Questions tagged [asymptotics]

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

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Notation for dominating (or uniformly bounded) functions

While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function. A situation like this. For some true function $f:\mathbb{R} \to \...
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$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I want to prove this formula using Stirling's approximation or otherwise:$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ wher $s=\sigma+it$. Can someone please ...
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Asymptotic of countour integral with singularity

I need to find the asymptotic of the integral $$\int_0^{(1+)} \left[\frac{t(e^{-i\tau/n^2}-t)}{1-t}\right]^{in-\frac{1}{2}} dt, \quad n\to+\infty,$$ where the countour starts from $t=0$, goes around $...
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Finding an Upper bound for modulus of a function

Consider the integral $$f(s)=\int_{0}^{\infty}\frac{\left(\frac{1}{2}+ix\right)^{1-s}}{\cosh ^2(\pi x)} dx$$ where $s=\sigma+it$ and $\sigma,t\in\mathbb{R}$. Find a Big-O upper bound for the above ...
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On asymptotic of logarithm of modulus of a function

Question: I need to bound Riemann zeta function on the vertical line $s=1/2+\epsilon+it$ where $0\leq t\leq T+\epsilon$, $\epsilon>0$ is arbitrarily small and $T+\epsilon>3$ is not an ordinate ...
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trying to find asymptotic or approximate solutions to coupled diffusion-advection-decay equations

I am trying to solve a set of diffusion-advection equations with coupled decay (see unanswered coupled diffusion-advection-decay equations). The equations are as follows: \begin{align} u_t - \alpha ...
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Asymptotics of a convoluted sum involving $r^{-\alpha}$

Let $\alpha,\beta$ be real constants satisfying $0<\alpha\leq \beta$, and let $R>1$ be an integer. Prove that \begin{equation}\label{eq:lemma_convolution} \sum^{R-1}_{r=1}\frac{1}{r^\beta (R-r)...
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Is $n\cdot\frac{1}{2}^{nx}$ limited for $x\in [0,1]$ and $n\rightarrow \infty$?

I've got this qustion to answer. I know that answer is no. My first idea is to use Big O notation, then: $$ f(n) = n \implies O(f(n)) = n $$ $$ g(n) = \frac{1}{2}^{nx} \implies O(g(n))=?$$ And at the ...
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determining the limiting behavior of a function defined by a Fourier transform

I need to determine this limit value $$ \lim_{r\rightarrow 0 } \frac{\partial}{\partial r }\left\{ r \sum_{\mathbb{k} } e^{i \vec{k} \cdot \vec{r} } [k^2 +1 - k (k^2 + 2 )^{1/2}] \right \} .$$ ...
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How to obtain logarithmic asymptotic behavior for this integral?

Let $t>0$. Also, let $\delta >0$ be very small. A physics paper claims that $$\Re\int_0^t \int_0^t \frac{1}{\sinh^2(t_1-t_2-i\delta)} dt_1 dt_2 \approx -2 \log \left( \frac{\sinh t}{\delta}\...
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Steepest descent for Linearized KdV equation

I take steepest descent on Linearzied KdV equation, $$ u_t+u_{xxx}=0 $$ And by Fourier transform I know the phase is $$ i(k^3+k\frac{x}{t}) $$ I want to know asymptotic of the exponential integral $$ \...
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Proof of logarithmic asymptotic identity

I saw the following identity here: $$ \log(a\,b) = \Theta(\log(a + b)) = c\log(a + b), $$ for any $a,b>1$ and some constant $c>0$ independent of $a,b$. Could someone provide any hints why this ...
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Asymptotic formula for divergent series $\sum_{k=1}^N \frac{1}{k^\alpha}$ [duplicate]

``Experimentally'', I have found that: $$s_N = \sum_{k=1}^N \frac{1}{k^\alpha} \sim \frac{N^{1-\alpha}}{1-\alpha}$$ So I conjecture that: $$s_N = \frac{N^{1-\alpha}}{1-\alpha}f(N), \qquad 0 < \lim_{...
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Asymptotic growth of inverse function [closed]

