Transformation to Canonical Form for Uniform Asymptotic Expansions

Abstract:

The existence of a one-to-one analytic transformation z ↔ w is established which takes a function of the form

into the canonical form where z ∈ ℂ and t = (t, τ)∈ℂ². The coefficient functions α(t), β(t), A(t), and C(t) are analytic for small |t|, and satisfy α(0) = β(0) = A(0) = C(0) = 0. The function ψ(z, t) is analytic in both z and t for small |z| and |t|. The transformation z ↔ w is frequently needed in uniform asymptotic expansions of integrals.