Power Series Expansion for Fresnel Sine Integral Function

Theorem

 * $\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}$

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