Asymptotic Expansion for Sine Integral Function

Theorem

\(\ds \map \Si x\)

\(\sim\)

\(\ds \frac \pi 2 - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }\)

\(\ds \)

\(\sim\)

\(\ds \frac \pi 2 - \frac {\sin x} x \paren {\dfrac 1 x - \dfrac {3!} {x^3} + \dfrac {5!} {x^5} - \cdots} - \frac {\cos x} x \paren {1 - \dfrac {2!} {x^2} + \dfrac {4!} {x^4} - \cdots}\)

where:

$\Si$ denotes the sine 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): $\S 35$: Miscellaneous Special Functions: Sine Integral $\ds \map {Si} x = \int_0^x \frac {\sin u} u \rd u$: $35.12$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 36$: Miscellaneous and Riemann Zeta Functions: Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$: $36.12.$