Questions tagged [oscillatory-integral]

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17 views

Comparing the limit of symmetrical sequences of oscillatory integrals

Cosidering the limits as $\,n \, \rightarrow \, \infty$ of the following two sequences of integrals: $$ A_n \, = \,\int_{-\infty} ^{\infty} f(ix)\frac{n^{a+ix}}{a-ix}dx \; \; \; \; \; \; \; B_n \, = \...
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73 views

What does Hardy mean in this lemma?

I am concerned with the paper "Oscillating Dirichlet's Integrals" by G.H. Hardy (Quarterly Journal of Pure and Applied Mathematics, Vol. XLIV, pg 1-40). I don't understand Lemma 1, but I suppose this ...
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207 views

Lower Bound on Oscillatory Integral

Let $p,y\in\mathbb{R}^d\setminus\{0\},\beta>0$ be given and fixed and define for all $\alpha>0$, $$I(\alpha) := \int_{x\in\mathbb{R}^d}\exp(\mathrm{i}\alpha p\cdot x-\alpha\beta \|x-y\|^2)f(x)\...
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47 views

A highly oscillatory integral

I have an integral which I am supposed to solve numerically and can find an approximated answer to it. The integral in question is $$\int_{-\infty}^{\infty}\sin^2{\frac1{x}} \, \mathrm{d}x=\pi~.$$ I ...
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28 views

Gronwall's Lemma and Non-Stationary Phase Lemma

Suppose we have a function $f \in C^1$ satisfying $$f(t) = \frac{1}{\epsilon}\int_0^t \sin(s/\epsilon) f(t + \epsilon \sin(s/\epsilon))ds + \phi(t), \quad \epsilon \in (0,1]$$ Where $\phi$ is some ...
2
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0answers
79 views

How to use Van der Corput's lemma to get the following estimates?

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrodinger Equation, and the link of the article is there Spherically Averaged ...
10
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1answer
235 views

Approximating $\displaystyle \int_{-\infty}^{\infty}\frac{e^{R+i\,u}}{\ln(R+i\,u)}du$

Working with a special function related with reciprocal gamma function, I've found the following integral $$\int_{-\infty}^{\infty}\frac{e^{R+i\,u}}{(R+i\,u)\,\ln^2(R+i\,u)}du$$ with $R>1$. ...
8
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3answers
272 views

Fourier transform of $e^{-i\lambda\sqrt{1+x^2}}$ - asymptotics for $\lambda$?

As the title says: I want to compute the Fourier transform (in the distributional sense) of $f(x)=e^{-i\sqrt{1+x^2}}$, $x\in \mathbb{R}^n$ - say $n=1$ for the moment. I have no idea how to get it done:...
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68 views

Stationary phase approximation higher order

I want to evaluate the following integral for large $z:$ \begin{eqnarray} I(r,z)=\int_{0}^{\infty}d\rho e^{i\sqrt{1-\rho^2}z} J_0(r\rho)J_1(R\rho) \label{eq:fourier:integral} \end{eqnarray} I ...
2
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0answers
24 views

Analytic form of the infinite integration with oscillatory integrand?

I recently have a problem as follows $\int_0^\infty \cos \bigl[ { k \cdot t \over {\sqrt{1 + k^2}} } \bigr] \cdot \cos \bigl[ k \cdot x \bigr] \, dk$ Here, x and t are spatial and temporal constant....
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33 views

Stationary phase for retarded potentials in electromagnetism

I want to apply something like a stationary phase approximation to the following expression $\int_V d^3x' \frac{B(x')}{|x-x'|}e^{ik|x-x'|}$ with $x\in \mathbb{R}^3$, $k\rightarrow \infty$ and $B$ is ...
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0answers
66 views

Integration by parts for an oscillatory integral

I want to understand a certain integration by parts argument in Treves' book Introduction to Pseudodifferential and Fourier Integral Operators (It appears in the proof of Theorem 4.1). Here is a self-...
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1answer
40 views

Estimating an integrand

Given an integral $$\int_1^T \frac{f(t)}{t} \, {\rm d}t$$ where $f(t)$ is oscillating and I want to make an estimate I can do the following $$\left|\int_1^T \frac{f(t)}{t} \, {\rm d}t\right| \leq \...
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11 views

What is a proper setting for oscillatory integrals and fourier inversion for estimating?

