  The fine structure interval of P states in hydrogenlike systems can be
determined theoretically with high precision, because the energy levels of P
states are only slightly influenced by the structure of the nucleus. Therefore
a measurement of the fine structure may serve as an excellent test of QED in
bound systems or alternatively as a means of determining the fine structure
constant $\alpha$ with very high precision. In this paper an improved analytic
calculation of higher-order binding corrections to the one-loop self energy of
3P and 4P states in hydrogen-like systems with low nuclear charge number $Z$ is
presented. A comparison of the analytic results to the extrapolated numerical
data for high $Z$ ions serves as an independent test of the analytic
evaluation. New theoretical values for the Lamb shift of the P states and for
the fine structure splittings are given.
