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The behavior for large h and
k of the integral
![∫
a
I(h,k) = 0 f(t)exp[− hϕ(t)+ kψ (t)]dt](a190x.png)
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
- 0 < liminf kνh−μ and limsupkνh−μ < ∞,
- kνh−μ → 0,
- kνh−μ →∞ and ψ(μ)(0) < 0, or
- kνh−μ →∞ 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∕(ν−μ) with
coefficients depending on kνh−μ. In case (2) it is shown that I(h,k) is asymptotic to
a double power series in h−1∕ν and kh−μ∕ν. In case (3) it is shown that I(h,k) is
asymptotic to a double power series in k−1∕μ and hkν−μ. In case (4) it is
shown that there exist two parameters σ, τ tending to zero as h,k →∞
such that exp(σ−2)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.
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