Asymptotic Analysis and Singular Perturbation Theory

Integrating the power series expansion of e−t2 term by term, we obtain the power
series expansion of erf x,
erf x = 2
√ x −
π
1
3 x3 + . . . + (−1)n
(2n + 1)n! x2n+1
+ . . . ,
which is convergent for every x ∈ R. For large values of x, however, the convergence
is very slow. the Taylor series of the error function at x = 0. Instead, we can use
the following divergent asymptotic expansion, proved below, to obtain accurate
approximations of erf x for large x:
erf x ∼ 1 − e−x2
√ π
∞
n+1 (2n − 1)!!
(−1)
2n 1
xn+1 as x → ∞, (2.4)
n=0
where (2n − 1)!! = 1 · 3 · . . . · (2n − 1). For example, when x = 3, we need 31 terms
in the Taylor series at x = 0 to approximate erf 3 to an accuracy of 10 −5 , whereas
we only need 2 terms in the asymptotic expansion.
Proposition 2.10 The expansion (2.4) is an asymptotic expansion of erf x.
Proof. We write
erf x = 1 − 2
√ π
and make the change of variables s = t 2 ,
For n = 0, 1, 2, . . ., we define
erf x = 1 − 1
√ π
Fn(x) =
∞
x
e −t2
dt,
∞
x2 s −1/2 e −s ds.
∞
x2 s −n−1/2 e −s ds.
Then an integration by parts implies that
Fn(x) = e−x2
− n +
x2n+1 1
Fn+1(x).
2
By repeated use of this recursion relation, we find that
erf x = 1 − 1
√ F0(x)
π
= 1 − 1
e
√
π
−x2 1
−
x 2 F1(x)
= 1 − 1
√ e
π
−x2
1 1
−
x 2x3
+ 1 · 3
F2(x)
22 24