1
vote
0answers
43 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 ...
1
vote
2answers
34 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
1
vote
1answer
84 views

About asymptotic behaviour of a divergent integral.

I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. ...
2
votes
1answer
44 views

Gaussian integral asymptotics

I am trying to derive the asymptotics of $$\int_{2\sqrt{m}}^{\infty}e^{-\frac{x^2}{4}}x^mdx$$ as $m\to\infty$ with no success. I tried integrating by parts, but could get no nice expression. Any help ...
0
votes
0answers
33 views

Asymptotic expansion of the floor function at infinity

Is it possible to study the behavior of the floor function at infinity by estimating its growth? The floor function has countably many discontinuities at integers, so I'm afraid that these ...
4
votes
1answer
112 views

Asymptotic estimate of an oscillatory differential equation

Let $f\in C^1(\mathbb{R} ,\mathbb{R} )$ and satisfying the condition: $$ \forall x >0, \quad f(x)>0, \forall x<0 , \quad f(x)<0 $$ Let $(\alpha, \beta) \in \mathbb{R^2}$. ...
1
vote
0answers
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 ...
3
votes
2answers
76 views

Asymptotics of coefficients

This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. ...
3
votes
2answers
59 views

Asymptotic expansion of $\sum_{k=0}^{\infty} k^{1 - \lambda}(1 - \epsilon)^{k-1}$

I'm seeing a physics paper about percolation (http://arxiv.org/abs/cond-mat/0202259). In the paper the following asymptotic relation is used without derivation. $$ \sum_{k=0}^{\infty} k P(k) (1 - ...
4
votes
1answer
36 views

asymptotics of inverse function

Suppose $f:[0,\infty)\to [0,\infty)$ is strictly increasing with $f(0)=0$ and it's given explicitly as a combination of elementary functions. How do you find the asymptotics of $f^{-1}(x)$ as $x\to 0$ ...
1
vote
3answers
118 views

Equation with the big O notation

How I can prove equality below? $$ \frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}), $$ where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$. It is clearly that it is true. But I ...
0
votes
1answer
158 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
1
vote
2answers
125 views

Derivative of $n^{\log n}$?

What would be the derivative of $n^{\log n}$? I have to prove that $(\log n)^n$ = $\omega$($n^{\log n}$). I am trying to implement L'Hopital rule.
3
votes
1answer
67 views

Cesaro means and equivalent sequences

Let $(u_n)$ be a sequence of complex numbers that converges in mean (Cesaro convergence). Let $(v_n)$ be a sequence such that $v_n\sim u_n$. Does the sequence $(v_n)$ converge in mean? Here is ...
2
votes
1answer
38 views

$x^2-\log x = u $ asymptotic behaviour

Find the asymptotic behaviour as $u \to \infty$ of the solutions of $x^2-\log x = u$. Is there a standard method to solve this kind of problems? May the fact that we obviously know the derivative of ...
6
votes
1answer
90 views

Limits of $\sum_{m=1}^{+\infty} \sum_{n=1}^{+\infty}e^{-mn x}$ at $0$ and $\infty$

Let $f(x) = \sum_{m=1}^{+\infty} \sum_{n=1}^{+\infty}e^{-mn x}$ for $x > 0$. Prove that $f(x) \sim e^{-x}$ as $x \to \infty$ and $\lim_{x\to 0} x\cdot (f(x) + \frac{1}{x}\log x)= \gamma$ where ...
2
votes
1answer
88 views

Asymptotic Expansion of a nearly divergent integral

I want to understand the asymptotic behavior of an integral of the form $$ I_f(\epsilon) = \int_0^1 \frac{\log(1/x)}{\sqrt{x}\sqrt{x+\epsilon}} f(x) dx $$ as $\epsilon \to 0^+$ for a generic function ...
1
vote
0answers
55 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 ...
2
votes
1answer
130 views

Derivative of big O symbol

Let's only work with functions $f(x)$ that have a series expansion at $x=0$. Is it true that: $$ {d O(1)\over d x} = O(1) $$ for all such functions $f(x)$? Here $O$ is the big-O notation and we are ...
0
votes
0answers
35 views