Let $f, g:\mathbb{R_+}\to\mathbb{R}_+$ be continuous monotonic functions, such that $f(x), g(x)\to+\infty$ as $x\to+\infty$. If $f(x)=\omega(g(x))$, is it true that $f^{-1}(x)=o(g^{-1}(x))$?
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Prove that given $V_{n}\sim\chi^2_{n}$. Then $\frac{V_{n}}{n}$ tends in probability to $1$ as $n\to\infty$. [closed]

Prove that given $V_{n}\sim\chi^2_{n}$. Then $\frac{V_{n}}{n}$ tends in probability to $1$ as $n\to\infty$. I´m doing this exercise and I does not know how to do it. Can anybody give me a hint? We ...
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"Asymptotically equal to" notation

It is unclear to me, how $\sim  $and $\simeq$ are different. I suppose that both of them mean that two expressions are asymptotically equal to each other. For example, could we write that $e \simeq \...
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Being $X_{n}$ distributed Poisson $(\lambda_{n})$.Prove that $X_{n}$ tends in probability to 0 Also being $Y_{n}=nX_{n}$ Prove that $Y_{n}$ tends to 0

$\lambda_{n}=1/n$ Also you have to prove that they tend to $0$ when $n$ tends to $\infty$. By definition $X_{n}$ tends to $0$ in probability when $P(|X_{n}-0|>\epsilon)$ tends to $0$ when $n$ tends ...
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1 vote
1 answer
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Difference between the usage of Big-Omega notation as used by Computer Scientists and Mathematicians.

I was reading Hoffstein¹, page 152. They define the counter part of the famous big-$\mathcal{O}$ notation: the big-$\Omega$, Similarly, we say that $f$ is big-$\Omega$ of g and write, $$f(X)=\Omega(g(...
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Is $2^{O(\sqrt{\log (\log^3 n)})} = 2^{O(\sqrt{\log \log n})}$?

As the question states, I am reading somewhere that: $$2^{O(\sqrt{\log (\log^3 n)})} = 2^{O(\sqrt{\log \log n})}$$ I don't quite see why, though, as $\log^3 n \neq O(\log n)$?
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Illegal transformation of condition for asymptotes?

Obviously, the condition for two functions to be asymptotic is $\lim \bigl(f(x) - g(x)\bigr) = 0$. But shouldn't that be equivalent to: \begin{align} &&\lim f(x) - \lim g(x) &= 0 \\ \...
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Prove that $X_{n}$ converges in probability and distribution to $X$.

I´m learning about convergency in probability of a family of random variables. I think I´m not understanding the concept very well, because I can apply it on an exercise when $X_{n}$ converges to a ...
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Can we find a constant $h$ such that $g(x)>f(x)$ for all $x>h$

Let us consider the two integer valued functions: $$f(x)=ax^2+bx+d$$ $$g(x)=3^{x}$$ where $a,b,d$ are positive integers. My question is: Can we find a constant $h$ such that $g(x)>f(x)$ for all $x&...
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Akra-Bazzi method for $n^2/\log_2(n)$ driving function

I came across $T(n) = 3T(n/3)+8T(n/4)+n^2/\log_2(n)$ with driving function $f(x) = \frac{x^2}{\log_2(x)}$ in the CLRS 4th ed. textbook and am having a tough time applying the Akra-Bazzi method to it. ...
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Check if growth rate is worse than quadratic?

Let's say I have collected a dataset for estimating algorithmic complexity: x, t ---- 1, 1 2, 4 3, 9 4, 16 where x is the input size and t is the elapsed time. ...
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1 answer
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Asymptotics of Laplace like integral with shrinking integration intervals

I want to find the asymptotics of the integral of the form $$I(M) = \int_0^1 x^M e^{-M f(\frac{x}{\ln M})} dx$$ as $M \to +\infty$. Assume also that $f$ is analytic and increasing near $0$, with power ...
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Root locus, deriving the behavior at infinity

