Derivative of Fresnel Cosine Integral Function

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Theorem

$\dfrac {\d \mathrm C} {\d x} = \sqrt {\dfrac 2 \pi} \cos x^2$

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


Proof

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

$\ds \map {\mathrm C} x = \sqrt {\dfrac 2 \pi} \int_0^x \cos u^2 \rd u$

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

$\dfrac {\d \mathrm C} {\d x} = \sqrt {\dfrac 2 \pi} \cos x^2$

$\blacksquare$