# Questions tagged [asymptotics]

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

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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 ...
1 vote
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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 \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 ...
1 vote
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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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### 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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### 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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### 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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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 $\rho=1-\... • 135 0 votes 0 answers 28 views ### Conditions and correct interpretation of Borel summation Hello to the community. In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ... 1 vote 2 answers 47 views ### Possible growth rates of a matrix entry with respect to exponentiation Let$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)\$. ...
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 ...