I'm reading Control Systems Engineering of Nise, Appendice M.1, Derivation of the Behaviour of the Root Locus at Infinity (Kuo, 1987). At one point, we have; $$ f(s) = (1+\frac{b1-a1}{s})^{1/n} $$ is ...
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1 answer
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Comparing $n \log_2 (n) /2$ and $\log_2 n!$ asymptotically

I want to compare $n \log_2 (n) /2$ and $\log_2 n!$. I calculated $\lim_{n\rightarrow\infty} \frac{\frac{n\log_2(n)}{2}}{\log_2 n!} = \infty$ and $\lim_{n\rightarrow\infty} \frac{\log_2 n!}{\frac{n\...
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2 votes
2 answers
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Exponential sum behaves like linear term for large $t$

I've done some calculations on interesting mathematical objects and came to the conclusion that they would behave nicely as expected if we would have that $$t \sim 2\sum_{n=1}^\infty \exp(-\pi n^2/t^...
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1 answer
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Relationship between the trace distance and the operator norm for some time dependent integral operator with kernel $K(x,y,t)$.

Let us assume that $L_{K_{t}}$ is a positive, compact operator with time dependent kernel $K(x,y,t)$. I know that for self adjoint operators $$ \|A\| = max_{n} |\lambda_{n}|$$ where $\lambda_{n}$ are ...
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$\frac{\pi}{2}\tan{\frac{\pi s}{2}} = O(\log\vert s\vert)$?

In order to proof some estimations of $\frac{\zeta'}{\zeta}$ a textbook derived the formula $$-\frac{\zeta'}{\zeta}(1-s)=-\log(2\pi) - \frac{\pi}{2}\tan{\frac{\pi s}{2}}+\frac{\Gamma'}{\Gamma}(s)+\...
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3 votes
2 answers
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sequence such that $x_{n+1}=n(x_n-n)$

Let $(x_n)$ be a sequence such that $x_{n+1}=n(x_n-n)$ Prove that $x_n=O(n)$ if and only if $x_1=2e$ If we have $x_n=O(n)$ then clearly $x_n=n+O(1)$ so they are equivalent. I don't get how $x_1$ ...
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Asymptotics of Bessel function $J_n(z)$ for $|z| \gg n$ but $|z|\not\gg n^2$

Let $\{n_j\}_{j=1}^\infty$ be a sequence of integers such that $n_j \to +\infty$, let $z_j = x_j + iy_j$ be a sequence of complex numbers approaching the real line, i.e. $x_j \to + \infty$ and $y_j \...
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What proportion of numbers less than $x^a$, $a>0$, are also on the order of it?

Let $x,\ x_0,$ and $y$ all be positive. I say that a number $y$ is on the order of $x$ if $Ay\leq x \leq By$ for all $x>x_0$ where $A$ and $B$ are positive constants. What proportion of numbers ...
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1 answer
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Follow up question: asymptotics of a two dimensional integral

This is a follow up question of Asymptotics of a two dimensional integral about asymptotics of integrals. The problem is to find the leading order term of this integral. $$\int_0^1d\epsilon\int_{-\...
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Comparison of convergence rates

I'm currently trying to compare two different types of assumptions on the convergence rate of a sequence. Let $(a_k)_{k \in \mathbb{N}}$ denote a sequence of positive real numbers. The following ...
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2 votes
2 answers
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Equivalent of a recurrence sequence $I_n=\frac{4n-3}{4n}I_{n-1}$

I have this sequence $(I_n)$ defined by $I_0=\frac{\pi}{2\sqrt2}$ and for all $n\ge1$, $I_n=\frac{4n-3}{4n}I_{n-1}$ (for those interested, it comes from $I_n=\int_0^1 \frac{t^n}{t^{3/4}(1-t)^{1/4}}{\...
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1 vote
1 answer
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How to show that $\forall n \geq 1$, $\frac{n^{2n}}{n!^2} \geq (\frac{n+1}{n})^{n^2-n}$?

