Linked Questions

81 votes
9 answers
13k views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

I have seen the Fresnel integral $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
Argon's user avatar
  • 25.3k
44 votes
7 answers
34k views

Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.

$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does. I'm trying to understand why and would like some help in working towards a formal proof.
Jozef's user avatar
  • 7,080
33 votes
5 answers
4k views

Computing the best constant in classical Hardy's inequality

Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327) If $p>1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then $$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^...
Giuseppe Negro's user avatar
4 votes
3 answers
179 views

What is the exact value of $\int^\infty_0\frac{\sin^2 x}{x^{5/2}}\,dx$

I would like to get the exact value of the following integral. $$\int^\infty_0\frac{\sin^2 x}{x^{5/2}}\,dx$$ I was able to prove the convergence as well. But I don't how to compute its exact value....
Guy Fsone's user avatar
  • 23.8k
3 votes
3 answers
173 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. $$\lim_{R\to\infty}\...
Thomas Russell's user avatar
4 votes
3 answers
187 views

Show that $\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$ for $n\ge 5.$

Question is to show that $$\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$$ when $n\geq5$, $x \in \mathbb R $ and $i$ is an imaginary unit. My effort: $$|\int_{-n}^{n}e^{iy^2}dy|\leq \int_{-n}^{n}|e^{...
Yulia23576's user avatar
3 votes
1 answer
344 views

A complex square root in the Schrödinger kernel

Consider the initial-value problem for the Schrödinger equation $$\tag{IVP} \begin{cases} i\frac{\partial u}{\partial t}+\Delta u=0 & x\in \mathbb{R}^n,\ t\in \mathbb{R}\setminus \{0\} \\ &\\ ...
Giuseppe Negro's user avatar
1 vote
1 answer
393 views

Driving Fresnel Integral $\int_{-\infty}^\infty \sin(x^2)\rm dx$ and $\int_{-\infty}^\infty \cos(x^2)\rm dx$ through polar coordinates

I am trying to derive the Fresnel Integrals $$\int_{-\infty}^\infty \sin(x^2)\rm dx ~~~~and ~~~~~\int_{-\infty}^\infty \cos(x^2)\rm dx$$ through the Gaussian Integral via $I=\int_{-\infty}^\infty e^{...
aleden's user avatar
  • 4,007
3 votes
1 answer
146 views

How do I prove the relation: $\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$

I want to prove the following relation: $${\Gamma_{1/2}}=\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$$ I noticed that: $$\frac{\intop_{x=-\infty}^{+\...
LithiumPoisoning's user avatar
3 votes
1 answer
263 views

Is this conjecture about Gaussian integral right or not? $ \int_{-\infty}^{\infty} e^{ix^2} dx=\sqrt{i \pi} = \frac{(1+i)\sqrt{\pi}}{\sqrt{2}}$?

For $a \in \mathbb{R},a>0 $ the Gaussian integral is $$ \begin{equation} \int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi} {a}} . \hspace{1cm} (1) \end{equation} $$ What happens if we ...
Breaking M's user avatar
  • 1,050