Transformation to canonical form for uniform asymptotic expansions

The existence of a one-to-one analytic transformation z ↔ w is established which takes a function of the form f(z, t) = α(t) z + β(t) logz + z + z²ψ(z, t) into the canonical form f(z, t) = A(t) w + β(t) log w + C(t) + w, where zε{lunate}C and t=(t,t)ε{lunate}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. © 1990.