代写MA3AM/MA4AM ASYMPTOTIC METHODS PROBLEM SHEET 2代写数据结构语言程序

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 + . . .;

(c) the roots of

ϵx3 + x − 2 = 0

are x = 2 − 8ϵ + . . . and x = ±iϵ−1/2 + 1 + . . .; and

(d) the roots of

ϵx3 + (x − 2)2 = 0

are x = 2 ± 2 √2iϵ1/2 + . . . and x = −ϵ −1 − 4 + . . ..