Asymptotic evaluation of two families of integrals

Abstract

In this paper the integrals f mv(τ) = ∝ 0 ∞ exp[−(t + τ)]t v( ln t) m(t + τ) −1 dt andg mv(τ) = ∝ 0 ∞ exp[− ¦ − τ ¦]t v( ln t) m(t − τ) −1 dt are investigated for positive real values of the variable τ. Here, m is a nonnegative integer, v is a complex variable with Re( v) > −1. Both integrals are related to the complex integral Φ mγ ( z) = ∝ 0 ∞exp[−( t − z)] t − γ (ln t) m ( t − z) −1 dt with 0 ⩽ Re( γ) < 1, the behavior of which is analyzed in detail. The results are applied to obtain asymptotic representations for f mn ( τ) and g mn ( τ), m and n both nonnegative integers, near τ = 0. The latter integrals play a role in the study of the equations of neutron transport and radiative transfer.