Fresnel Sine Integral Function is Odd

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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

$\blacksquare$


Sources