Derivative of Fresnel Sine Integral Function

Theorem

 * $\displaystyle \frac {\d \operatorname S} {\d x} = \sqrt {\frac 2 \pi} \sin x^2$

where $\operatorname S$ denotes the Fresnel sine integral function.

Proof
We have, by the definition of the Fresnel sine integral function:


 * $\displaystyle \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$

By Fundamental Theorem of Calculus (First Part): Corollary, we therefore have:


 * $\displaystyle \frac {\d \operatorname S} {\d x} = \sqrt {\frac 2 \pi} \sin x^2$