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Questions tagged [oscillatory-integral]

For questions about definite integrals $\int_a^b f(x)\,dx$ where the integrand is oscillating, often of the form $f(x) =g(x) e^{i h(x)}$.

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Asymptotic of integral involving a theta function [closed]

I would appreciate some help with theta functions. Consider $\theta_3(u,q)$: $$ \theta_3(u,q) = 1 + 2 \sum_{n = 1}^{+\infty} q^{n^2} \cos(2 n u) $$ I am interested in the asymptotic of the following ...
Cozy's user avatar
  • 1
2 votes
1 answer
69 views

Intuitively, why does $I(\lambda)$ decay as $\lambda \to \infty$ if $\Phi$ is not constant?

I'm quoting a few lines from Sogge's Fourier Integrals in Classical Analysis. Stationary phase is of central importance in classical analysis since integrals of the form \begin{equation} I(\lambda) = ...
stoic-santiago's user avatar
5 votes
1 answer
201 views

The asymptotic of an integral $I$

Consider the integral $$ I(\lambda)=\int_0^1 \frac{1}{\sqrt{v}}\,\left( \int_{-\infty}^{+\infty} \frac{e^{i\lambda u (u^2-v)}}{\sqrt{u^2+ 4v}}\,\varphi(u,v)\, du\right) dv, $$ where $\varphi\in C_0^\...
cbi's user avatar
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3 votes
4 answers
127 views

The oscillatory integral $\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z \sqrt{b + x}) - \cos(z \sqrt{b - x})}{x} dx \right| < \infty$

I am trying to prove the boundedness of certain oscillatory integrals and I can not deal with the following situation, which I have reduced to a specific example. I claim that $$ \sup_{b, z > 0} \...
Robert Wegner's user avatar
0 votes
0 answers
34 views

Integrating products of many oscillating functions

I'm working with the circular random matrix ensembles, in particular the circular unitary ensemble (CUE). For unitaries drawn from the CUE with dimension $N$, the distribution of its eigenvalues is ...
miggle's user avatar
  • 285
1 vote
1 answer
67 views

Oscillatory integral and Riemann integral

Consider the summation, with parameter $a \ge 0$ and non-negative integer $M = 1, 2, 3...$ $$S (a, M) = \sum_{m=1}^M \frac{2}{1+M} \sin\left( \frac{\pi m}{M+1}\right) \sin\left( \frac{\pi m M}{M+1} \...
Nigel1's user avatar
  • 655
0 votes
0 answers
16 views

Characterisation of what can be written as an oscillatory integral?

Let $\chi$ be a $C_c^\infty$ function which equals $1$ on $|x|<1.$ Define $$ I_{\Phi, \epsilon} = \int e^{i\Phi(x,\theta)}a(x, \theta) \chi(\epsilon \theta) d \theta $$ for a phase function $\Phi$ ...
Ma Joad's user avatar
  • 7,534
0 votes
2 answers
76 views

inequality $|x + 3\eta^2| \geq |x| + |\eta|^2$ [closed]

I'm learning the basics of Fourier Analysis. Reading a book, I've found this inequality: " If $x\geq -3$ and $|\eta | >2$ we have $|x + 3\eta^2| \geq |x| + |\eta|^2$. I've tried to solve this ...
N230899's user avatar
  • 115
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0 answers
19 views

Next term in the Stationary Phase Lemma expansion in dimension 2

Consider two functions $f\in\mathcal{S}(\mathbb{R}^2)$ and $\phi\in C^\infty(\mathbb{R}^2)$ satisfying that $\nabla\phi(x_0,y_0)=0$ and $\mathrm{det}\text{ }\mathrm{Hess}\text{ }\phi(x_0,y_0)\neq 0$, ...
W2S's user avatar
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0 answers
42 views

Applications of highly oscillatory integrals

I was reading a series of articles on numerical integration of highly oscillatory functions, e.g., S. Olver, Numerical approximation of highly oscillatory integrals S. Xiang, H. Wang, Fast ...
Vl F's user avatar
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2 votes
1 answer
61 views

$L^1$ norm of spherical/circular Dirichlet kernel

I'm currently studying a particular Fourier multiplier and I came across the following question. The cubic $d$-dimensional Dirichlet kernel is \begin{equation} D_n(x)=\prod_{i=1}^d D_n^1(x_i), \end{...
Francesco_Trig's user avatar
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0 answers
42 views

Oscillatory Integrals near the Riemann singularity

The question comes from E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, we concerned about the highly oscillatory distribution $$ D(x)=\mathrm{p.v.} \...
InnocentFive's user avatar
1 vote
1 answer
59 views

Numerically compute an oscillating series

I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
knuth's user avatar
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8 votes
0 answers
111 views

