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

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11
votes
0answers
373 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
8
votes
0answers
121 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
8
votes
0answers
109 views

Given the first $n$ primes, find the least common multiple of $p_1 - 1$, $p_2 - 1$, …, $p_n - 1$

Given the first $n$ primes, we can label the $k$th prime as $p_k$. So, what is the least common multiple(LCM) of {$p_1 - 1$, $p_2 - 1$, $p_3 - 1$, ..., $p_n-1$}? In other words, if we subtract $1$ ...
8
votes
0answers
159 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by ...
8
votes
0answers
187 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega ...
7
votes
0answers
109 views

Heat equation asymptotic behaviour 2

Let $D$ be the domain defined as $D := \{ (x,t): t \in [0,1) , \; x < (1-t)^\alpha \}$. Let $u(x,t)$ satisfy the heat equation $u_t = \frac{1}{2}u_{xx}$ in $D$, with initial condition: ...
7
votes
0answers
51 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
7
votes
0answers
148 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
7
votes
0answers
257 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
7
votes
0answers
78 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
6
votes
0answers
43 views

Comparing asymptotic forms of series

I've run into some asymptotic analysis in research here and there and largely it feels pretty magical to me. My research has led me to consider the following question which I haven't the slightest ...
6
votes
0answers
53 views

Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
6
votes
0answers
121 views

Asymptotic behavior a recursion involving min/max

Usually when I face solving recursions I use generating functions but I'm not aware of any "tools" to use when min/max expressions are involved. For example, I have the following recursive term: ...
6
votes
0answers
95 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
votes
0answers
208 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I don´t know how to solve this and as far as could see, this ...
6
votes
0answers
263 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
5
votes
0answers
109 views

Boundary Layer, leading order, Pertubation Theory, Differential Equations

I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.: Find the leading order of the problem: $\varepsilon ...
5
votes
0answers
56 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
5
votes
0answers
125 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
5
votes
0answers
104 views

Binomial asymptotic.

Is there any "direct" proof of the following asymptotic inequality: let $\alpha\le 1$ and consider $$Q_n(x)=\sum_{k=1}^n\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+k-1)}{k!}x^k$$ Then, $$\int_0^1 ...
5
votes
0answers
185 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
4
votes
0answers
35 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
4
votes
0answers
71 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
4
votes
0answers
129 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
4
votes
0answers
56 views

A general question of asymptotics

I am very desperately longing to know if there is a explicit relationship between $$F(n)=f(1)+f(2)+...+f(n)$$ and $$G(x)=\sum_{k=1}^{\infty}f(k)x^k$$ Assuming we can let $f$ be a sufficiently well ...
4
votes
0answers
66 views

Looking for a closed form for the quotient of a sequence of compositions of $\exp()$-function

Related to that previous question I have another still open detail problem. Consider the sequence of evaluations at some given $x$ $$ \small \begin{array} {} z_0 &=& e^x \\ z_1 ...
4
votes
0answers
226 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
4
votes
0answers
73 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
198 views

How many edges does an Erdős-Rényi graph have to have, to almost surely have a component with multiple cycles?

An Erdős-Rényi graph is a random graph, selected according to the distribution obtained one where we have some number $n$ of nodes, and some probability $p$ of each potential edge being ...
4
votes
0answers
272 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
4
votes
0answers
79 views

Dominated convergence on $e^{-n^2 t} t^{s/2-1}$

I am trying to apply the Dominated Convergence Theorem to show that $$\sum_{n\ge 1} \int_0^1 e^{-n^2 t} t^{s/2-1}dt= \int_0^1 \sum_{n\ge 1}e^{-n^2 t} t^{s/2-1}dt$$ as soon as $s>1$. I've ...
4
votes
0answers
553 views

When do floors and ceilings matter while solving recurrences?

I came across places where floors and ceilings are neglected while solving recurrences. Example from CLRS (chapter 4, pg.83) where floor is neglected: Here (pg.2, exercise 4.1–1) is an example ...
4
votes
0answers
241 views

How fast does $\binom{n}{k}$ grow when $k \le n/2$?

How fast does $\binom{n}{k}$, $n$ fixed, grow when $k \le n/2$? Especially, I'm interested in the growth of the "inverse" of binomial coefficient $B_n(x) := \min \{k:\binom{n}{k} \ge x\}$. EDIT: ...
4
votes
0answers
109 views

Asymptotic Analysis of Complex Integrals

Let $\gamma\subset B(0,2)\subset \mathbb{C}$ a smooth Jordan curve envolving the origin and $\phi:\mathbb{C}\setminus (-\infty,0)\to \mathbb{C}$ an analytic function having finite boundary values, ...
3
votes
0answers
28 views
+50

Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
3
votes
0answers
85 views

Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where ...
3
votes
0answers
50 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
3
votes
0answers
50 views

Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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 ...
3
votes
0answers
50 views

Asymptotic expansion of a Fourier Transform as $\omega\rightarrow 0$

First of all, I do apologise if the question is not formulated in precise mathematical terms, but as a physics student I lack a formal background on rigorous functional analysis. Suppose we have a ...
3
votes
0answers
38 views

Merten's function

I am tasked with applying the Wiener-Ikehara Theorem to achieve a bound of little o(x) on Merten's function $\sum_{n=1}^x \mu (n)$. My problem is the Wiener-Ikehara Theorem applies to Dirichlet series ...
3
votes
0answers
36 views

Applying function to both sides of asymptotic expression

I apologize in advance if this has been asked elsewhere, but I couldn't find it. This seems like it should be a pretty simple question, but I'm drawing a blank. If you know that $f(x) \sim g(x)$, ...
3
votes
0answers
34 views

Proof $\mathcal{O}(f(n)) = \mathcal{O}(g(n)) \iff f(n) \in O(g(n)) \land g(n) \in \mathcal{O}(f(n))$

There is an exercise that ask me to prove this logic formula about the complexity of algorithms: $\mathcal{O}(f(n)) = \mathcal{O}(g(n)) \iff f(n) \in O(g(n)) \land g(n) \in \mathcal{O}(f(n))$ ...
3
votes
0answers
47 views

How does the Stokes phenomenon appear in the asymptotic expansion of $\int_0^\infty \frac{e^{-zt}}{1+t^4} dt$ for $z \to \infty$?

Consider the asymptotic $z \to \infty$ behaviour of the function $$ \tag 1 I_1(z) \equiv \int_0^\infty \frac{e^{-zt}}{1+t^4} dt.$$ This converges for $\Re(z) > 0$, and the asymptotic expansion $$ ...
3
votes
0answers
102 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
3
votes
0answers
33 views

Computing the asymptotic spectrum of a negative distance kernel

Consider the following integral operator: $$K(f) : x \mapsto\int_0^1 K(x,x')f(x') dx', \quad \text{where} \quad K(x,x') = - |x-x'|^{3/2}.$$ The kernel is sometimes referred to as a negative ...
3
votes
0answers
87 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
3
votes
0answers
110 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
3
votes
0answers
74 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...