My goal is to get the dispersive estimate of Schrodinger operator for free particle via the method of stationary phase; indeed $\|u(t,x)\|_{L^\infty_x}\le C_d t^{-d/2}\|u_0(x)\|_{L^1_x}$. When ...
1
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0answers
37 views

An asymptotic for a simple oscillatory integral

Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\infty}\int_{0}^{\infty} \psi(x,y) \, \mathrm{e}^{\dot{\imath}\phi(x,y)}dxdy$$ where 1) the phase $\phi$ and amplitude $\psi$ are smooth 2)...
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23 views

Oscillatory Cancellation of Integral

Under which conditions for a real function $f(x)$ is $$ \left| \int_{1}^{x} \left\{ f\left(\frac{x}{w}\right) - f\left(\left\lfloor \frac{x}{w} \right\rfloor\right) \right\} \cos\left(w\right) \, {\rm ...
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29 views

Oscillating function with periodic change of sign and substraction of values

I'm pretty close defining a function that meets my criteria for oscillation of set of integer values. But I'm missing something that you can probably easily fix. $$f(n) = (-1)^{\lfloor\frac{n}{6}\...
2
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2answers
113 views

Inverse of a Fourier Transform is $L^1 \cap L^{\infty}$

I was reading a Navier-stokes paper by Kato, and he affirm that if $k\geq0$ than $$\mathcal{F}^{-1}(| \xi |^ke^{-|\xi|^2})\in L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$$ (here $\mathcal{F}^{-1}$ ...
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0answers
66 views

Highly oscillating integral [closed]

I am trying to integrate a highly oscillatory integral. the integrand is a function of 4 variables: f1, f2, x2, y the integrand is ...
3
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1answer
50 views

Rigorous derivation of the long-time limit of oscillatory integrals

I am trying to estimate the following integrals in the limit $t\to+\infty$: $\displaystyle\int_{-\infty}^{+\infty}\mathrm d\omega\,f(\omega)\frac{1-\cos(\omega t)}{\omega^2}$ and $\displaystyle\int_{-...
1
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1answer
48 views

Is $\int_{A_t} \frac{1}{|x_j|^{p}}\, dx \leq \sum_{j=1}^{d} \int_{A_{j,t}}\frac{1}{|x_j|^p}\, dx \leq \int_{A_t} \frac{1}{|x|^p} \,dx$?

Let $t>0$ and $A_t= \{ x\in \mathbb R^{d}: |x|>t\},$ $A_{j,t}=\{x\in A_t: |x_j|> |x_l| \ \text{for all} \ l\neq j \}.$ Question: (1) Can we say $A_t \subset \cup_{j=1}^d A_{j,t}$? (...
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0answers
38 views

How to disprove an equality involving a double integral

I want to show that the following equality does not hold: \begin{equation}\label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
2
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0answers
65 views

Non-vanishing of K-Bessel function

I don't know much about the spectral theory of $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ nor do I know much about Bessel functions, hence the following question. Suppose $f$ is Maass form of ...
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34 views

Problem regarding to solve “6 nonlinear dynamic system of first order differential equation” analytically ( approximated )

I have some problem to solve, Problem is regarding to solve "6 nonlinear dynamic system of first order differential equation" analytically ( approximated answer):- 1) Can I solve this system of ...
1
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0answers
59 views

Double wells and triple wells quantum oscillators

The Hamiltonion for the double well and triple wells can be written in form $H=H_0-x^2+x^4$ and $H=H_0+x^2-x^4+x^6$, repectively. Can anyboody guide which "basis function" can be used to construct the ...
2
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2answers
90 views

Compute the main order asymptotics of the integral $\int_0^{\infty}e^{itx-e^x}dx$, as $\mid t \mid \to \infty$.

I have found this integral interesting because it does not fall into any of the Fourier/steepest descent/integration by parts methods to compute asymptotics! The Mellin transform of this function is ...
0
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0answers
45 views

Asymptotic analysis of $\int_0^{\infty}f(x)\cos(nx)dx$, as $n\to \infty$ where $f$ is a decaying-to-zero integrable function

This may be recognised as the real part of a certain Fourier transform, which in turn is known to $\to 0$ as $n\to \infty$, by the Riemann-Lebesgue lemma. From here i am interested in the main ...
1
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1answer
128 views

Evaluating $\int_0^\infty dq\, \frac{\sin(qr)}{r} \left(\frac{q}{\sqrt{q^2+m^2}}-1\right)$

I am trying to evaluate, at least in the limit $|x|\to0$, $$ F_m(r)\equiv\int \frac{d^3q}{(2\pi)^3}\left(\frac{1}{2\sqrt{\mathbf q^2-m^2}}-\frac{1}{2|\mathbf q|}\right) e^{i\mathbf q \cdot \mathbf x}\,...
4
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1answer
133 views

Asymptotic evaluation of an oscillatory integral

First of all, I am a physicist, so please excuse me if I make basic mistakes in the following, I will try to be as rigorous as possible. In my research, I recently came across the following integral ...
1
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1answer
103 views

Strange oscillations in Matlab

I have created a midpoint algorithm to solve 2nd order ODEs in Matlab. And now Im comparing my solver with built in - ode23s. I have used a harmonic motion described as a 2nd order ODE, the .m file ...
10
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1answer
156 views

Why is this integral is super-exponentially small?

Consider the integral $$I_n^{(a,b)} = \int_{-1}^1 (1-x)^a\,(1+x)^b\, P_n(x)\, dx,$$ where $P_n(x)$ is the $n$-th Legendre polynomial. Here's a plot of $|I_n^{(50,20)}|$ for $n=0,\dots,70$: (I just ...
0
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1answer
50 views

How to prove that arc-length of x cos(1/x) is divergent?

