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MA3AM/MA4AM ASYMPTOTIC METHODS
PROBLEM SHEET 2
1. Consider the integral
where 0 < ϵ ≪ 1. Show that
and hence, using the result e −xx n dx = n! for n ∈ N0, that
where
By bounding |RN| above, deduce that
as ϵ → 0, i.e. that I(ϵ) ∼ (−1)n ϵ n (2n)!.
2. Show that the roots of the following cubic equations are as given, for 0 < ϵ ≪ 1.
(a) The roots of
x3 − (2 + ϵ)x 2 − (1 − ϵ)x + 2 + 3ϵ = 0
are x = 1 + 2/3ϵ + . . ., x = −1 − 6/1ϵ + . . ., and x = 2 − 3/1ϵ + . . .;
(b) the roots of
x3 − (3 + ϵ)x − 2 + ϵ = 0,
are x = 2 + 9/1ϵ + . . . and x = −1 ± √2/3ϵ 1/2 + . . .;