Proving a Certain Inequality that Involves the Sinc Function

Could someone kindly show me how to rigorously prove that there exists a constant $ C > 0 $ such that $$ \forall N \in \mathbb{N}: \quad \sup_{x \in \mathbb{R}} \sum_{\substack{k \in \mathbb{Z} \\ ...
5
votes
6answers
244 views

Asymptotic solution to the integral $\int_{-\pi/2}^{\pi/2} (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x$

Recently, I have posted a question on how to find a reduction formula for the trigonometric integral $$\int (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x.$$ This problem seems to be tough, however. When ...
1
vote
1answer
136 views

Calculating expected value of distance in a circle-circle intersection

Consider two circles $c_1$ and $c_2$ both of radius $r$ located in 2-D plane such that the distance between their centers is $r$. Assume a point is randomly and uniformly chosen within their ...
0
votes
2answers
64 views

Meaning of $O(n)$ in an expression

As my mathematical knowledge is increasing, I have been seeing more and more of $O(n)$ implementation in expressions. Here is what I mean. Example: $$z^{q_{N+1} + q_N} w^{q_{N+1} + q_N} (-1)^N (w-1)/w ...
4
votes
1answer
140 views

How to analyze the asymptotic behaviour of this integral function?

Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$ Without working out ...
7
votes
1answer
143 views

Approximate $\int_0^{\pi /2} \frac{ds}{\sqrt{1-x\sin^2s}}$

I am trying to approximate the following integral $$K(x)=\int\limits_0^{\pi /2} \frac{ds}{\sqrt{1-x\sin^2s}}$$ with $0<x<1$. I need to show that for x close to one that $K(x)\sim ...
9
votes
1answer
93 views

Find asymptotic of recurrence sequence

Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$. The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$. ...
0
votes
1answer
33 views

Behavior of the following function at x=0 singularity

I am trying to do the following integral: \begin{equation} \int\frac{dx}{x^{2p}(x-1)^{2q}} \end{equation} for positive $2p$ and $2q$. I want to understand how does this function blow up (the $x$ ...
2
votes
2answers
115 views

When does L'Hopital's rule pick up asymptotics?

I'm taking a graduate economics course this semester. One of the homework questions asks: Let $$u(c,\theta) = \frac{c^{1-\theta}}{1-\theta}.$$ Show that $\lim_{\theta\to 1} u(c) = \ln(c)$. Hint: ...
0
votes
1answer
180 views

Explanation of the binomial theorem and the associated Big O notation

I'm currently following the MIT Single Variable lectures online and the professor states that the binomial theorem for the expansion $(x + \Delta x)^{n} = x^{n} + nx^{n-1}\Delta x + O((\Delta ...
4
votes
1answer
124 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
2
votes
1answer
69 views

Asymptotic solving of a hyperbolic equation

The solition and anti-solition nonlinear equation is given as: My problem is that, how do we get the next equation after considering asyptotic behaviour? Resource: (solition) at page 38
1
vote
0answers
147 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 ...
3
votes
2answers
70 views

Prove: For all $n\geq 1$, $a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}$.

Set $S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}$, k is a positive integer, sort elements of $S$ increasingly, that $a_1<a_2<a_3<\text{...}<a_n<\text{...}$. Prove: ...
7
votes
1answer
382 views

How to find asymptotics?

The function $\Phi:(0,\infty) \mapsto \mathbb{R}$ is defined as follows. We put $\Phi(x):=1$ if $x \ge 1$. Let the function $\Phi$ satisfy $$\Phi(x)=\int_0^x \Phi\left(\frac t {1-t}\right) \frac {dt} ...
3
votes
1answer
74 views

How to find asymptotics of integrand?

Let $ f \in C ([0, \infty)) $ be s. t. $$f(x) \int_0^x f(t)^2 dt \to 1, x \to \infty.$$ How to prove that $f(x) \sim \left( \frac 1 {3x} \right)^{1/3} $ as $x \to \infty?$
2
votes
2answers
70 views

How can an oblique asymptote be $y = x$ , as $x\to \infty$?