I tried using the fact that $(n+1)/n \leq 2$ and thus $\frac{n^{2n}}{n!^2} \geq 2^{n^2-n}$ but this does not seem to be true.
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2 votes
1 answer
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Computing the asymptotics of a principal value integral

I have been looking at the following principal value integral (with $x>0$ and $0< \sigma < 1$) $$ \mathrm{P.V.} \int_0^\infty \frac{v^{-\sigma} e^{-x v}}{1 - v} dv, $$ and I would like to ...
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1 vote
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Asymptotics of function from asymptotics of antiderivative

If $f$, $g$ are positive functions, $f(x)\sim g(x)$ for large $x$ means that $\lim_{x\rightarrow \infty}f(x)/g(x)=1$. In the book "The Riemann Zeta function" by H. M. Edwards, in Chapter 4, ...
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2 votes
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Flipping $n$ sticky coins until they are all heads

Consider a collection of $n$ coins, of which $k$ initially are heads. These coins are "sticky" and, when flipped, remain the same with probability $p > 1/2$, and flip over with ...
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Why does a given image value of a function $x H^3(x)$ where $H(x)$ contains hypergeometric functions have preimage value?

Prof. Edward B. Bender is a famous combinatorial mathematician. In his paper of A Survey of Asymptotic Behaviour of Maps published in Journal of Combinatorial Theory, there is a part of analyzing the ...
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Prove or disprove this statement about asymptotic relation between functions.

Let $f,g,h: \mathbb{N} \to \mathbb{N}$ be an increasing functions. Prove or disprove these statements: a) if $f(n)=\Theta(g(n))$ then $\max\{f(n),g(n)\}=\Theta(\min\{f(n),g(n)\})$. b) if $f(n)=\...
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1 answer
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Find mle of theta from some mixed density

I'm trying to find a mle from: $$ P_\theta(x) = (1 + \theta) I(0 \leq x \leq 1/2) + (1 - \theta) I(1/2 < x \leq 1)$$ Then, \begin{align*} L(\theta) &= \Pi_{i=1}^n P_\theta(x_i) = (1+\theta)^k (...
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2 votes
1 answer
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Asymptotics of a two dimensional integral

I am working on the following integral $$\int_0^1d\epsilon\int_{-\epsilon}^\epsilon dt\left(\sqrt{1-(\rho+t)^2}-\sqrt{1-\rho^2}\right)e^{-N t^2},$$ where $\rho=1-\epsilon$, $N\rightarrow \infty$. The ...
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4 votes
2 answers
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Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

Define $f(n)$ to be: $$ \sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d} $$ But $\sigma_0(d) = 2^{\omega(d)}$ for any $d \mid n\#$ a primorial, so: $$ f(n) = \prod_{p \text{ prime} \\ p \leq n} ...
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Generalized Master Theorem (Divide-and-Conquer) using Ceil / Floor

I'm a bit tired of virtually all books deriving the master theorem always using their own variation: They sometimes use inequalities $T(n)< T(\frac{n}{b})+f(n)$, sometimes are more sloppy and use $\...
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2 votes
2 answers
65 views

Asymptotics of an integral with singular derivation

I want to evalute the leading order term of the following integral as a series of $1/N$ and $\epsilon$, $\int_{-\epsilon}^\epsilon dt (\sqrt{1-(\rho+t)^2}-\sqrt{1-\rho^2})e^{-N t^2}$, where $\rho=1-\...
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Conditions and correct interpretation of Borel summation

Hello to the community. In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ...
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1 vote
2 answers
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Possible growth rates of a matrix entry with respect to exponentiation

Let $A = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$, so $A^n = \begin{pmatrix}1 & n \\ 0 & 1\end{pmatrix}$. Thus, $(A^n)_{1,1} = 1 = \Theta(1)$, and $(A^n)_{1,2} = n = \Theta(n)$. ...
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2 answers
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Expansion of logarithm with little o

Prove that $$ \ln (1+z+o(z))=z-\frac{z^{2}}{2}+o\left(z^{2}\right), \quad z \rightarrow 0. $$ My attempt is to use Taylor's expansion, so we get $$\ln (1+z+o(z))=z+o(z)+o(|z+o(z)|^2).$$ What are ...
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