  I calculate the 1-loop self-energy of the lowest Landau level an electron of
mass m in a strong, constant and uniform external magnetic field B, beyond its
always used truncation at (ln L)^2, L=|e|B/m^2. This is achieved by evaluating
the integral deduced in 1953 by Demeur and incompletely calculated in 1969 by
Jancovici, which I recover from Schwinger's techniques of calculation. It
yields \delta m \simeq (\alpha*m/(4*\pi))*[(\ln L -\gamma_E -3/2)^2 -9/4
+\pi/(\beta-1) + \pi^2/6 +\pi*\Gamma[1-\beta]/L^{\beta-1}
+(1/L)*(\pi/(2-\beta)-5) +{\cal O}(1/L^{>= 2})] with \beta \approx 1.175 for 75
=< L =< 10000. . The (\ln L)^2 truncation exceeds the precise estimate by 45%
at L=100 and by more at lower values of L, due to neglecting, among others, the
single logarithmic contribution. This is doubly unjustified because it is large
and because it is needed to fulfill appropriate renormalization conditions.
Technically challenging improvements look therefore necessary, for example when
resumming higher loops and incorporating the effects of large B on the photonic
vacuum polarization, like investigated in recent years.
