# All questions with tag [math: asymptotics]

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Consider an integral $$I(x) = \int\limits_{\mathbb{R}^n} e^{i\xi x } \delta(\xi^2-k^2)\chi( (\xi-k,\gamma)) \, d\xi$$ where $x, k, \gamma \in \mathbb{R}^n$, $|\gamma| = 1$ and $(x,y)\equiv xy = x_1 y_1 + \ldots + x_n y_n$. Here $\delta$ is the Dirac delta, $\chi$ is the Heaviside step function: $$\chi(t) = 1_{\left\{ t \geqslant 0 \ri... 2022-07-25 17:46:14 1 ## Interesting Recurrence Relation T(n) = T(\sqrt{n}) + T(n-\sqrt{n}) + n I found an interesting recurrence that I do not know how to solve. I think this has to do with quicksort with pivots at rank \sqrt{n}. I do not know how to approach this problem nor found any helpful resources about it. Here is the recurrence:$$T(n)=T(\sqrt{n})+T(n−\sqrt{n})+n$$Any help would be much appreciated. Thanks! Let's say the base c... 2022-07-25 17:16:30 0 ## How can Big-O be proved using derivatives? Say we have:$$f(n) \in O(g(n))$$By definition we require to show that:$$0 \le f(n) \le c\cdot g(n) $$for some c&gt;0 and also for all n&gt;n_0. This is generally uncomplicated when sensible and also polynomial features are entailed, yet if features have logarithms and also square origins, I get perplexed and also not exactly sur... 2022-07-25 17:14:03 0 ## Next asymptotic term of the average order of sigma$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n). $$Is the next asymptotic term recognized? That is, exists a monotonic raising function f(x) such that$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+f(n)+o(f(n)) $$? (I would certainly presume something like f(x)=cx if f exists.) At the same time, exist monotonic raising features f... 2022-07-25 17:13:44 4 ## Big O Notation reliability? Is Big - O symbols constantly trusted? For example: Algorithm A: n * 10^{100} = \mathcal{O}\left(n\right) Algorithm B: n^{1.001} = \mathcal{O}\left(n^{1.001}\right) According to Big - \mathcal{O} symbols, Algorithm A would certainly be extra reliable, yet for all sensible objectives Algorithm B is extra reliable. In a scenario simi... 2022-07-25 17:13:11 1 ## WKB approximation question I read some things on asymptotic evaluation, yet just how do you obtain from the 1st line to the 2nd line? y \sim \frac{1+x}{2\lambda}\exp\left(\frac{\lambda x}{1+x}\right) - \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)\\y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty I see th... 2022-07-25 16:39:35 1 ## Prove that \log(n) = O(\sqrt{n}) How to confirm \log(n) = O(\sqrt{n})? Just how do I locate the c and also the n_0? I recognize to start, I require to locate something that \log(n) is smaller sized to, yet I m having a tough time thinking of the instance. 2022-07-25 16:31:02 1 ## properties of a real analytic function If there are a distance r&gt;0 and also constants M,C\in\mathbb R for all y\in U with$$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb N_0^n$$after that f\in C^\infty(U) is actual analaytic. Yet I do not have any kind of suggestion just how to confirm this. I feel in one's... 2022-07-25 13:19:23 1 ## Disproving a big O equation As a homework assignment I am trying to prove/disprove the next statement: Let f(x)=O_a(g(x)), then \forall A,B\in\mathbbR\rightarrow A\cdot f(x)=O_a(B \cdot g(x)) Which I think is wrong and thus trying to disprove. So assuming I'm right, my question is logical: we know that there exist a C_1 \gt 0 such that \|f(x)\| \le C_1 \cdot \|g(... 2022-07-25 13:11:21 1 ## Big-O notation always holds for this two functions? For 2 any kind of features f(n) and also g(n) constantly holds: f(n) = O(g(n)) or g(n) = O(f(n)) Right? Many thanks 2022-07-25 12:34:10 1 ## expansion of \int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt in inverse powers of p This question relates to this answer I gave to a question about the integral$$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$I derived an expansion in inverse powers of p and then realized that I don't know how to justify it rigorously or how to determine its radius of convergence. I substituted u=\sqrt pt and applied$$\left(1+...
2022-07-25 12:28:10
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Help me please with the adhering to indispensable. I've asked this inquiry prior to https://math.stackexchange.com/q/140460/30554, yet it ends up that it was wrong inquiry. I need to get an asymptotic restriction (with $k$ mosts likely to $\infty$, and also $q$ dealt with) of it. For $q\ge 2, t\ge 0, k \in Z$ $$\int_0^{\infty}\left(1-\frac{t^2}... 2022-07-25 07:55:44 0 ## Asymptotic bounds: \ll vs. \ll_{\epsilon}? I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e. \log n \ll_\epsilon n^\epsilon which means that \log n \leqslant C_\epsilon n^\epsilon for sufficiently large n, where the constant C_\epsilon depends only on the consta... 2022-07-25 07:51:22 1 ## Big O Notation question I am attempting to recognize the Big - O and also little - O symbols, so I outlined 2 charts which I have actually uploaded listed below, yet I still do not actually get the principle of it. Just what does the O\left(\frac{1}{x^6}\right) term do? 2022-07-25 07:49:20 0 ## Describe growth of \epsilon n For all \epsilon we have that f(n)\le \epsilon n where n is an all-natural number. What can we claim concerning the development of f(n)? Plainly f(n)=O(n), can we claim anything sharper? 2022-07-25 07:48:41 2 ## Can a function "grow too fast" to be real analytic? Does there exist a continual function \: f : \mathbf{R} \to \mathbf{R} \: such that for ¢ all real analytic features \: g : \mathbf{R} \to \mathbf{R} \:, for all actual numbers x, ¢ there exists an actual number y such that \: x &lt; y \: and also \: g(y) &lt; f(y) \:? 2022-07-25 07:47:36 1 ## asymptotic limit at the integral I would love to get an asymptotic restriction at the adhering to indispensable: for p\ge 2, n \in N, t \ge 0$$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t  I assume replacement $t=\frac 12 \sqrt{(n+1)!}y$ need to function. Yet after the substittution, I do not recognize what to do. Thanks for your...
2022-07-25 07:37:16
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## Why is $\log(n!)$ $O(n\log n)$?

I assumed that $\log(n!)$ would certainly be $\Omega(n \log n )$, yet I read someplace that $\log(n!) = O(n\log n)$. Why?
2022-07-24 06:40:44
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## With probability $o(1)$

I am not exactly sure just how to read little/big O expressions in probability theory: What does a declaration like "with probability $1-o(1)$" suggest? Does it suggest with high probability?
2022-07-24 03:47:12
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## Compactly supported function whose Fourier transform decays exponentially?

It is popular since a function can not be compactly sustained both on the room side and also the regularity side (so - called unpredictability concept). On the various other hand a function can have rapid degeneration on both sides, as an example guassian function $e^{-x^2}$. My inquiry is whether the intermediate instance exists. Extra specific...
2022-07-24 03:36:51