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

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17
votes
2answers
872 views

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

How to show that $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$ No clue at all. Need help! Appreciated!
0
votes
1answer
15 views

$ n - \sqrt{n}$ $\Theta$ Complexity

$ n - \sqrt{n} \leq n - \sqrt{n} + \sqrt{n}=n=O(n)$ But I don't know what I should do about $\Omega(.) , \Theta(.)$ Should I try to solve it with lim?
5
votes
1answer
59 views

Decay of Fourier Transform

I encountered the following statement, and I cannot see why it is true(if it is). Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: ...
0
votes
0answers
14 views

Additive error in Stirling's Approximation

I know that Stirling's Approximation is asymptotic to $n!$ as in the ratio approaches $1$ (which is the definition). But as far as I have noticed, the additive error diverges. Is this a common ...
3
votes
1answer
29 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
4
votes
3answers
654 views

Formally proving that a function is $O(x^n)$

Say I have a function \begin{equation*} f(x) = ax^3 + bx^2 + cx + d,\text{ where }a > 0. \end{equation*} It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can ...
13
votes
1answer
262 views

Asymptotic expression of $\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi$

How to derive the following asymptotic expression ($|\omega| \ll D $)? $$P.V.\int_{- D}^{D} d\xi \frac{\tanh(\beta \xi)}{\xi -\omega} \approx 2 \ln\left(\frac{D}{\sqrt{\omega^2+T^2}}\right),\ \ \ ...
0
votes
0answers
41 views

Node potentials of minimum cost flow successive shortest path algorithm

I have a simple directed graph $G(V,E)$ that has a source $s$ and sink $t$. Each edge $e$ of $G$ has positive integer capacity $c(e)$ and positive integer cost $a(e)$. I am trying to find the minimum ...
0
votes
0answers
27 views

Prove or provide a counter example: $f(n) \notin o(g(n))$ and $f(n) \notin \omega(g(n)) \implies f(n) \in \theta (g(n))$

$f(n) \notin o(g(n))$ and $f(n) \notin \omega(g(n)) \implies f(n) \in \theta (g(n))$ Without giving me the answer, please tell me how would you manipulate this? I know that $f(n) \in \theta (g(n)) ...
3
votes
2answers
43 views

Limit with polylog

How do you show the following limit? $$\lim_{x\to\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac12x^2=\frac{\pi^2}3$$ Where $\operatorname{Li}_n(x)$ is the polylogarithm. This question is ...
0
votes
0answers
34 views

Asymptotic analysis of Integrals of powers of sine and their application to intersections of hyperspheres

I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the ...
4
votes
1answer
70 views

Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?

$$\int_0^m \frac{n^x}{\Gamma(x+1)}dx:n,m \in \mathbb{R}$$ I'm dubious as to whether there's a closed form for the above, if there is I'll be very happy. Otherwise: Is there a closed form for ...
0
votes
1answer
23 views

comparing expressions confusion

This formula is actually from a big $O$ notation example, but I am confuse about the mathematical formula. I read that: if $n$ and $c$ are $1$, $3n^2 - 100n + 6$ is not a big o of $n^3$ or $cn^3 ...
15
votes
1answer
151 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
3
votes
1answer
97 views

$\sum \limits_{n \geq 0}a_n \frac{x^n}{n!}=e^{x+x^2/2}$ implies $a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 }$

Prove the following asymptotic formula for the exponential generating function coefficients of $e^{x+x^2/2}$: $\; \; a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 }$ Stanley ...
1
vote
0answers
37 views

Using singularity analysis to find the main asymptotic term of the Catalan Numbers

Using singularity analysis to find the main asymptotic term of the Catalan Numbers \begin{align} C_n = [z^n]\frac{1-\sqrt{1-4z}}{2z} \end{align} Can someone please explain to me the general concept ...
2
votes
0answers
40 views

asymptotic expansion of hermite functions

Does anybody know how to proof the first asymptotic expansion of this page: http://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion ? (and how the physicist use this asymptotic ...
1
vote
2answers
31 views

Is this inductive big O proof possible / Does this question make sense?

Prove that $\sum_{i=j}^k \frac 1i$ is $O(\ln(k)-\ln(j-1))$ using induction for all $i$. The way I understand this question, it's nonsense - $i$ is the iteration variable, not something that can be ...
0
votes
2answers
30 views

Analysis of Algorithms - Big O Notation Equivalences - Limits

Please see below block question from review for test. True Or False? Justify Your answers A) is 2^(n+1) = O($2^n$) B) is 2^2n = O($2^n$) C) is log($n^2$) = O(logn) D) is ...
6
votes
1answer
3k views

Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
0
votes
0answers
28 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
0
votes
0answers
10 views

WKB for a sixth order eigenvalue problem

I have the following 6th order eigenvalue problem: $$ (D^2 - \alpha^2)^3 y(x) = -\alpha^2 \lambda Q(x) \, y(x), \quad 0 < x < 1, \quad \text{+ BCs}, $$ where $D = \mathrm{d}/\mathrm{d} x $, ...
2
votes
1answer
114 views

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$. I have already proved that $1/\zeta(s) = \sum_{m=1}^{\infty} \mu (m)/m^s$. But how do I show that if $k\ge 1$, $1/\zeta (2k) = ...
3
votes
2answers
71 views

Question about steps/derivation regarding Laplace method.

