Graph rules for functions of the time-evolution operator

An expansion of functions of the evolution operators U(t,$t_0$) in terms of powers of the coupling constant is given. The coefficients of the Nth power can be determined from graphical rules closely related to the Feynman rules. As a particular case the logarithm of the evolution operator is studied. This leads to an exponential representation of U(t,$t_0$) which constitutes for every order of the coupling constant a unitary approximation.