Gronwall lemma with highly oscillatory kernel

As a toy model for a larger problem, I want to show that if $A,B\geq 0$ and $u(t)$ satisfies $$|u(t)|\leq A+\left|\int_0^t B\cos(s^2)u(s)ds\right|$$ then $u$ satisfies a bound like $$|u(t)|\leq AC$$ ...
kieransquared's user avatar
2 votes
0 answers
23 views

Resource recommendation: multidimensional stationary phase method on polygons

Consider the oscillatory integral of the form $$I(\lambda) = \int_D a(\mathbf x) e^{i\lambda \mathbf x^T A \mathbf x} d\mathbf x,$$ where $D\subset \mathbb R^n$ is a box (or more generally a polygon), ...
Laplacian's user avatar
  • 2,108
0 votes
1 answer
95 views

Dirac delta doublet function in simple harmonic oscillation. Conditions imposed?

I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative of dirac delta) at t=0. $$f = \delta'(t)$$ I've already considered the case for a dirac ...
zzz's user avatar
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0 votes
1 answer
109 views

Upper bound of an oscillatory integral

Suppose that $\mathrm{f}:[\mathrm{a}, \mathrm{b}] \rightarrow \mathbb{R}$ is a smooth, convex function, and there exists a constant $\mathrm{t}>0$ such that $f^{\prime}(x) \geq t$ for all $x \in(a, ...
Snowball's user avatar
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1 vote
0 answers
95 views

Integrals with massive amount of cancellation (positive and negative portions of integrand cancel out almost perfectly)

In Gamelin's Complex Analysis, there are exercises/examples (pg. 201-202) of the form $$\int_{-\infty}^\infty \frac{P(x)}{Q(x)} \cos(ax) dx$$ The first "trick" is to use complex analysis: ...
D.R.'s user avatar
  • 8,945
2 votes
3 answers
198 views

Simple harmonic oscillator position function

We know from Hooke's law $$F=-kx $$ and $$ md^2x/dt^2 = -kx$$ therefore $$x''+w^2x=0$$ we must get $$x(t) =A\cos(wt)$$ but I don't know how I know how to derive the position function from graph, but ...
mark's user avatar
  • 105
1 vote
0 answers
41 views

Prove that the nonzero solutions to $x^{\prime \prime}+(1+\cos (t \sin (t))) x=0$ are oscillatory

Prove that the nonzero solutions to $x^{\prime \prime}+(1+\cos (t \sin (t))) x=0$ are oscillatory, that is $x(t)$ has infinitely many zeros. I think that I have to use Sturm Picone's Comparison ...
Ri-Li's user avatar
  • 9,098
0 votes
0 answers
107 views

Integral of functions that have oscillating discontinuous points(not finite) aren't differentiable?

I know that integral of removable discontinuous functions are differentiable but jump discontinuous aren't. However, When 2xsin(1/x)-cos(1/x) is integrand, which is derivative of x^2sin(1/x) has no ...
HYUN-HO WOO's user avatar
2 votes
1 answer
235 views

Estimate of an oscillatory integral in Stein's book

In Stein's book "Harmonic Analysis, Real-variable methods, orthogonality and oscillatory integrals", the author claims on page 335 that if $\eta\in C^{\infty}_{c}(\mathbb{R})$ and $l\geq 0$, ...
Sqrt's user avatar
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1 vote
0 answers
53 views

$\int_{-\infty}^\infty e^{i\lambda x^2} x^l e^{-x^2}\, dx \sim \lambda^{-(l+1)/2} \sum_{j=0}^\infty c_j^{(l)} \lambda^{-j}$ for even $l\in \mathbb N$

We aim to show $$\int_{-\infty}^\infty e^{i\lambda x^2} x^l e^{-x^2}\, dx \sim \lambda^{-(l+1)/2} \sum_{j=0}^\infty c_j^{(l)} \lambda^{-j} \tag{9}$$ for even non-negative integers $l$, as done in ...
stoic-santiago's user avatar
1 vote
1 answer
92 views

$O(\lambda^{-1/k})$ estimate for $\int_a^b e^{i\lambda \phi(x)}\, dx$ given $|\phi^{(k)}(x)| \ge 1$ for fixed $k$

In what follows, $\phi$ is a single-variable real-valued function. The excerpt is from Chapter 8, of Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. ...
stoic-santiago's user avatar
1 vote
0 answers
35 views

Integral of mean square integral (coming from the circle method)

Question: I want to bound \[ \sum _{q\leq \gamma }\int _{\pm 1/q\gamma }\left |\int _{x}^{2x}e(u\beta )u^{iT}du\right |^2d\beta \] or maybe just \[ \int _{\pm \delta }\left |\int _x^{2x}e(u\beta )u^{...
tomos's user avatar
  • 1,662
0 votes
0 answers
98 views

Is there more than one way to derive an energy function from differential equations?

