Derivative of Fresnel Sine Integral Function

Theorem

$\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$

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

Proof

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

$\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$

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

$\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$

$\blacksquare$