answers
127 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
7
votes
0answers
209 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
7
votes
0answers
221 views

Summation of $1/(xy)$ over a triangular region

Let us consider a lattice formed by all points with integer and positive coordinates on a Cartesian plane, and where $K$ is the maximal value for the x-axis. Let us assign to each lattice point the ...
7
votes
0answers
60 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
6
votes
0answers
85 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
votes
0answers
192 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
5
votes
0answers
49 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
5
votes
0answers
100 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
5
votes
0answers
103 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
5
votes
0answers
175 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
5
votes
0answers
92 views

Binomial asymptotic.

Is there any "direct" proof of the following asymptotic inequality: let $\alpha\le 1$ and consider $$Q_n(x)=\sum_{k=1}^n\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+k-1)}{k!}x^k$$ Then, $$\int_0^1 ...
5
votes
0answers
198 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
4
votes
0answers
44 views

Boundary Layer, leading order, Pertubation Theory, Differential Equations

I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.: Find the leading order of the problem: $\varepsilon ...
4
votes
0answers
51 views

A general question of asymptotics

I am very desperately longing to know if there is a explicit relationship between $$F(n)=f(1)+f(2)+...+f(n)$$ and $$G(x)=\sum_{k=1}^{\infty}f(k)x^k$$ Assuming we can let $f$ be a sufficiently well ...
4
votes
0answers
60 views

Looking for a closed form for the quotient of a sequence of compositions of $\exp()$-function

Related to that previous question I have another still open detail problem. Consider the sequence of evaluations at some given $x$ $$ \small \begin{array} {} z_0 &=& e^x \\ z_1 ...
4
votes
0answers
58 views

Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?

$$\int_0^m \frac{n^x}{\Gamma(x+1)}dx:n,m \in \mathbb{R}$$ I'm dubious as to whether there's a closed form for the above, if there is I'll be very happy. Otherwise: Is there a closed form for ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
174 views

Double harmonic summation

I am interested in determining an asymptotic formula for the double summation of $1/(ab)$, where $a$ is an odd integer ranging between 1 and $k/\sqrt{j}$, $b$ is an odd integer ranging between $a$ and ...
4
votes
0answers
101 views

How many edges does an Erdős-Rényi graph have to have, to almost surely have a component with multiple cycles?

An Erdős-Rényi graph is a random graph, selected according to the distribution obtained one where we have some number $n$ of nodes, and some probability $p$ of each potential edge being ...
4
votes
0answers
150 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
4
votes
0answers
76 views

Dominated convergence on $e^{-n^2 t} t^{s/2-1}$

I am trying to apply the Dominated Convergence Theorem to show that $$\sum_{n\ge 1} \int_0^1 e^{-n^2 t} t^{s/2-1}dt= \int_0^1 \sum_{n\ge 1}e^{-n^2 t} t^{s/2-1}dt$$ as soon as $s>1$. I've ...
4
votes
0answers
216 views

How fast does $\binom{n}{k}$ grow when $k \le n/2$?

How fast does $\binom{n}{k}$, $n$ fixed, grow when $k \le n/2$? Especially, I'm interested in the growth of the "inverse" of binomial coefficient $B_n(x) := \min \{k:\binom{n}{k} \ge x\}$. EDIT: ...
4
votes
0answers
94 views

Asymptotic Analysis of Complex Integrals

Let $\gamma\subset B(0,2)\subset \mathbb{C}$ a smooth Jordan curve envolving the origin and $\phi:\mathbb{C}\setminus (-\infty,0)\to \mathbb{C}$ an analytic function having finite boundary values, ...
3
votes
0answers
19 views

Computing the asymptotic spectrum of a negative distance kernel

Consider the following integral operator: $$K(f) : x \mapsto\int_0^1 K(x,x')f(x') dx', \quad \text{where} \quad K(x,x') = - |x-x'|^{3/2}.$$ The kernel is sometimes referred to as a negative ...
3
votes
0answers
66 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
3
votes
0answers
128 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
3
votes
0answers
67 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
3
votes
0answers
61 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
3
votes
0answers
60 views

Asymptotics of partitions in at most n parts, bounded by r

For every positive integers $n,r,w$ define $$ p_w(n,r)=\#\{ (i_1,...,i_r) | \, 0\leq i_1 \leq \dots \leq i_r\leq n, \, i_1+\dots+i_r=w\} $$ as the number of partitions of $w$ in at most $r$ piece ...
3
votes
0answers
43 views

Growth rate of $\exp(log^{a}(x))$ slower then any power of $x$.

So I'm trying to show that for $0<a<1$ and for $\epsilon >0$ that $\exp((\log x)^{a})=\mathcal{O}(x^{\epsilon}).$ So this amounts to showing that ...
3
votes
0answers
83 views

Asymptotic solution for $T(n) = 6T(n/4) + n \lg n$

I am given that $T(n) = 6T(n/4) + n \lg n$ and want to find $\Theta(T(n))$. Below is what I have typed up for my solution so far; I asked my professor because I was unsure as to how I could assure ...
3
votes
0answers
88 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
3
votes
0answers
61 views

Limit of a sum (no probabilities)

Show that $$\lim_{n\to+\infty}\left(\frac{2}{3}\right)^n\sum_{k=0}^{[n/3]}\binom{n}{k}2^{-k}=\frac{1}{2}$$ without using probabilities. $[\;\cdot\;]$ denotes the integer part.
3
votes
0answers
71 views

Definition of small $o$

In one of my homework assignments (in intro to applied mathematics) there is the following definition: Given two functions $f(\epsilon),g(\epsilon)$ that are defined in $D=(0,\epsilon)$ we say ...
3
votes
0answers
153 views

Saddle point and stationary point approximation of the Airy equation

Happy New Year to you all. Let $$\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $$f(x) \approx ...
3
votes
0answers
73 views

How to explore the asymptotic of an iteration

Definition Given that $f:\Bbb R\to\Bbb R$ is a real-valued function. The iteration of $f$, say $f^n$, is defined here: $f^0(x)=x$ $f^n(x)=f\left(f^{n-1}(x)\right)$ for any positive integer $n$. ...
3
votes
0answers
57 views

Is there a common name for $O(x^{cx})$ type functions?

Is there a common name for the growth rate of functions that are asymptotically on the order of $x^{cx}$, for some $c$? The term super-exponential is much too general. The factorial function grows in ...
3
votes
0answers
403 views

Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
3
votes
0answers
92 views

Products of primes of the form $an + b$

What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)? Surely (except for the trivial cases) they are of order strictly between that of he ...
3
votes
0answers
342 views

When do floors and ceilings matter while solving recurrences?

I came across places where floors and ceilings are neglected while solving recurrences. Example from CLRS (chapter 4, pg.83) where floor is neglected: Here (pg.2, exercise 4.1–1) is an example ...
3
votes
0answers
85 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
3
votes
0answers
145 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
3
votes
0answers
187 views

proving the saddle point method in a specific case

$1:$ the problem Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by: $\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$ I want to ...
2
votes
0answers
26 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 ...
2
votes
0answers
13 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 ...