# Fresnel Sine Integral Function is Odd

## Theorem

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

where:

$\operatorname S$ denotes the Fresnel sine integral function
$x$ is a real number.

## Proof

 $\ds \map {\operatorname S} {-x}$ $=$ $\ds \sqrt {\frac 2 \pi} \int_0^{-x} \sin u^2 \rd u$ Definition of Fresnel Sine Integral Function $\ds$ $=$ $\ds -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \sin {\paren {-u}^2} \rd u$ substituting $u \mapsto -u$ $\ds$ $=$ $\ds -\sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$ $\ds$ $=$ $\ds -\map {\operatorname S} x$ Definition of Fresnel Sine Integral Function

$\blacksquare$