Asymptotic Expansion for Fresnel Sine Integral Function

Theorem

 * $\displaystyle \map {\operatorname S} x \sim \frac 1 2 - \frac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\frac 1 x - \frac {1 \times 3} {2^2 x^5} + \frac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\frac 1 {2 x^3} - \frac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$

where
 * $\operatorname S$ denotes the Fresnel sine integral function
 * $\sim$ denotes asymptotic equivalence as $x \to \infty$.