Linked Questions

68
votes
8answers
7k 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 ...
31
votes
6answers
17k 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.
4
votes
1answer
7k views

Integrating Fresnel Integrals with Cauchy Theorem?

In regards to the above proof, I'm a little confused as to how the last conclusion was made -- How does the fact that $$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}$$ to conclude that: $$\int_0^{...
1
vote
4answers
278 views

Find $\int_{0}^{\infty} \sin x^{2}\,dx$

This problem is from my textbook of complex analysis. I have attempted this as: let $$u=x^{2}$$ then $$dx=\frac{du}{2\sqrt{u}}$$ therefore $$\frac{1}{2}\int_{0}^{\infty} \frac{\sin u}{\sqrt{u}}\,du $$ ...
1
vote
1answer
1k views

Fresnel Integrals via Differentiation under the Integral Sign

I've been trying to compute $\int_{-\infty}^{\infty}sin( x^2)dx$ via the feynman method with no luck. I was able to compute the Gaussian integral. The trick failed for fresnel integrals. Any ...
0
votes
1answer
276 views

How to integrate $\sin (x^2)$ function? [closed]

what are the steps to integrate the following - Not numerical method , but the integrated function is required. $$ \int \sin(x^2)\,dx.$$
1
vote
2answers
110 views

$\lim\limits_{t \to \infty} f'(t) = 0 \impliedby \lim\limits_{t \to \infty} f(t)=c $ is not true!

Suppose $f(t)$ is continuously differentiable and $c$ is a finite constant. We know that \begin{equation} \lim\limits_{t \to \infty} f(t)=c \implies\lim\limits_{t \to \infty} f'(t) = 0 \quad \text{...
1
vote
0answers
423 views

Integral of $\sin(x^2)$ From 0 to Infinity

In my complex analysis final yesterday we computed $$\int_{-\infty}^\infty{\sin(x^2)dx}=\int_{-\infty}^\infty{\cos(x^2)dx}=\sqrt{\pi/2}$$ using contour integration. It seems similar to the Gaussian $$\...
6
votes
0answers
279 views

Historical context: The Fresnel integrals

The evaluation of the Fresnel integrals has been done a plethora of times both on this site, and numerous other places. The two main ways of evalutating these integrals has either been with some ...
1
vote
1answer
51 views

Help on contour integral on another answer on this site

Do you mind expanding on the part along the diagonal in the first answer by Robjohn to this question proof? Particularly how to achieve (3). I am trying use a parameterization for $z=xe^{i \pi/4}$ ...