Definite Integral to Infinity of Sine of a x^2

Theorem

 * $\displaystyle \int_0^\infty \map \sin {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$

where $a$ is a positive real number.

Proof
We have, by Euler's Formula: Corollary:


 * $\map \exp {-i a x^2} = -i \map \sin {a x^2} + \map \cos {a x^2}$

As $\map \sin {a x^2}$ and $\map \cos {a x^2}$ are both real for real $a, x$, we therefore have: