Other Properties

We also note that the odd symmetry of the Fresnel integrals allows other observations among which are:

$$\int_{-u}^0 \cos\left(\frac{\pi}{2}x^2\right) = C(u)$$ (8)

$$\int_{-u}^0 \sin\left(\frac{\pi}{2}x^2\right) = S(u)$$ (9)

$$\int_{-u}^u \cos\left(\frac{\pi}{2}x^2\right) = 2C(u)$$ (10)

$$\int_{-u}^u \sin\left(\frac{\pi}{2}x^2\right) = 2S(u)$$ (11)

$$\int_{-\infty}^\infty \cos\left(\frac{\pi}{2}x^2\right) = 1$$ (12)

$$\int_{-\infty}^\infty \sin\left(\frac{\pi}{2}x^2\right) = 1.$$ (13)