# Sine Integral Function is Odd

## 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

$\blacksquare$