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There are two well known asymptotic expansions of the Dawson integral $F(x)$ and the function $\exp(x^2)\operatorname {erfc}(x)$ as $x \rightarrow \infty$:

$$ F(x)\sim (1/2)(1/x+1/(2x^3)+ 3/(4x^5)+\cdots) $$ and $$ \exp(x^2)\operatorname {erfc}(x)\sim(1/\sqrt{\pi})(1/x - 1/(2x^3) + 3/(4x^5)+\cdots). $$

I am looking for the original publications where these expansions were derived for the first time. I would appreciate any pointers. I am also interested in other publications dealing with the issue of why the Laplace transformation cannot (or perhaps can?) be used to derive such expansions.

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  • $\begingroup$ Likely known prior to the 1950s. W. R. Wilcox, "Asymptotic Computation of the Repeated Integrals of the Error Function Complement", Mathematics of Computation, Vol. 18, No. 85, 1964, pp. 98-105, cites H. S. Carslaw & J. C. Jaeger, Conducting Heat in Solids, second edition, Oxford 1959 for the following $$\mathrm{erfc} \: x \cong \frac{\exp(-x^{2})}{x\sqrt{\pi}} \left[1-\frac{1}{2x^2}+\frac{1\cdot3}{2^{2}x^{4}}-\frac{1\cdot3\cdot5}{2^{3}x^{6}}+\ldots \right]$$ $\endgroup$
    – njuffa
    Sep 4, 2022 at 20:58
  • $\begingroup$ See Error function etymology: Why the name?, which mentions the paper On a class of definite integrals by Glaisher (1871). In this paper several earlier references/authors to work on the error function are given. Although the actual expansion you are asking about doesn't appear to be in this paper, it was surely known at this time (perhaps in this book), and probably the expansion was known to Euler, if not earlier. $\endgroup$
    – Dave L Renfro
    Sep 4, 2022 at 21:12
  • $\begingroup$ A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, New York: McGraw-Hill 1953, point (p. 147) to the following for the introduction of $\mathrm{erfc}$ (albeit using different notation): Christian Kramp, Analyse des Refractions Astronomiques et Terrestres, Strasbourg: P.J. Dannbach & Leipzig: E.B. Schwickert, 1799 (scan). $\endgroup$
    – njuffa
    Sep 4, 2022 at 22:59
  • $\begingroup$ Looking at Kramp (1799), he examined integrals $\int t^{m-1} e^{-t^{n}} \, dt$ and notes that for $m=1$ " ... le résultat remarqable $\int e^{-tt} = \frac{1}{2}\sqrt{\pi}$". FWIW, Erdélyi et al. (1953) have $$\mathrm{Erfc} \ x = \int_{x}^{\infty} e^{-t^{2}} \, dt = \frac{1}{2}e^{-x^{2}} \left[ \sum_{m=0}^{M-1} \frac{{\left(-1\right)}^m {\left(\frac{1}{2}\right)}^m}{x^{2m+1}} + O \left({|x|}^{-2M-1} \right) \right]$$ while A&S (1964), 7.1.23 have $$\sqrt{\pi}ze^{z^{2}} \mathrm{erfc} \ z \sim 1+ \sum_{m=1}^{\infty} {\left(-1\right)}^m \frac{1\cdot3\ldots (2m-1)}{{\left(2z^{2}\right)}^m}$$. $\endgroup$
    – njuffa
    Sep 5, 2022 at 0:16
  • $\begingroup$ Frank Gilman, "Evaluation of the Probability Integral", Annals of Mathematics, Second Series, Vol. 9, No. 4, Jul. 1908, pp. 167-172 has: $$\mathrm{erfc} \: x = \int_{x}^{\infty} e^{-x^{2}} \, dx = \frac{e^{-x^{2}}}{2x} \left[ 1 - \frac{1}{2x^2}+\frac{1\cdot3}{(2x^2)^2} = \frac{1\cdot3\cdot5}{(2x^2)^3} + \frac{1\cdot3\cdot5\cdot7}{(2x^2)^4} - \ldots\right]$$ and states that this is "commonly used". So it looks like you would want to look at the literature between 1799 and 1908 to find a first use of the asymptotic series (caveat: may well precede the use of the name "error function"). $\endgroup$
    – njuffa
    Sep 5, 2022 at 1:48

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