Questions on the Fresnel integrals.

learn more… | top users | synonyms

6
votes
2answers
804 views

Trig Fresnel Integral

$$\int_{0}^{\infty }\sin(x^{2})dx$$ I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you. I have not done complex analysis (only ...
37
votes
3answers
1k views

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...
1
vote
1answer
517 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 ...
6
votes
2answers
1k views

Are Complex Substitutions Legal in Integration?

This question has been irritating me for awhile so I thought I'd ask here. Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals: ...
2
votes
3answers
213 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...