  We describe a nonperturbative (in Zalpha) numerical evaluation of the
one-photon electron self energy for 3P_{1/2}, 3P_{3/2}, 4P_{1/2} and 4P_{3/2}
states in hydrogenlike atomic systems with charge numbers Z=1 to 5. The
numerical results are found to be in agreement with known terms in the
expansion of the self energy in powers of Zalpha and lead to improved
theoretical predictions for the self-energy shift of these states.
