ABSTRACT
Example: Integration by parts and elementary truncation to the least term. A solution of the differential equation
f ′ − 2xf + 1 = 0 (3.1) is related to the complementary error function:
f(x) =: E(x) = ex 2 ∫ ∞ x
e−s 2 ds =
√ pi
2 ex
2 erfc(x) (3.2)
Let us find the asymptotic behavior of E(x) for x → +∞. One very simple technique is integration by parts, done in a way in which the integrated terms become successively smaller. A decomposition is sought such that in the identity fdg = d(fg) − gdf we have gdf ¿ fdg. Although there may be no manifest perfect derivative in the integrand, we can always create one, in this case by writing e−s
2 ds = −(2s)−1d(e−s2). We have
E(x) = 1 2x
− e x2
s2 ds =
1 2x
− 1 4x3
+ 3ex
s4 ds = ...