I don't know how this energy function (screenshot below) comes from the oscillator equation. I know you can get it from $E =\frac{{\dot x}^2}{2} - \int \ddot x (x)dx$, which is conservative (meaning $...
user3146's user avatar
  • 695
3 votes
2 answers
99 views

Harmonic series with sign alternates every $n$ terms.

Let $A(1)=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$ Let $A(2)=\frac{1}{1}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\dots$ Let $A(3)=\frac{1}{1}+\...
Hussain-Alqatari's user avatar
2 votes
0 answers
28 views

Separating a harmonically driven ODE into oscillating and non-oscillating components

Consider the first-order ODE $$ x'(t) = f(t) \cos(\omega t) - \gamma x(t) $$ whose exact solution is $$ x(t) = x(0) e^{-\gamma t} + \int_0^t e^{-\gamma (t-t')} \cos(\omega t') f(t')~dt' $$ For simple ...
Endulum's user avatar
  • 276
0 votes
0 answers
25 views

Show that x(t) is bounded by |x(t)|≤ 2A/βb

Assume that the system is underdamped, starts from rest, and the force is bounded (|f(t)| < A for all t). Given the formula for the oscillator: $$ x(t)= \frac{1}{\beta m}\int_{0}^{t}f(t-v)e^{-b/2m}...
Grant Ballard's user avatar
4 votes
0 answers
230 views

The direction of the steepest descent path at the saddle point (Picard-Lefschetz theory)

I am having to perform oscillatory integrations like $e^{iS}$ using Picard-Lefschetz theory. One can write this as $e^{h+is}$ where $h(x,y)=-{\rm Im}(S(x,y))$ is the Morse function. To perform these ...
Faber Bosch's user avatar
5 votes
2 answers
158 views

Is subset of $L^1$ functions bounded by $1 / x^2$ compact?

I was given the following exercise Given $E \subset L^1([1, +\infty))$ as follows $$ E = \left\{ f \in L^1([1, +\infty)) \;\middle|\; \forall x \in [1, +\infty) \quad |f(x)| \leq \frac{1}{x^2} \right\...
aziis98's user avatar
  • 145
1 vote
1 answer
70 views

Is the estimate $\int_0^\infty \frac{e^{ikx}}{x^\alpha} f(x)dx = o\left(\frac{1}{k^{1-\alpha}}\right)$ true?

Let $0<\alpha<1$. First, we have an estimate $$\int_0^\infty \frac{e^{ikx}}{x^\alpha} dx = O\left(\frac{1}{k^{1-\alpha}}\right), \quad k\to\infty,$$ obtained by substituting $kx=x'$. It is ...
Laplacian's user avatar
  • 2,108
1 vote
0 answers
100 views

Fourier transform of the Bochner-Riesz multipliers

How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define: $\hat{m_{\lambda}}(x)=\int_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \xi}d\xi$, ...
Dapao Zhang's user avatar
1 vote
0 answers
27 views

Finding 2nd Order Linear ODE for known signal of summed oscillators starting at different times?

I have a signal which is a sum of $N$ cosines that oscillate between 0 and 1 and that start at different times. More specifically, I have $$N(t) = \sum_{n=1}^N H(t - t_n) \Big[\frac{1}{2} + \frac{1}{2}...
Rylan Schaeffer's user avatar
4 votes
1 answer
443 views

Convergence (?) of an improper oscillatory integral

Is the integral $$I = \int_0^\infty \text{d}x \, \exp\left( i x^2 \right)$$ well-defined? I would think it is, since I understand this object as the limit $$\lim_{a \rightarrow \infty} I(a)$$ where $$...
Latrace's user avatar
  • 308
1 vote
2 answers
53 views

Find $\lim_{m \to \infty} \int_{-\infty}^{\infty} \sin(e^t) e^{-(t - m)^2} dt$

I would like to know if it's the case that $$\lim_{m \to \infty} \int_{-\infty}^{\infty} \sin(e^t) e^{-(t - m)^2} dt = 0.$$ For each fixed $t$, the integrand approaches zero as $m \to \infty$, so one ...
JZS's user avatar
  • 4,904
0 votes
0 answers
38 views

Complex integration of an oscillatory function

I've been trying to find $\int_{-\infty}^{\infty}e^{i(x^{3}+y x^{2}+y^{2} x)}dx$ and $\int_{-\infty}^{\infty}x e^{i(x^{3}+y x^{2}+y^{2} x)}dx$. I've been doing the integral numerically but I'm not ...
123infinity's user avatar
1 vote
1 answer
179 views

A consequence of Van der Corput Lemma

Let $\phi (x) = x^2$, $k,\lambda\in \mathbb{R}$, and $[a,b]\subset\mathbb{R}$ and define the integral $$I(\lambda;k)=\int_{a}^{b}e^{i(\lambda\phi(x)-kx)}dx$$ I am trying to prove that $|I(\lambda;k)|\...
medvjed's user avatar
  • 145
0 votes
1 answer
165 views

What is the heuristic 'integration by parts' in computing $\int_{\mathbb R} x e^{ix\xi} d\xi = 0$?

