Fresnel Cosine Integral Function is Odd

Theorem

$\map {\operatorname C} {-x} = -\map {\operatorname C} x$

where:

$\operatorname C$ denotes the Fresnel cosine integral function

$x$ is a real number.

Proof

\(\ds \map {\operatorname C} {-x}\)

\(=\)

\(\ds \sqrt {\frac 2 \pi} \int_0^{-x} \cos u^2 \rd u\)

Definition of Fresnel Cosine Integral Function

\(\ds \)

\(=\)

\(\ds -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \cos {\paren {-u}^2} \rd u\)

substituting $u \mapsto -u$

\(\ds \)

\(=\)

\(\ds -\sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u\)

\(\ds \)

\(=\)

\(\ds -\map {\operatorname C} x\)

Definition of Fresnel Cosine Integral Function

$\blacksquare$

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 $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $36.23.$