# Questions tagged [oscillatory-integral]

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### 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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### 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 ...
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### 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 ...
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### 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 $R>1$. ...
3answers
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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