In my Calculus book, an oblique asymptote defined as: Oblique Asymptote: the function $y = f(x)$ has an oblique asymptote $y = mx + n$, if: $$\lim_{x\to \infty} {f(x) \over x} = m$$ ...
2
votes
2answers
68 views

Big-$O$ inside a log operation

I would appreciate help in understanding how: $$\log \left(\frac{1}{s - 1} - O(1)\right) = \log \left(\frac{1}{s - 1}\right) + O(1)\text{ as }s \rightarrow 1^+$$ I thought of perhaps a Taylor series ...
3
votes
1answer
60 views

Approximating an integral with elementary functions

Consider the integral $$\int_1^\infty\frac{\exp(-nx)}{x}dx$$ We get:$$\int_1^\infty\frac{\exp(-nx)}{x}dx=n\int_ n^\infty\frac{\exp(-x)}{x}=nE_1(n)$$ My question is, can we approximate this integral ...
7
votes
1answer
79 views

An asymptotic integral inequality

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $g(x)=xf(x)-\int_0^xf(t)\ dt$, and we have $f(0)=0$ and $g(x)=O(x^2)$ as $x\to0$. Is it true that $f(x)=O(x)$ as $x\to0$ ?
0
votes
2answers
81 views

Why is big-Oh multiplicative?

If $f$ is $O(g)$ over some base, this means that $f(x) = \beta(x)g(x)$, where $\beta$ is eventually bounded. So this means that eventually, $f$ is at most $c$ times $g$, where $c$ is some constant. ...
3
votes
1answer
172 views

Asymptotics of a summation over real valued functions

Let $f$ and $g$ be integrable in $[0,1]$ and $(-\infty, \infty)$ respectively. Let $a_k$ be a divergent series of positive terms and $S_k = a_1 + a_2 + \ldots + a_k$ such that the following ...
10
votes
8answers
828 views

Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$.

I can't seem to find a good way to solve this. I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one. Any ...
2
votes
0answers
43 views

How prove this $\frac{|x-z|}{|x-y|}=1+\frac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$

prove that $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$ for $|x|\longrightarrow \infty$ where $$\hat{x}=\dfrac{x}{|x|}$$ This problem from book,following is my idea: ...
18
votes
6answers
1k views

Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?

I am searching for the value of $$\sum_{n=k+1}^{\infty} \frac1{n^3} \stackrel{?}{=} \sum_{n = 1}^{\infty} \frac1{n^3} - \sum_{n=1}^{k} \frac1{n^3} = \zeta(3) - \sum_{n=1}^{k} \frac1{n^3}$$ For which ...
9
votes
2answers
454 views

Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$$ No clue at all. Need help! Appreciated!
1
vote
1answer
335 views

Meaning of algebraic decay

I am reading the paper here and I am running into a few roadblocks. One of them was resolved here and now I am stuck at another. (Pg 177) Suppose $\alpha\in(0,2)$ and $t_i=ih$ for some fixed $h>0$ ...
3
votes
1answer
229 views

Techniques for asymptotic growth comparison between complicated expressions

For the following functions: $$\frac{2^n}{n + n \log n}$$ and $$4^{\sqrt{n}}$$ I'd like to compare their asymptotic growth as $n \to \infty$. Is there any other way to do that other than using ...
1
vote
1answer
113 views

Difficulty proving / disproving the following equalities relations ( Big Ω)

I have left with some functions I can't find witenesses for proving/disproving Big Ω equalities relations. Here are the three relations: $ \sum\limits_{i=1}^{n} (i^3 - i ^2) = \Omega(n^4) $ ...
1
vote
0answers
29 views

Evaluating a simple sum bound

I'm trying to evaluate and prove a simple statement but It seems really raw/bad solution. I would like to advise with you if this is the right way because It is really getting more complicated than It ...
2
votes
1answer
91 views

Asymptotic behavior of $\cos(\sqrt{4n+1}x)-\cos(\sqrt{4n+\alpha}x)$

While reading a paper in physics i came across asymptotic behavior of $\cos(\sqrt{4n+1}x)-\cos(\sqrt{4n+\alpha}x)$ and it was written this is equal to $O(n^{-1/4})$ for any real $\alpha$. I tried to ...