Question

Find Laplace transform F(p) for function S(t) - sin-Frenele integral

Answer 1

$$S(t) = \int_0^t \frac{\sin u}{\sqrt{2\pi u}} \, du = \left[ v = u^2 \right] = \sqrt{\frac{2}{\pi}} \int_0^{\sqrt{t}} \sin\left(v^2\right) dv$$ $$S(0) = 0$$ $$\int_0^\infty \sin(v^2) \, dv = \frac{1}{2} \sqrt{\frac{\pi}{2}}$$ $$S(\infty) = 1/2$$

By integration theorem:

$$S(t) \rightarrow \frac{\sqrt{\sqrt{p^2 + 1 - p}}}{2p\sqrt{p^2 + 1}}$$