Limit to Infinity of Fresnel Cosine Integral Function
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Theorem
- $\ds \lim_{x \mathop \to \infty} \map {\mathrm C} x = \frac 1 2$
where $\mathrm C$ denotes the Fresnel cosine integral function.
Proof
\(\ds \lim_{x \mathop \to \infty} \map {\mathrm C} x\) | \(=\) | \(\ds \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \cos u^2 \rd u\) | Multiple Rule for Limits of Real Functions, Definition of Fresnel Cosine Integral Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 2 \pi} \int_0^\infty \cos u^2 \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 2 \pi} \times \frac 1 2 \sqrt {\frac \pi 2}\) | Definite Integral to Infinity of $\map \cos {a x^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Fresnel Cosine Integral $\ds \map {\mathrm C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $35.23$