Questions tagged [asymptotics]
For questions involving asymptotic analysis, including function growth, Big-$O$, Big-$\Omega$ and Big-$\Theta$ notations.
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Constant term in the Euler-Maclaurin expansion of $s_n=\sum_{k=1}^n \tfrac{1}{k+1/2}$
This question is a followup from my previous post based on the Euler-Maclaurin formula: How to find the correct constant term with Euler-Maclaurin formula, $\sum_{j=1}^n j\log j$.
This time I am ...
- 2,247
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Informations about a sequence from tail behaviour
Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.
(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$
(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to \frac{1}{...
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Are there well-established methods for the asymptotic analytical evaluation of Fourier series?
To obtain asymptotic expansions of integrals which depend on a large parameter, such as
$$\int_{-\infty}^{+\infty} f(x)\;e^{i\lambda h(x)}{\rm d}x$$
($\lambda\to\infty$), one appeals to the methods of ...
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Diophantine approximation and asymptotic for $\dfrac{1}{\sin(n\pi\sqrt{3})}$
I have this exercise and i proved the lemma but i couldn't use it to prove the asymptotic formula i couldn't plug the sin function into the inequality because it change variations maybe some choice of ...
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Approximating the Prime Counting Function as $\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}$
Approximating the Prime Counting Function as $\boxed{\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}}$
Intro________________
In a unrelated topic I was viewing how the mechanical statistics ...
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Variance of asymptotic Travelling Salesperson Problem
Consider N realisations of a uniform distribution on a bounded area in R^2 (e.g., the circle (0,1)). I know that when N is large, the length of a TSP visiting all of those points becomes "...
4
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How to prove $\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz$ in the sense of Borel summation?
As the title shows, I would like to prove this identity in the sense of Borel summation,
$$\sum_{n=0}^\infty (-1)^n f_n=-\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\frac{f_z}{\sin(\pi z)}dz,$$
providing ...
- 43
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Density or growth rate for Eisenstein integers by products and doing $2 x + k$
Consider these two sets of Eisenstein integers.
SET 1 :
constructed by these rules :
a) any unit is in the set.
b) if $x$ is in the set, then so is $2 x + 7$.
c) if $x$ and $y$ are in the set then so ...
- 16.4k
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Landau Notation Problem
I have this function
$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
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+100
'deducing' a bound using the first order taylor series. How to make it more precise?
So, I just saw a ‘proof’ that the generalized birthday problem has a median of C*sqrt(n). Though the probability in question is interesting, this question is more about calculus and maybe asymptotics ...
- 1,376
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Understanding asymptotic analysis
I have tried to write a "proof" for finding the range that c can be when analysing a function asymptotically. I am trying to understand the maths behind Big O and Big Omega. Can someone ...
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Stochastic convergence with and without rate
I have a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$. If I know that there exists a sequence of strictly positive real numbers $(\epsilon_n)_{n \in \mathbb{N}}$ which is such that $\...
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Find the dominating term
I know that, $$a+b=\Theta(\max\{a, b\}).$$
I need to understand the dominating term in this expression,
$$\mathcal{O}(m^{\frac{2}{3}}n^{\frac{2}{3}}+m+n).$$
We know that,$$m^{\frac{2}{3}}n^{\frac{2}{...
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Asymptotics of modified Bessel function of second kind
Let denote $K_\nu$ the modified Bessel function of second kind of argument $\nu\in(0,\infty)$. It is kown that for $x\in\mathbb{R}$, we have:
$$K_\nu(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}$$
as $x\to+\...
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What does "any polynomial dominates any logarithm" mean here?
My textbook states that
any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$
However, it wasn't clear to me what the ...
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