I am reading something on the Laplace method of integrals and I don't understand part of it's argument. It gives the integral $$\int_{-3}^4 e^{-\lambda x^2}\log(1+x^2)dx$$ and finding the leading ...
0
votes
0answers
37 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
-1
votes
1answer
200 views

Asymptotic behavior of the solution of $x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$ near $0$

Can you help me find the leading asymptotic behaviors about the irregular singular point $x=0$ of $$x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$$ So far I have got $y(x) = ...
1
vote
1answer
38 views

Interpreting little-$o$ notation

This is the integrand of a complex integral: $$\frac{o(\zeta - z)}{\zeta - z}$$ The ensuing discussion says that this can be made as small as desired [by confining $\zeta$ close to $z$]. In general ...
3
votes
3answers
69 views

Wouldn't each addition take time $O(n)$?

I am going over the asymptotic runtime of regular matrix multiplication. Here is a lecture slide I am referencing(too much to type out, shown below), from Algorithms Everything makes sense up ...
1
vote
0answers
24 views

help with an asymptotic for a certain product

I'm having difficulty finding an asymptotic formula for the following product: $$ k^{\alpha}\prod_{1 \leq i \leq N \atop i \neq k} (i^{\alpha}-k^{\alpha})$$ where $N$ is a parameter tending to ...
0
votes
1answer
16 views

Asymptotic expansion for the inverse of a matrix-valued function

suppose I have an asymptotic expansion for a matrix-valued function $\psi : \mathbb{C} \to \mathbb{C}^{2 \times 2}$ : $$\psi(\lambda) \sim I + \frac{m_1}{\lambda} + \frac{m_2}{\lambda^2} + \cdots \ \ ...
5
votes
1answer
128 views

Integral asymptotic expansion of $\int_0^{\pi/2} \exp(-xt^3\cos t)dt$ as $x \to \infty$

I have the integral $$I(x)=\int_0^{\pi/2}\exp(-xt^3\cos t)dt$$ and I want to derive the first two terms in the asymptotic expansion for $x\rightarrow \infty$, which should give me ...
14
votes
2answers
496 views

Asymptotic integral expansion of $\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$ for $x \to \infty$

I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like: ...
2
votes
0answers
21 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
1
vote
1answer
74 views

Find upper bound time complexity of recurrence function using iterative method

I want to find the upper bound time complexity of this function. I know how this is done using the induction method, but I can't find clear steps on how to solve it using the iteration method. ...
0
votes
1answer
13 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
3
votes
0answers
19 views

Big-O Notation exponentials

I'm learning about Big-O notation for algorithm runtime, and I need some help understanding one part. I read that for the constant, c, does not matter as the function increases rapidly. Does that ...
0
votes
0answers
27 views

how to estimate $\sum (n,i)$?

How to estimate $\sum\limits_{i=-A}^{A} \binom{n}{i}$ ? This probably relates to the central limit theorem and the proof of it. But I want good estimates, also for small values (not just limits to ...
1
vote
1answer
46 views

Approximation of combination $ {n \choose k} = \Theta \left( n^k \right) $?

Is it a valid to say $$ {n \choose k} = \Theta \left( n^k \right) $$ for any $n$ and $k$? If so, how to prove it? Note: $k$ is not a function of $n$. Note: Observed it here (page 5): ...
1
vote
0answers
21 views

Prerequisites to reading *Convergence of Probability Measures* by Patrick Billingsley.

I want to improve myself in asymptotic theory regarding the realm of probability. I tried reading Convergence of Probability Measures by Patrick Billingsley but right off the bat the De ...
4
votes
1answer
94 views

Method of Steepest descents integral

I am looking to evaluate the following asymptotic integral: Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of steepest descents ...
1
vote
1answer
31 views

Elementary proof of an estimate on $\zeta$

I saw somewhere that If $s=\sigma +it$ where $\sigma >0$ and $t\in \mathbb R$,$x\geq |t|/\pi\implies \zeta(s)=\displaystyle \sum_{n\leq x} \frac{1}{n^s}+\frac{x^{1-s}}{s-1}+O(x^{-\sigma})$ ...
1
vote
0answers
34 views

A combinatorial way to understand $\sum \log^2 n $

Stirling's formula has many derivations using the factorial function: $$ \log N! = \sum_{n=1}^N \log n = \sum_{n=1}^N \sum_{m=1}^n \bigg( \log m - \log (m-1) \bigg) = \sum_{n=1}^N \sum_{m=1}^n - ...
3
votes
1answer
266 views

A proof of Stirling's Formula

I need to gain understanding of a proof of Stirling's formula. I have read through Tim Gowers' and Terence Tao's but I'm struggling to follow them. How rigorous is this proof, if at all? Thank you. ...
0
votes
0answers
42 views

Approximation of integral as integral range tends to 0

I would like to approximate $\int_{0}^{x}t^{-2}e^{t}\mathrm{d}t$ (maybe find the first two terms) as $x\rightarrow0$. I can't seem to do it by "divide and conquer" or any method. Any suggestion would ...
0
votes
0answers
10 views

Code that Creates a Monotonic Array

I'm not sure whether this belongs on StackOverflow or here. Suppose I have an infinite (indexed from $-\infty$ to $\infty$) array of random doubles x[]. They are ...
0
votes
1answer
25 views

Stokes phenomenon of the Airy function

I am now trying to understand what the so-called stokes phenomenon means. In this page Stokes phenomenon, it reads that `` For large $x$ of given argument the solution (of the Airy equation)can be ...
0
votes
2answers
27 views

Finding the roots and the rescaling of an equation

This question is taken from Hinch's book on perturbation. I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$ I have found to possible ...
0
votes
1answer
59 views

What algorithm can sort the first sqrt(n) elements of an array in O(n) time?

I want to partially sort an array of $n$ elements to get the first $\sqrt{n}$ elements sorted, and it has to be done in $O(n)$ time. The complexity $O(n)$ seems to imply that it is necessary to go ...
0
votes
0answers
13 views

(Empirical likelihood method) Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the equation (1) \begin{equation} ...
2
votes
1answer
77 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...