A function $$f(x)=x\cos\left(\frac1x\right), \,\, x \in (0,1) $$ And I want to prove that length of the graph of $f$ over the interval $(\alpha,1)$ is divergent as $\alpha\to 0$. I try to use a ...
2
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1answer
129 views

Simplifying oscillating limit-integrals by substituting Dirac Delta functions

I have to simplify an expression which looks vaguely like: $$\int_0^{\pi} e^{ir\sin{\theta}} F(\theta)\,d\theta$$ where $r$ is very very big. If $F(\theta)$ is asymptomatic, for very large $r$ the ...
2
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1answer
94 views

General theory of divergent, highly oscillatory integrals

I'm interested in evaluation of integrals of the form $$\int e^{i S(x)}O(x) dx,$$ where $S(x)$ is a polynomial of degree higher than one and $O(x)$ is a polynomial or more generally, polynomialy ...
3
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3answers
128 views

Nature of the series $\sum_{n\ge0}\int_0^{\pi/2}\cos^n(t)\sin(nt)\,dt$

Consider, for all $n\ge0$ : $$u_n=\int_0^{\pi/2}\cos^n(t)\sin(nt)\,dt$$ Does the series $\sum_{n\ge0}u_n$ converge ?
1
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0answers
110 views

Asymptotic of fourier transform of a oscillatory kernel

I am reading Harmonic analysis written by Stein. In page 426, there is a result about the asymptotic of the fourier transform of $\frac{e^{2\pi i |\xi|}}{|\xi|^\alpha}$ at the unit sphere. Suppose $\...
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2answers
53 views

Why does this sequence not oscillate?

I'm confused on why this sequence converges to 0 rather than diverge because of an oscillating series, since the result would be negative if n is odd and positive if n is even Edit for the downvote: ...
1
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1answer
2k views

Integrals of rapidly oscillating functions

Why is it the case that if the integrand of a given integral is rapidly oscillating, then all the contributions to the integral cancel out apart from where the function is stationary? For example, ...
1
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0answers
34 views

Fast numerical oscillatory integral with fixed support points.

I have to numerically evaluate $$\int_0^\beta d\tau e^{i \nu_n \tau} G(\tau)$$ for a large number on $\nu_n$. $G(\tau)$ is expensive to evaluate. Currently I'm using Quadpacks dqawo (double ...
0
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1answer
29 views

Prove the following inequality (Oscilations)

For $f,g: \mathbb{R}^n\to [-M,M]$, Prove: $Osc_{fg}\leq M(Osc_f+Osc_g)$ Where $Osc_f(U)= supf(x)-inf f(x)$ (for $x\in U\subset\mathbb{R}^n)$ I tried using the identity $4fg=(f+g)^2-(f-g)^2$ but ...
1
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4answers
72 views

Explain how second order differential equations of the form $\ddot{y}+y=0$ exhibit osciallatory dynamics

I'm trying to build a skillset for research in computational neuroscience (and loving math even more as I go along) and have just jumped into the world of differential equations – very simple ones. ...
2
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1answer
80 views

Decay property of oscillatory integrals in $\mathbb{R}^n$

We know that an oscillatory integral in $n$ dimensions is an integral of the form \begin{equation*} I(\lambda)=\int_{\mathbb{R}^n}e^{i\lambda\phi(x)}f(x)dx \end{equation*} where $\phi\in C^{\infty}$ ...
2
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1answer
374 views

An oscillatory integral

I would like to know if there is a way of expressing the integral $$G(a) = \int_{-\infty}^{\infty} \sin(ax)\frac{\sqrt{x^2 + 1}}{x} \; dx$$ in terms of known functions. Numerically I have observed ...
2
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1answer
114 views

Uniformly boundedness of an oscillatory integral

Let $f\in H^1(\mathbb{R}^3)$. Define, for $M>0$, $$I(M)=\int_{B(0,M)}e^{i|y|^2}f(y)dy$$ where $B(0,M)$ is the ball centered in the origin and of radius $M$ in $\mathbb{R}^3$. Is it true that $|I(...
1
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0answers
36 views

The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq C\...
2
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0answers
36 views

Singularity of an oscillatory integral

Given $x\in\mathbb{R}^3\setminus \{0\}$, consider the following integral: $$I(x):=\int_{\mathbb{R}^3}\frac{e^{-i|x-y|^2}}{|y|} \, dy$$ Now $I(x)$ diverges as $x$ approaches to $0$, and it seems to me ...
4
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0answers
115 views

Asymptotics of Null Solution to Heat Equation

In the book J. Rauch, Partial Differential Equations, the author claims that for $\alpha\in(1/2,1)$, the function $u$ defined by $$u(x,t)=\int_{-\infty}^{\infty}e^{-(i\tau)^{\alpha}}e^{x(-i\tau)^{1/2}}...
6
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2answers
190 views

How to get the asymptotic form of this oscilatting integral?

So the integral is like this: $$\int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\left|\frac{1-x}{1+x}\right|\right)^2+\pi^2\right]}\mathrm{d}x$$ The question is how to get the asymptotic form of ...