# Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]]
(* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *)


What is the reason the term -I Exp[-x^2] √π is present here? The function being expanded is real-valued, and anyways the term decays faster than any of the polynomial terms given here, so why include it in the series, when I specified that I need terms up to the order 8?

The issue is reproducible both in version 8 and 10.2.

• FunctionExpand[DawsonF[x]] probably gives a bit of a clue – chuy Apr 22 '16 at 18:23
• To add to chuy's comment, note that the known asymptotic expansions for the imaginary error function (see e.g. the Wolfram Functions site) necessarily involve a complex term. – J. M. will be back soon Apr 22 '16 at 22:52
• I still do not get it. Why there should be an imaginary term, if I'm interested in the asymptotic expansion when $x$ approaches the positive infinity along the real axis? – Vladimir Reshetnikov Apr 23 '16 at 3:08