Fresnel integrals

Remarks

Example: $S$ and $C$ are odd, since the integrands are even, and $S(0) = C(0) = 0$. The curve $t \mapsto (C(t), S(t))$ has velocity $(\cos t^2, \sin t^2)$ of length $1$, so $t$ is its arc length, and its direction angle $t^2$ gives curvature $2t$. This curve is the Euler spiral.

Non-example: $x \mapsto \int_0^x \sin(t)^2\, dt = \frac{x}{2} - \frac{\sin 2x}{4}$ is not a Fresnel integral; the square must be inside the sine.

As $x \to \infty$, $S(x) \to \sqrt{\pi/8}$ and $C(x) \to \sqrt{\pi/8}$, as one sees by rotating the contour of the Gaussian integral. Many references use the normalisation $\int_0^x \sin\left(\frac{\pi t^2}{2}\right) dt$ and $\int_0^x \cos\left(\frac{\pi t^2}{2}\right) dt$ instead; then both limits are $\frac12$.