  We obtain a variety of series and integral representations of the digamma
function $\psi(a)$. These in turn provide representations of the evaluations
$\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$.
The approach is through a limit definition of the zeroth Stieltjes constant
$\gamma_0(a)=-\psi(a)$. Several other results are obtained, including product
representations for $\exp[\gamma_0(a)]$ and for the Gamma function $\Gamma(a)$.
In addition, we present series representations in terms of trigonometric
integrals Ci and Si for $\psi(a)$ and the Euler constant $\gamma=-\psi(1)$.
