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

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15 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 ...
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1answer
13 views

Determine the values of real parameters …

We have the function $$f:R\setminus \{-1 \}\to{R}$$ $$\color{maroon}{f(x)={x^2+ax+b\over x+1}}$$ $${a,b\in R}$$ For which values of parameters $a$ and $b$ the function admits an oblique asymptote ...
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9 views

Find the values of the parameters for which the function admits an oblique asymptote…

can you please help me solve this exercise: Find the values of real parameters $a$ and $b$ so that the function $$\color{maroon}{f(x)={(ax^3+bx^2)}^{1/ 3}}$$ admits an oblique asymptote: ...
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1answer
29 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
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1answer
28 views

How do you define computational complexity abstractly?

Let the problem we're studying be $f : X \to Y$. Say, I don't know what I want to define time-complexity with respect to, I just know I have a map $|\cdot| : X \to \Bbb{R}$, such that $|\cdot| \geq ...
3
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2answers
33 views

Short argument for asymptotic value of parameter integral

I want to find the main term of the asymptotic expansion for $x\to 0^+$ for $$f(x)=\int_0^{\pi/2} \dfrac{\cos t}{t+x}dt.$$ Now, clearly, the problem is at $t=0$ and the cosine is almost $1$ ...
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2answers
28 views

Discrete Mathematics: Prove that f(x) is in O(x)

Prove that $$\frac{2x^{2}+x}{x+1}$$ is in $O(x)$
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1answer
37 views

asymptotics of this sum $ x \to 0 $

given the sum $$ \sum_{n=0}^\infty \frac{\exp(-nx)}{n+a} =f(x) $$ what would be the asymtptic of this series ?? for $a=1$ i believe this series goes as $ f(x) \sim \frac{1}{x}+ \gamma $ for every ...
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1answer
47 views

Bound summation of successive square roots

What is a tight upper bound for $f(n)$ where $f(n) = f(\sqrt{n}) + \frac{1}{n}$. One can easily find the following upper bound $O(\lg \lg n)$, however I'm interested in a tight bound. Regards.
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0answers
26 views

Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
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1answer
46 views

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$.

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$. I'm trying to prove the statement by building on my observation that $(1-\frac{1}{n})^n$ ...
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0answers
64 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 ...
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0answers
34 views

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

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 ...
0
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1answer
22 views

Interpreting expression with big-O notation in the exponent ($f(x) = x^{1+O(1)}$)

How should one interpret the notation $f(x) = x^{1+O(1)}$? I'm a bit confused as to what this means. Does it merely suggest that f(x) grows as some integer power of x?
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2answers
20 views

Big Omega — n, n + 100

Given $f(n) = n$ and $g(n) = n + 100$, it seems that f(n) is $O(g(n))$ when $C = 1$ and $k= 0$. That is, for every $n$ from $0$ to infinity, g(n) is strictly larger than f(n). Now, concerning ...
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1answer
36 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
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0answers
23 views

prove that θ(n-1)+ θ(n)= θ(n)

I need to prove that statment with big Theta defenition.. θ(n-1)+ θ(n)= θ(n) I have tried many things but cant prove that with defenition of theta
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1answer
36 views

Prove that |O(2n)-O(n)|=O(n)

I need to prove that statement with the defenition of big O |O(2n)-O(n)|=O(n) Does it can be proven? or not? if i can, so how..in which way? i tried almost ...
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2answers
34 views

Big-O notation and polynomials

In my text, I am given that the sum of the first n positive integers can be understood in terms of big-O notation. ''Since each of the integers in the sum of the first $n$ positive integers does not ...
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0answers
30 views

Free lecture notes to Carl Bender's Mathematical Physics video lecture course?

I am just watching Carl Bender's Mathematical Physics video lecture course (about asymptotics and its application in physics) http://www.perimeterscholars.org/328.html which is great! Are there any ...
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3answers
67 views

Taking Limits with Binomial Coefficients

I am interested in taking the following limit: \begin{equation} \lim_{n \to \infty}\frac{{n/2 \choose m}}{n \choose m}. \end{equation} Provided that $m$ is fixed the solution is: \begin{equation} ...
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2answers
51 views

Expanding $\ln(1+f(x))$ around $f(x)=0$ when we do not know whether there is an $x$ such that $f(x)=0$.

I want to expand $\ln(1+f_T(x,\theta))$ about $1+f_T(x,\theta)=1$. What I have in mind is something like $$ \ln(1+f_T(x,\theta))=\ln(1)+f_T(x,\theta)-\frac{1}{2} \frac{1}{1+\tilde{f}} ...
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0answers
13 views

How to determine sub-exponential time growth?

I'm a little bit confused of sub-exponential time growth; consider the definition from Hoffstein's book An Introduction to Mathematical Cryptography: Given input of $k$ bits, then if an algorithm ...
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0answers
42 views

Integral asymptotics

Is there some kind of a variation of the Laplace's method or some other formula for the asymptotics of integrals of a type $$\int_a^bf(x)e^{mp(x)}\cos(mq(x)+x/2)dx, \ m\to\infty.$$ Here $f,p,q$ are ...
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1answer
20 views

Prove the following $\frac{\Omega(f(n))}{\Omega(g(n))} \subseteq \Omega(\frac{f(n)}{g(n)})$

I want to prove the following: $$\frac{\Omega(f(n))}{\Omega(g(n))} \subseteq \Omega(\frac{f(n)}{g(n)})$$ I wonder if its true? What about using $n$ and $n^2$? Any suggestions? Thanks!
2
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1answer
28 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
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1answer
13 views

Order functions by speed of their asymptotic growths

We are given list of functions. Task is to sort it by the speed of their asumptotic growth in ascending order. Yes, it's a homework. I already spent some solid amount of time calculating limits. I ...
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1answer
76 views

What is $O\Big((n+1)!\Big)$?

