Derivative of Fresnel Cosine Integral Function

Theorem

 * $\displaystyle \frac {\d \operatorname C} {\d x} = \sqrt {\frac 2 \pi} \cos x^2$

where $\operatorname C$ denotes the Fresnel cosine integral function.

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


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

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


 * $\displaystyle \frac {\d \operatorname C} {\d x} = \sqrt {\frac 2 \pi} \cos x^2$