The Reflection Identity of the Gamma Function via Complex Integration
The reflection identity of the gamma function is:
\begin{equation} \Gamma( z ) \Gamma( 1-z ) = \frac{\pi}{\sin( \pi z ) } \end{equation}
We can start from the definition of $\beta( 1- \alpha, \alpha ) $
\begin{align} \Gamma( 1- \alpha ) \Gamma( \alpha ) &= \beta ( 1- \alpha, \alpha ) = \int^\infty_0 \frac{t ^{1- \alpha-1}dt}{( 1+ t )^{1- \alpha+ \alpha} } \\ &= \int^\infty_0 \frac{t ^{-\alpha} dt}{1 + t} \\ &= \int^\infty_{-\infty} \frac{e ^{-\alpha x} e^x dx}{1+ e^x} = \int^\infty_{-\infty} \frac{e ^{( 1-\alpha )x }}{ 1 + e^{x}} = I \end{align}