This question is about heuristics to understand with the tempered distribution given by the oscillatory integral $\int xe^{ix\xi} d\xi$ (all integrals are over $\mathbb R$). Source is M. Zworski's ...
Calvin Khor's user avatar
  • 35.1k
0 votes
1 answer
517 views

Using the Riemann-Lebesgue lemma to find the integral of a very oscillating function

I'm a bit confused with how the Riemann-Lebesgue lemma is used to show that the integral of a very oscillating function approaches zero as the number of oscillation increases. To be more specific, let ...
zola's user avatar
  • 49
-2 votes
1 answer
201 views

How to integrate sinc function numerically [closed]

How to compute (numerically) $$ F(x) = \int_{-\infty}^x \dfrac{\sin(t)}{t} dt $$
user2052436's user avatar
3 votes
1 answer
138 views

How to do a fast numerical computation of an oscillatory integral including $\operatorname{HeunC}$ function using Mathematica?

I am trying to numerically compute the following integral in Mathematica $$\int_{1}^{1000} dx \,(x+2)e^{-2Ia(x+2)}\operatorname{HeunC}[-4Ib,-4Ib,1+4Ib,1,-4Ib,-x/2]$$ where $\operatorname{HeunC}$ is ...
HadamardN2's user avatar
7 votes
1 answer
388 views

Prove that the integral of a rapidly oscillating function is $0$

Claim: If $n$ is a nonzero integer, then $$\int_{-\pi/2}^{\pi/2} \exp[{2in(x+\tan x)}] \ \mathrm{d}x = 0$$ Using Euler's identity $e^{ix}= \cos x + i \sin x$, the fact that $\sin$ is an odd function ...
Siupa's user avatar
  • 337
2 votes
1 answer
105 views

What does it mean by $f$ stays in a bounded set in $C^{k+1}(X)$?

I am trying to understand the statement of Theorem 7.7.1 in The analysis of linear partial differential operator I by Hormander. What exactly is meant by $f$ stays in a bounded set in $C^{k}(X)$? I ...
Johnny T.'s user avatar
  • 2,913
2 votes
2 answers
224 views

Decay of fractional derivative of Schwartz function

Let $\phi$ be a Schwartz function and let $\alpha>0$. I want to analyze the decay as $x\rightarrow\infty$ of: $$\int_\mathbb{R}e^{2\pi i x\xi}|\xi|^\alpha\phi(\xi)\,d\xi$$Heuristically, for $\xi\ll ...
user293794's user avatar
  • 3,738
8 votes
2 answers
331 views

Oscillatory integral with absolute value

Suppose $f:[0,1]\to[0,+\infty)$ is a continuous function. Are there general conditions under which the following limit exists? $$\lim_{n\rightarrow \infty} \int_0^1 |\cos(nf(x))|dx$$ Obviously if $f$ ...
Cantor's user avatar
  • 2,010
0 votes
1 answer
63 views

Numerically computing an oscillatory integral

I want to numerically compute integrals of the form $$F(s) = \frac{1}{2\pi} \int_0^A \frac{e^{isu}}{1 + B\cos(r u)}\,du$$ for fixed $A,r>0, 1 > B >0$ for a finite number of values of $s\in \...
Erdberg's user avatar
  • 57
0 votes
0 answers
45 views

How to check the following inequality?

Assume $\chi_+$ is a smooth cut-off function away from origin, more precisely, it is defined as below: \begin{equation} \chi_+(\lambda)= \begin{cases} 1, & \lambda - \mu > 2\lambda_1, \\ 0, &...
Tao's user avatar
  • 353
0 votes
2 answers
90 views

How to integrate an integral which contain an oscillatory term $e^{\imath x\cdot p}$ [closed]

I need to evaluate a integral $$f(p)=\int_0^\infty dx \frac{x^2}{2\sqrt{x^2+a^2}}e^{-\imath x\cdot p}, $$I tried a lot, but unable to find a method to integrate due to the presence of oscillatory term ...
QFT addict's user avatar
0 votes
0 answers
67 views

Asymptotic expansion for $\int e^{i\lambda x^2}e^{-x^2}x^j\text{d}x$

I want to prove Theorem 2.5 on page 9 from these lecture notes by following the guidelines provided in Exercise 13 on page 18. Using complex integration one shows that \begin{equation}\int_{\mathbb{R}}...
LordOfNumbers's user avatar