Particular values of the gamma function

The gamma function, denoted as \(\Gamma(z)\), is a generalization of the factorial function that extends its definition to all complex numbers except the non-positive integers. It is defined for positive real numbers \(z\) by the following integral: \[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt \] The gamma function has several important values, particularly at positive integers and half-integers.