31A (Asymptotic Methods)

Watson's lemma gives, under integrability away from zero and the stated endpoint asymptotic expansion,

I (λ) \sim n = 0 \sum \infty \frac{a _{n} Γ ( n β + 1 )}{λ ^{n β + 1}} .

(1)

Indeed each power integrates over the positive half-line to

Γ (n β + 1) λ^{- (n β + 1)}

; replacing a fixed upper endpoint by infinity has exponentially small error. A remainder

o (t^{Nβ})

near zero gives

o (λ^{- Nβ - 1})

after splitting off a small endpoint neighbourhood and rescaling

u = λ t

For Laplace's method, assume the phase and amplitude are smooth near the nondegenerate maximum, as required for a full expansion. Localize to a small neighbourhood of

c

: outside it the unique maximum and continuity make the exponential smaller by a factor

e^{- λ ε}

. Set

ζ = sgn (t - c) 2 (ϕ (c) - ϕ (t)) .

(2)

Near

c

this is a smooth increasing coordinate, with

ζ^{'} (c) = ∣ ϕ^{''} (c) ∣

. It need not be monotonic on the entire integration interval. In the localized integral put

A (ζ) = F (t (ζ)) d t / d ζ

. Extending its Gaussian integration to both infinite endpoints incurs exponentially small error. Taylor expansion of

A

and integration of even Gaussian moments gives

J (λ) \sim e^{λ ϕ (c)} 2 π \sum_{j \geq 0} \frac{A ^{(2 j)} ( 0 )}{2 ^{j} j ! λ ^{j + 1/2}} .

(3)

Here

A (0) = F (c) / ∣ ϕ^{''} (c) ∣

, giving the leading term $e^{λ ϕ (c)} F (c) 2 π / (λ ∣ ϕ^{''} (c) ∣)$ . The asymptotic-equivalence notation for that term presumes

F (c) \neq = 0

; if it vanishes the series determines the first nonzero term. The maximum condition alone, without amplitude regularity, would not imply a full expansion.

For the gamma function, set