Asymptotic Analysis and Singular Perturbation Theory

3.2 Perturbed Gaussian integrals
Consider the following integral
I(a, ε) =
∞
−∞
exp − 1
2 ax2 − εx 4
dx, (3.4)
where a > 0 and ε ≥ 0. For ε = 0, this is a standard Gaussian integral, and
I(a, 0) = 1
√ 2πa .
For ε > 0, we cannot compute I(a, ε) explicitly, but we can obtain an asymptotic
expansion as ε → 0 + .
First, we proceed formally. Taylor expanding the exponential with respect to ε,
exp − 1
2 ax2 − εx 4
1 −
= e 2 ax2
1 − εx 4 + 1
2! ε2x 8 + . . . + (−1)n
ε
n!
n x 4n
+ . . . ,
and integrating the result term-by-term, we get
I(a, ε) ∼ 1
√
2πa
where
1 − ε〈x 4 〉 + . . . + (−1)n
ε
n!
n 〈x 4n 〉 + . . .
〈x 4n 〉 =
∞
−∞ x4n 1 − e 2 ax2
∞ 1
e− 2
−∞ ax2
dx
dx .
We use a special case of Wick’s theorem to calculate these integrals.
Proposition 3.3 For m ∈ N, we have
where
Proof. Let
〈x 2m 〉 =
(2m − 1)!!
am ,
(2m − 1)!! = 1 · 3 · 5 . . . (2m − 3) · (2m − 1).
J(a, b) =
∞ 1
e− 2
−∞ ax2 +bx dx
∞ 1
e− 2
−∞ ax2 dx .
, , (3.5)
Differentiating J(a, b) n-times with respect to b and setting b = 0, we find that
〈x n 〉 = dn
J(a, b) .
dbn Writing
b=0
1 −
e 2 ax2 +bx −
= e 1
2 a(x−b)2 + b2
2a
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