Derivative of Fresnel Sine Integral Function
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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:
- $\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$
$\blacksquare$