Asymptotic Expansion for Fresnel Cosine Integral Function

From ProofWiki
Jump to navigation Jump to search

Theorem

\(\ds \map {\operatorname C} x\) \(\sim\) \(\ds \frac 1 2 + \frac 1 {\sqrt {2 \pi} } \paren {\map \sin {x^2} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {2^{3 n} n! x^{4 n + 1} } } - \frac 1 2 \map \cos {x^2} \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {2^{3 n} n! x^{4 n + 3} } } }\)
\(\ds \) \(=\) \(\ds \frac 1 2 + \frac 1 {\sqrt {2 \pi} } \paren {\map \sin {x^2} \paren {\frac 1 x - \frac {1 \times 3} {2^2 x^5} + \frac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} - \map \cos {x^2} \paren {\frac 1 {2 x^3} - \frac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }\)

where:

$\operatorname C$ denotes the Fresnel cosine integral function
$\sim$ denotes asymptotic equivalence as $x \to \infty$.


Proof


Sources