Class Bn-functions property of Feynman integrals

The class B n -functions property of Feynman integrands (and hence also of absolutely convergent Feynman integrals) is proved as functions of internal and external momenta and the masses involved in the theory. The author shows how one may find real constants b r >1, r=1, ..., k, which set energy scales, such that for certain parameters eta r ,r=1, ..., k, with eta r >or=b r , the absolute value of a Feynman integrand J may be bounded, with respect to the eta r , with the so called power asymptotic coefficients denoting estimated degrees of J with respect to the parameters eta r , respectively. This basic result is a key one in studying the asymptotic behaviour of (renormalised) Feynman amplitudes and in establishing the convergence proof of renormalisation.