Sine Integral Function is Odd

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Theorem

$\displaystyle \map \Si {-x} = -\map \Si x$

where:

$\Si$ denotes the sine integral function
$x$ is a real number.


Proof

\(\displaystyle \map \Si {-x}\) \(=\) \(\displaystyle \int_0^{-x} \frac {\sin u} u \rd u\) Definition of Sine Integral Function
\(\displaystyle \) \(=\) \(\displaystyle -\int_0^x \frac {\map \sin {-u} } {-u} \rd u\) substituting $u \mapsto -u$
\(\displaystyle \) \(=\) \(\displaystyle -\int_0^x \frac {\sin u} u \rd u\) Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle -\map \Si x\) Definition of Sine Integral Function

$\blacksquare$


Sources