  The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau
\!\cdot\! \vec \varphi\,) = \cos|\vec \varphi\,| + i\vec \tau \!\cdot\!
\hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with
eight real parameters. The resulting analytical formula involves the sum over
three real roots of a cubic equation, corresponding to the so-called
irreducible case, where one has to employ the trisection of an angle. When
going to the special unitary group $SU(4)$ with 15 real prameters, the
analytical formula involves the sum over four real roots of a quartic equation.
The associated cubic resolvent equation with three positive roots belongs again
to the irreducible case. Furthermore, by imposing the pertinent condition on
$SU(4)$ matrices one can also treat the symplectic group $Sp(2)$ with ten real
parameters. Since there the roots occur as two pairs of opposite sign, this
simplifies the analytical formula for $Sp(2)$ matrices considerably. An outlook
to the situation with analytical formulas for $SU(5)$, $SU(6)$ and $Sp(3)$ is
also given.
