Asymptotic Expansion for Cosine Integral Function

Theorem

$\ds \map \Ci x \sim \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }$

where:

$\Ci$ denotes the cosine integral function

$\sim$ denotes asymptotic equivalence as $x \to \infty$.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$: $35.16$