What is $f(n) = (n+1)!$ which is also $f(n) = (n+1)n!$ in terms of big-O notation? My guess is $O(n \cdot n!)$ but I am not sure. I only know it is certainly $f(n) \in O(n^n)$.
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2answers
85 views

Use the Euler-Maclaurin summation formula to estimate a summation

$\sum_{k=0}^n \frac{1}{1+\frac{k}{n}}$ How can we estimate it to order $O(n^{-5})$ ?
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0answers
22 views

Asymptotic distribution of the t statistic

I want to find the asymptotic distribution of the t statistic when (a) we assume equal variances between groups with sample sizes m and n (b) assume unequal variances between groups For part (a) it ...
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1answer
20 views

Tight bound of worst case performance of algorithm

I am trying to find the "tight bound of an algorithm for the worst case run time. I have found that the upper bound of the worst case is O(n), I have also found that the lower bound for the worst case ...
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0answers
38 views

Is $\log^* (n+1)^{n+2} \in O(\log^* n)$?

I would like to know if $\log^* (n+1)^{n+2} \in O(\log^* n)$, where $\log^*$ is the iterated logarithm. I tried doing: $ \log^* (n+1)^{n+2} =\\ \log^{*}(\log(n+1)^{n+2})-1 =\\ \log^{*}((n+2) \cdot ...
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2answers
37 views

Asymptotics - Big Omega

I have a question about Asymptotics involving big Omega... How do I need to approach this equation in order to prove it? $$n \cdotΩ(f(n)) = Ω(n\cdot f(n))$$ Thank you very much for your answers!
4
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0answers
53 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 ...
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1answer
55 views

Prove $\Omega(f(n)) \subset \Omega(g(n)), iff : g(n)\in \mathcal{O}(f(n)) \wedge f(n) \not\in \mathcal{O}(g(n))$

I want to prove the following $$\Omega(f(n)) \subset \Omega(g(n)), iff : g(n)\in \mathcal{O}(f(n)) \wedge f(n) \not\in \mathcal{O}(g(n))$$ What I did so far is: $$t(n)\in\Omega(f(n)) \rightarrow ...
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0answers
22 views

Asymptotic estimate for an expression of

\begin{equation} A = \frac{(\frac12-\frac{1}{n})(\frac12-\frac{2}{n})...(\frac12-\frac{t-1}{n})}{(\frac{1}{2}+\frac{1}{n})(\frac{1}{2}+\frac{2}{n})... (\frac{1}{2}+\frac{t}{n})} \end{equation} Can we ...
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1answer
39 views

asymptotic estimate for this expression

How can I compute an asymptotic estimate for following expression? \begin{equation} A = ...
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0answers
23 views

asymptotic estimate for binomial coefficient

How can we compute the asymptotic estimate for binomial coefficient $Q = \binom{n}{s-t}$ with the $n,s \gg 1$ and $ t \ll n$ and $ t \ll s$
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0answers
15 views

How to compute an asymptotic expansion for $\sum_{i \ge 0} a_in^{-i} $ to a relative error of $O(n^{-2})$? [duplicate]

How to compute an asymptotic expansion for $\sum_{i \ge 0} a_in^{-i} $ to a relative error of $O(n^{-2})$
3
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1answer
61 views

On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding ...
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1answer
42 views

Compute summation with a relative error of O(n^-2)

$a(n) = \sum_{i \geq 0} a_i n^{-i}$, how can we compute the value of $a(n)^n$ with a relative error of $O(n^{-2})$?
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0answers
19 views

asymptotics big Omega and O

I have a problem at Asymptotycs.. there is given to me that : $$f(n)+g(n) \in \Omega(t(n))$$ I need to prove with the defenition that: $$\operatorname{abs}(f(n)-g(n))∈O(t(n))$$ How I start ...
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2answers
25 views

$$\\|O(2n) - O(n)|=O(n)$$

I need to prove or contradict:$$\\|O(2n) - O(n)|=O(n)$$ I try: $$\\f(n)=1.5n\in O(2n),g(n)=0.25n\in O(n),h(n)>0\in O(n) : \\ |1.5n - 0.25n|=h(n)\\1.h(n)=1.25n \in O(n)\\ but: 2. h(n)=-1.25n \notin ...
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0answers
61 views

How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...
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1answer
24 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
3
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0answers
39 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 ...
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0answers
26 views

Stochastic big O notation

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
2
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0answers
18 views

Asymptotic behaviour of oscillating integral

I'm interested in the big $x$ ($x \to \infty$) behaviour for the following integral $\int_{-\infty}^{\infty} \frac{dk}{\sqrt{k^2+1}} \frac{e^{-\sqrt{k^2+1}/2}}{1-e^{-\sqrt{k^2+1}}} e^{ikx}$ After a ...
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0answers
24 views

Relations between stochastic Big O-Notation, limsup and lim?

I have to solve a list of exercises on probability theory but I'm having several problems in understanding the following questions; could you help me? Thank you! Let $||.||$ indicate the Euclidean ...
0
votes
0answers
38 views

Meaning of theta notation in summation.

Just a simple question: What does the big theta notation mean in this equation? $$S(n,p) = \sum_i^ni^p=\Theta(n^{p+1})$$