Solving the matrix exponential function for the groups SU(3), SU(4) and Sp(2)

Abstract: 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.