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