  In a kinematically complete experiment at the Mainz microtron MAMI, pion
angular distributions of the $^3$He(e,e'$\pi^+)^3$H reaction have been measured
in the excitation region of the $\Delta$ resonance to determine the
longitudinal ($L$), transverse ($T$), and the $LT$ interference part of the
differential cross section. The data are described only after introducing
self-energy modifications of the pion and $\Delta$-isobar propagators. Using
Chiral Perturbation Theory (ChPT) to extrapolate the pion self energy as
inferred from the measurement on the mass shell, we deduce a reduction of the
$\pi^+$ mass of $\Delta m_{\pi^+} = (-1.7^{+ 1.7}_{- 2.1})$ MeV/c$^2$ in the
neutron-rich nuclear medium at a density of $\rho = (0.057^{+ 0.085}_{-
0.057})$ fm$^{-3}$. Our data are consistent with the $\Delta$ self energy
determined from measurements of $\pi^0$ photoproduction from $^4$He and heavier
nuclei.
