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

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Simplify $\frac{n(k^2-1)}{2}$ to $ nk^2$

How does $\frac{n(k^2-1)}{2}$ become $nk^2$? I'm sorry for the stupid question but I'm at wits end and I have no idea how to go about this. Context Thanks
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40 views

Evaluating a Limit with Generalized Harmonic Numbers.

Using WolframAlpha, I could informally come up with the following result: $$ \lim_{n \rightarrow \infty} \frac{H_n^{(-\frac{1}{2})}}{n\sqrt{n}} = \frac{2}{3} $$ Allowing me to infer that ...
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61 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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113 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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50 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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49 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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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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86 views

Asymptotic estimate of coprime pairs of integers $\leq n$.

Let $M_{n} = \{(x,y) \in [n] \times [n]: xy \leq n^{2} \text{ and } gcd(x,y) = 1\}$, where $[n] = \{1, 2, \dots , n\}$. In other words, let $M_{n}$ be the set of pairs of coprime integers both $\leq ...
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77 views

Method of dominant balance and perturbation theory

We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters: $y=y_0+\epsilon ^1 y_1+\epsilon ^2 ...
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56 views

How to find Laplace approximation for following integral?

Let's have integral $$ I(x) = \frac{1}{2\pi} \int \limits_{-\pi}^{\pi}e^{xcos(\theta )}d \theta, \quad x \to +\infty . $$ How to use Laplace approximation for this integral and find first two ...
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38 views

Asymptotics for exponential integrals

Suppose I have a situation where I want to find an asymptotic expansion as $x \to \infty$ for an integral of the form: $$ \int_{a}^{b} f(t) e^{-\phi(t) x} \mathrm{d}t$$ Let us also suppose that ...
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33 views

$\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$?

I guess $\lim_{z \to x \pm i \infty} \Gamma(z) \zeta(z + \alpha) = 0$ where $x$ and $\alpha$ are real numbers. The guessing is from numerical experiments and I know $\Gamma(z)$ vanishes exponentially ...
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36 views

Question about asympotic expansions!! please help!

Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion $$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac ...
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27 views

How does one prove that $n^{-100} = \omega(2^{\sqrt{\log{n}}})$?

I feel that $2^{\sqrt{\log{n}}}$ could be dramatically simplified, but I'm sure how. Aside from plugging in huge values to test the functions, any ideas on how I can prove this relationship?
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28 views

Frechet differentiability, asymptotic normality

I try to prove the asymptotic normality from the Frechet differentiability. Consider $$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$ and ...
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24 views

Prove that if $k^2=o(n)$, then $n(n-1)(n-2)…(n-k+1) \approx n^k$

Prove that if $k^2=o(n)$, then $n(n-1)(n-2)...(n-k+1) \approx n^k$ Do I start by dividing both sides by n^k and collecting terms, perhaps? Not sure. Not entirely sure about the relevance of ...
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42 views

Computational Complexity

My question is very basic, it is just so that I have a basic grasp of the terminology of algorithm speed. When someone says an algorithm speed is $O(n^2)$ they say that the number of steps of this ...
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41 views

Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
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Newton polygon and asymptotic behavior near a singular point

As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to ...
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33 views

Can I use the Big-O (Landau) notation to “segment” the set of positive increasing real functions?

Let functions $f(n)$ and $g(n)$ be increasing in $n$. I am trying to say the following precisely: As $n\rightarrow\infty$, if $f(n)$ is "smaller" than $g(n)$ then $A$ is true, and if $f(n)$ is ...
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Can an entire $f$ satisfy $x>k | f(x+yi)=\ln(x+yi+z)+o(1) $?

Let $z$ be a complex number. Let $i$ be the imaginary unit. Let $x,y,k$ be positive real numbers. Consider $$x>k | f(x+yi)=\ln(x+yi+z)+o(1) $$ true for all $x>k,y$ and some $k,z$. Is there ...
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41 views

Asymptotics for prime factors

Am I correct in assuming that the same result: $$ N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ \sim \frac{x}{\log x}\frac{(\log_2 x)^{k-1}}{(k-1)!}\ (x \rightarrow \infty) $$ also holds for: $$ ...
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56 views

Is it possible to find the least common divisor of a two numbers that are not relatively prime in polynomial time?

