Limit to Infinity of Fresnel Sine Integral Function

Theorem

$\ds \lim_{x \mathop \to \infty} \map {\mathrm S} x = \frac 1 2$

where $\mathrm S$ denotes the Fresnel sine integral function.

Proof

\(\ds \lim_{x \mathop \to \infty} \map {\mathrm S} x\)

\(=\)

\(\ds \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \sin u^2 \rd u\)

Multiple Rule for Limits of Real Functions, Definition of Fresnel Sine Integral Function

\(\ds \)

\(=\)

\(\ds \sqrt {\frac 2 \pi} \int_0^\infty \sin u^2 \rd u\)

\(\ds \)

\(=\)

\(\ds \sqrt {\frac 2 \pi} \times \frac 1 2 \sqrt {\frac \pi 2}\)

Definite Integral to Infinity of $\map \sin {a x^2}$

\(\ds \)

\(=\)

\(\ds \frac 1 2\)

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Fresnel Sine Integral $\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.20$
• 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: Fresnel Sine Integral $\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $36.20.$