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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 witheight real parameters. The resulting analytical formula involves the sum overthree real roots of a cubic equation, corresponding to the so-calledirreducible case, where one has to employ the trisection of an angle. Whengoing to the special unitary group $SU(4)$ with 15 real prameters, theanalytical formula involves the sum over four real roots of a quartic equation.The associated cubic resolvent equation with three positive roots belongs againto the irreducible case. Furthermore, by imposing the pertinent condition on$SU(4)$ matrices one can also treat the symplectic group $Sp(2)$ with ten realparameters. Since there the roots occur as two pairs of opposite sign, thissimplifies the analytical formula for $Sp(2)$ matrices considerably. An outlookto the situation with analytical formulas for $SU(5)$, $SU(6)$ and $Sp(3)$ isalso given.

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