Asymptotic Expansion for Fresnel Sine Integral Function
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This theorem requires a proof.
also establish general term and write sums with $\sum$
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Theorem
- $\displaystyle \map {\operatorname S} x \sim \frac 1 2 - \frac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\frac 1 x - \frac {1 \times 3} {2^2 x^5} + \frac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\frac 1 {2 x^3} - \frac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$
where
- $\operatorname S$ denotes the Fresnel sine integral function
- $\sim$ denotes asymptotic equivalence as $x \to \infty$.
Proof
also establish general term and write sums with $\sum$
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Fresnel Sine Integral $\displaystyle \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.19$