Fresnel Cosine Integral Function of Zero

Theorem

$\map {\operatorname C} 0 = 0$

where $\operatorname C$ denotes the Fresnel cosine integral function.

Proof

By Fresnel Cosine Integral Function is Odd, $\operatorname C$ is an odd function.

Therefore, by Odd Function of Zero is Zero:

$\map {\operatorname C} 0 = 0$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Fresnel Cosine Integral $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $35.23$
• 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 Cosine Integral $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $36.23.$