Asymptotic Expansion for Fresnel Sine Integral Function

Theorem

$\map {\mathrm S} x \sim \dfrac 1 2 - \dfrac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\dfrac 1 x - \dfrac {1 \times 3} {2^2 x^5} + \dfrac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\dfrac 1 {2 x^3} - \dfrac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$

where

$\mathrm S$ denotes the Fresnel 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): $\S 35$: Fresnel Sine Integral $\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.19$