Integrals of the Fresnel integrals

Asked Apr 12 '23 at 15:23

Active Apr 12 '23 at 15:23

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I find in an engineering book the Integrals of the Fresnel integrals and i don't know how to prove the expressions given bellow.

$$C_I(t)=\int_{0}^{1}C(u)du=tC(t)-\frac{1}{\pi}sin(\frac{\pi t^2}{2})$$ and $$S_I(t)=\int_{0}^{1}S(u)du=tS(t)+\frac{1}{\pi}cos(\frac{\pi t^2}{2})-\frac{1}{\pi}$$

Where the Fresnel integrals are: $$C(t)=\int_{0}^{t}\cos(\frac{\pi u^2}{2})du$$ $$S(t)=\int_{0}^{t}\sin(\frac{\pi u^2}{2})du$$

I tried to solve the integrals as they are but i know they are a special type of integrals and i don't know how to start properly the proof. Thank you !

asked Apr 12 '23 at 15:23

Alexandru Harai

2

There is something strange here. Your integrals are definite integrals between $0$ and $1$, so they must be numbers. What is $t$ in $C_I(t)$? – GReyes Apr 12 '23 at 15:36
2

Integration by parts seems like a reasonable approach. I think you meant the upper limit to be $t$, not $1$? – user170231 Apr 12 '23 at 15:37
Well, the only information that book provides about this t is that, t are a parameter that is used in the parametric equations of the clothoid or euler spiral . – Alexandru Harai Apr 12 '23 at 15:39
Or are they integrals from $0$ to $t$ instead? – GReyes Apr 12 '23 at 15:39
I know that all papers about Fresnel integrals say that are improper integrals but in this book they appear with 0,t limits – Alexandru Harai Apr 12 '23 at 15:42
1

You can consider the integrals from $0$ to $t$, and then $C(t)$ and $S(t)$ give a parametrization of the Euler spiral. – GReyes Apr 12 '23 at 15:44
The upper limits must be typos. See integral of $C$ and integral of $S$ from $0$ to $t$. – user170231 Apr 12 '23 at 16:00

Integrals of the Fresnel integrals

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