Near zero, the absolute value of the integrand is comparable to

t^{z - 1} ∣ lo g t ∣,

which is integrable for

z > 0

. At infinity, the factor

e^{- t}

dominates every power of

t

and

lo g t

, so the integral also converges there.

Direct use of Laplace's method is inconvenient because the phase

z lo g t - t

does not have the form

z ϕ (t)

with a fixed phase: its stationary point

t = z

z \to \infty

. Rescaling is needed to fix the maximum.

ii

Solution

t = zs

. Then

Γ^{'} (z) = e^{z l o g z} \int_{0}^{\infty} \frac{l o g z + l o g s}{s} e^{z (l o g s - s)} d s .

The fixed phase

ϕ (s) = lo g s - s

has its unique maximum at

s = 1

, with

ϕ (1) = - 1, ϕ^{''} (1) = - 1.

The amplitude at the maximum is

lo g z

. The interior-maximum form of Laplace's method gives

Γ^{'} (z) \sim e^{z l o g z} e^{- z} lo g z \frac{2 π}{z} .

Thus

Γ^{'} (z) \sim \frac{2 π}{z}, e^{z l o g z - z} lo g z,

a = 2 π

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