Questions on the Fresnel integrals.
16
votes
1answer
155 views
+100
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
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
2
votes
1answer
111 views
Definite integral involving Fresnel integrals
I am seeking to evaluate
$\int_0^{\infty} f(x)/x^2 \, dx$
with
$f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin
...
0
votes
1answer
60 views
Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts
$$
\int_0^1 \! C(x) \, \mathrm{d} x.
$$
where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$.
I am really not quite sure how to go about this one, especially given that it needs to be calculated ...
3
votes
2answers
341 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:
...
3
votes
1answer
361 views
Connect two curves with Euler spiral segment
Image of situation:
http://upload.wikimedia.org/wikipedia/commons/5/54/Easement_curve.svg
Let's say we have a straight line (blue) and a circular arc (green). My goal is to connect these two curves ...
