Asymptotic Analysis and Singular Perturbation Theory

Proof. Taylor’s theorem implies that for y ≥ 0 and N ∈ N
where
e −y = 1 − y + 1
2! y2 + . . . + (−1)N
N! yN + sN+1(y),
sN+1(y) =
1
(N + 1)!
dN+1 dyN+1 −y
e y=η y N+1
for some 0 ≤ η ≤ y. Replacing y by εx 4 in this equation and estimating the
remainder, we find that
where
e −εx4
= 1 − εx 4 + 1
2! ε2 x 4 + . . . + (−1)N
N! εN x 4N + ε N+1 rN+1(x), (3.8)
Using (3.8) in (3.4), we get
I(a, ε) =
N
n=0
It follows that
N
I(a,
ε) − anε
n
n=0
|rN+1(x)| ≤ x4(N+1)
(N + 1)! .
anε n + ε N+1
∞
≤ εN+1
∞
−∞
−∞
≤ ε N+1 1
(N + 1)!
1 −
rN+1(x)e 2 ax2
dx.
1 −
|rN+1(x)| e 2 ax2
dx
∞
−∞
x 4(N+1) 1 −
e 2 ax2
dx,
which proves the result.
These expansions generalize to multi-dimensional Gaussian integrals, of the form
I(A, ε) = exp − 1
2 xT
Ax + εV (x) dx
R n
where A is a symmetric n×n matrix, and to infinite-dimensional functional integrals,
such as those given by the formal expression
1
I(ε) = exp −
2 |∇u(x)|2 + 1
2 u2
(x) + εV (u(x)) dx Du
which appear in quantum field theory and statistical physics.
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