# Tagged Questions

1answer
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### Asymptotic expansion of an integral with exponential decay and highly oscillating kernel [on hold]

I would appreciate if one can get the leading term of the following integral: $$I(x) = \large{\int}_0^\infty \frac{g(s)}{\sqrt{s^2 + \frac 1 4}}e^{- i x s- m x\sqrt{s^2 + \frac 1 4}}ds$$ as ...
1answer
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### Switching Limits and summation

I'm currently working on proving some theorems and there is one recurring problem that I somehow can't solve. $a_n$ is a real sequence in either $[0,1]$ or $\mathbb{R}$ that approaches $0$. ...
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### Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
2answers
162 views

### Convergence of power towers

Let's define the sequence $\{s_n\}$ recursively as $$s_1=\sqrt2,\ \ \ s_{n+1}=\sqrt2^{\,s_n}.$$ Or, in other words, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
1answer
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### Proving an asymptotic run time is faster than another using L’Hôpital’s

I'm working on a problem: Show using L’Hôpital’s Rule that a running time of $n\log(n)$ is asymptotically faster than (i.e., little-oh of) a running time of $\frac{n^2}{\log(n)}$.` I suppose a ...
1answer
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### Monotonicity of $f(x)-g(x)$ where $g$ is asymptotically greater than $f$

If $g(x) \succ f(x)$ (or $\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=0$), will $g(x)-f(x)$ always be a strictly increasing function?
3answers
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### Taking Limits with Binomial Coefficients

I am interested in taking the following limit: $$\lim_{n \to \infty}\frac{{n/2 \choose m}}{n \choose m}.$$ Provided that $m$ is fixed the solution is: ...
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### 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 ...
2answers
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### 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
1answer
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### 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 $x = ia$. ...
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### 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 $x\to 0$ ...
3answers
166 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 ...
1answer
187 views

### Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that $$f(n) = {\mathcal O}(\log n),$$ but $$2^{f(n)} ≠ {\mathcal O}(n).$$ Is ...
2answers
154 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.
1answer
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### 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 ...
1answer
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### $x^2-\log x = u$ 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 ...
1answer
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### 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 ...
1answer
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### 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 ...
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### 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 ...
1answer
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### 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 ...
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1answer
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### 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 ...
1answer
143 views