Vol. 15, No. 2, 1965

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ISSN: 0030-8730
Laplace’s method for two parameters

R. N. Pederson

Vol. 15 (1965), No. 2, 585–596
Abstract

The behavior for large h and k of the integral

        ∫
a
I(h,k) = 0 f(t)exp[− hϕ(t)+ kψ (t)]dt

is considered under hypotheses which are fulfilled, for example, if f, ϕ, ψ are real analytic, ϕ is strictly increasing, and ϕ(0) = ψ(0) = 0. In most cases it is assumed that k = o(h) as h,k →∞. If ν and μ are the respective orders of the first nonvanishing derivatives of ϕ and ψ at the origin, it is found that the behavior

  1. 0 < liminf kνhμ and limsupkνhμ < ,
  2. kνhμ 0,
  3. kνhμ →∞ and ψ(μ)(0) < 0, or
  4. kνhμ →∞ and ψ(μ)(0) > 0.

In case (1) it is shown that I(h,k) is asymptotic to a power series in (k∕h)1(νμ) with coefficients depending on kνhμ. In case (2) it is shown that I(h,k) is asymptotic to a double power series in h1∕ν and khμ∕ν. In case (3) it is shown that I(h,k) is asymptotic to a double power series in k1∕μ and hkνμ. In case (4) it is shown that there exist two parameters σ, τ tending to zero as h,k →∞ such that exp(σ2)I(h,k) is asymptotic to a double power series in σ and τ. If μ ν it is proved that the coefficients of the above power series are unique.

Mathematical Subject Classification
Primary: 41.50
Milestones
Received: 18 February 1964
Published: 1 June 1965
Authors
R. N. Pederson