  We give an exhaustive presentation of the semi-analytical approach to the
model independent leptonic QED corrections to deep inelastic neutral current
lepton-nucleon scattering. These corrections include photonic bremsstrahlung
from and vertex corrections to the lepton current of the order ${\cal
O}(\alpha)$ with soft photon exponentiation. % A common treatment of these
radiative corrections in several variables -- leptonic, hadronic, mixed,
Jaquet-Blondel variables -- has been developed and double differential
cross-sections are calculated. In all sets of variables we use some structure
functions, which depend on the hadronic variables and which do not have to be
defined in the quark parton model. The remaining numerical integrations are
twofold (for leptonic variables) or onefold (for all other variables). For the
case of hadronic variables, all phase space integrals have been performed
analytically. Numerical results are presented for a large kinematical range,
covering fixed target as well as collider experiments at HERA or
LEP$\otimes$LHC, with a special emphasis on HERA physics.
