Asymptotic Expansion for Sine Integral Function

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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 $x \to \infty$.


Proof


Sources