We derive explicit formulas for calculating $e^A$, $cosh{A}$, $sinh{A}, cos{A}$ and $sin{A}$ for a given $2times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3times3$ matrix $A$. These formulas are expressed exclusively in te
rms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients.
We report on high resolution Fourier-transform spectroscopy of fluorescence to the a^3Sigma_u^+ state excited by two-photon or two-step excitation from the X^1Sigma_g^+ state to the 2^3Pi_g state in the molecule K_2. These spectroscopic data are comb
ined with recent results of Feshbach resonances and two-color photoassociation spectra for deriving the potential curves of X^1Sigma_g^+ and a^3Sigma_u^+ up to the asymptote. The precise relative position of the triplet levels with respect of the singlet levels was achieved by including the excitation energies from the X^1Sigma_g^+ state to the 2^3Pi_g state and down to the a^3Sigma_u^+ state in the simultaneous fit of both potentials. The derived precise potential curves allow for reliable modeling of cold collisions of pairs of potassium atoms in their ^2S ground state.
