Asymptotic Expansion for Sine Integral Function

Theorem

 * $\displaystyle \map \Si x \sim \frac \pi 2 - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }$

where:
 * $\Si$ denotes the sine integral function
 * $\sim$ denotes asymptotic equivalence as $x \to \infty$.