32D (Asymptotic Methods)

E (x) \sim e^{- x} x^{- 1} \sum_{n \geq 0} (- 1)^{n} n! x^{- n},

a_{n} = (- 1)^{n} n!

ii

t = x + u

gives

E (x) = e^{- x} x^{- 1} \int_{0}^{\infty} e^{- u} (1 + u / x)^{- 1} d u

. Expanding the last factor and applying Watson lemma yields the same coefficients

(- 1)^{n} Γ (n + 1)

Successive term magnitudes have ratio

(n + 1) / x

, so truncate near the least term,

n ≃ x - 1

. Stirling gives least relative term

2 π x e^{- x}

, hence an exponentially small relative remainder of that order.

Substitute

u = n t

and use the

π

-periodicity of

cos (n cos u)

to get

J_{n} = \int_{0}^{π} cos (n cos u) d u = π J_{0} (n)

. Stationary phase at

0, π

gives

J_{n} \sim \frac{2 π}{n} cos (n - π /4) .

Thus $A = 2 π / n$ , $α = n - π /4$ , and the next term is $O (n^{- 3/2})$ .

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