Asymptotic Expansion for Fresnel Cosine Integral Function

Theorem

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

where:
 * $\operatorname C$ denotes the Fresnel cosine integral function
 * $\sim$ denotes asymptotic equivalence as $x \to \infty$.