We perform an analysis of the form factors that rule the structure-dependent amplitude in radiative pion decay. The resonance contributions to pion -> e nu_e gamma decays are computed through the proper construction of the vector and axial-vector form factors by setting the QCD driven asymptotic properties of the three-point Green functions VVP and VAP, and by demanding the smoothening of the form factors at high transfer of momentum. A comparison between theoretical and experimental determination of the form factors is also carried out. We also consider and evaluate the role played by a non-standard tensor form factor. We conclude that, at present and due to the hadronic uncertainties, the search for new physics in this process is not feasible.Ministerio de Educación y Ciencia, FPA2004-00996. Generalitat Valenciana, GVACOMP2007-156. European Commission, EU MRTN-CT-2006-035482 (FLAVIAnet)-Peer reviewed