Questions on the Fresnel integrals.

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34
votes
3answers
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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 $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...
4
votes
2answers
661 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: ...
1
vote
1answer
238 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 ...