  We develop a simple computational tool for $SU(3)$ analogous to Bargmann's
calculus for $SU(2)$. Crucial new inputs are, (i) explicit representation of
the Gelfand-Zetlin basis in terms of polynomials in four variables and positive
or negative integral powers of a fifth variable (ii) an auxiliary Gaussian
measure with respect to which the Gelfand-Zetlin states are orthogonal but not
normalized (iii) simple generating functions for generating all basis states
and also all invariants. As an illustration of our techniques, an algebraic
formula for the Clebsch-Gordan coefficients is obtained for the first time.
This involves only Gaussian integrations. Thus $SU(3)$ is made as accessible
for computations as $SU(2)$ is.
