# Tagged Questions

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

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### 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 ...
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### 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 $n$. ...
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### 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 ...
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### 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 ...
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### 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 $\lambda$ ...
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### Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
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### Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} \right)...
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### higher-order (3+) Taylor expansion of a likelihood function

I was wondering what is the effect if I replace the second derivative of the log-likelihood ("Likelihood" hereafter) function with its expectation in a higher-order Taylor expansion of the likelihood ...
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### Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r'th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)}$$ I know this looks bizarre but kindly consider the argument below. I'm also ...
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### A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
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### asymptotics of Involutions recurrence relation

Consider the following recurrence relation where $t(n)$ is the number of involutions on $\{1,...,n\}$ $$(n+1)t(n)+t(n+1)-t(n+2)=0$$ When $n \rightarrow \infty$, Wimp and ...
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I'm reviewing for my qualifying exam and I'm stuck on part of a problem. Setup Suppose that $(X,Y)$ are two random variables with joint distribution $f(x,y\mid\alpha,\beta)=c(\... 0answers 45 views ### An advection problem with weak diffusion in asymptotic analysis. Consider the following advection problem with weak diffusion: $$\varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u,$$ for$−\infty < x < \infty$, and$t > 0$where$u(x, 0) = f(...
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I'm learning about the WKB method, and I'm applying it to an assignment. The assignment question asks to find the "leading order" WKB expansion for the particular equation. For WKB you make the ...
I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials). The asymptotic expansion is  f_n(x) \sim_{n \...