As the question states: Is it possible to find the least common divisor of two number that are not relatively prime in polynomial time? If so, how? Thanks!
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28 views

Find the order of the following expression as x->0

Could someone help me find the order of the following expression without using the quotient rule? $\frac{1-\cos(x)}{1+\cos(x)}$ I expanded the denominator and the numerator but not sure how I get to ...
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29 views

Asymptotically evaluating integrals with oscillatory behaviour in both numerator and denominator

I have come across an integral that I would like to asymptotically evaluate (to leading order at least) which I have seen no mention of in standard textbooks. I want to evaluate an integral of the ...
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65 views

How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...
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Convergence in distribution and independence

Let $\{X_n\}_{n\geq1}$ and $\{Y_n\}_{n\geq1}$ be two sequences of random variables. Let $X$ and $Y$ be random variables and suppose $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{d} Y$. Furthermore, ...
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30 views

How to find the influence function of $\int_{[0,t]}(1-F_\_)^{-1}dF$,i.e., cumulative hazard function

The common strategy is to replace $F$ with $(1-t)F+t\delta_x$ and then expand the integral. However, I am not sure how to deal with $F_\_$. It seems different from $F$.
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parametric integral and asymptotic representation

Here is a parametrial integral $$I(a)=\int_0^{\pi}\int_0^{\pi} ...
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61 views

Integration by parts in vector calculus

I have an axi-symmetric integral (the domain and all functions are axi-symmetric) in cylindrical coordinates which needs to be integrated by parts for use in a finite element code. The integral is ...
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Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
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38 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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28 views

Steepest descent?

Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral ...
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53 views

Boundary layer method

I am trying to solve the following differential equation using boundary layer method. $\psi ''(z) + \frac{1}{z} \psi'(z)(3 - \frac{4}{1+(\frac{z}{zc})^8})+ \frac{m^2}{1+(\frac{z}{zc})^8}\psi(z)=0$ ...
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54 views

Find asymptotics in a given form $n=(e+o(1))^{f(s)}$

Let $p\to\infty$, $s={\binom {p^4} p}$ and $n={\binom {p^4}{p^2}}$. Find a function $f(s)$ in the following form $$\large n=(e+o(1))^{f(s)}$$ I've tried to use the followinf asymptotics for ...
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60 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
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95 views

Questions about the superfactorial function.

N superfactorial or $n\$$ is defined as - $$n\$=\prod_{k=1}^n k!$$ Then is there any asymptotic formula for this? Are there any infinite series , integrals related to this function? Is there a ...
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61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
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Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
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215 views

Relation between the exponential function and the modified bessel function of second kind

I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the ...
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Aymptotic analysis for Following fucntions

Below is an excercise from algorithm design manual For each pair of expressions (A,B) below, indicate whether A is O, o, Ω, ω, or Θ of B. Note that zero, one or more of these relations may hold for a ...
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how many ways to make change, asymptotics

This is a simplified coin-changing question. Suppose the only coins available are all powers of $10$ dollars. How many ways are there to make change for $\$ 1000000$? In general, to make change for ...
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41 views

Asymptotic growth over an interval

Given a function $f(x)$, we can define the new function $$ A_f(t) = \limsup\limits_{x\to\infty}\ (f(x+t) - f(x)) $$ Is there a place that this transformation has been studied? Also, given a positive ...
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36 views

expected value tree structure

I'm trying to do a run-time analysis of an algorithm. The idea is a tree structure is created where any node can have two children. At each iteration of the algorithm there's a 50% chance that a node ...
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52 views

Prove or disprove asymptotic relation of two sets

I am looking for a while to prove or disprove: (preparing for finals) O(f(n)-g(n)) ⊂ |O(f(n)) - O(g(n))| where || is absolute value. Note that ⊂ is needed and not ⊆ I assumed the a subtraction ...
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48 views

What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
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41 views

Conditioned probability in certain matrices with entries 0,1,$-1$

Consider $2\times n$-matrices with entries 0, 1 or $-1$, such that the number of zeroes in both rows is the same. Let $P_n$ be the probability that the first non negative element of both rows is a ...
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138 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
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57 views

Asymptotic notation of the following function

I have two functions, $f(n)$ and $g(n)$, and I am trying to determine whether $f(n)$ is $O(g(n))$, $\Omega(g(n))$ or $\Theta(g(n))$. I am not sure about my answers. Help will be appreciated. a) ...