Dawson's Integral

An Integral which arises in computation of the Voigt lineshape:

$$D(x)\equiv e^{-x^2}\int_0^x e^{y^2}\,dy.$$ (1)

It is sometimes generalized such that

$$D_{\pm}(x)\equiv e^{\mp x^2}\int_0^x e^{\pm y^2}\,dy,$$ (2)

giving

$$D_+(x) = {\textstyle{1\over 2}}\sqrt{\pi}\,e^{-x^2}\mathop{\rm erfi}\nolimits (x)$$ (3)

$$D_-(x) = {\textstyle{1\over 2}}\sqrt{\pi}\,e^{x^2}\mathop{\rm erf}\nolimits (x),$$ (4)

where $\mathop{\rm erf}\nolimits (z)$ is the Erf function and $\mathop{\rm erfi}\nolimits (z)$ is the imaginary error function Erfi. $D_+(x)$ is illustrated in the left figure above, and $D_-(x)$ in the right figure. $D_+$ has a maximum at $D_+'(x)=0$, or

$$1-\sqrt{\pi}\,e^{-x^2}x^2\mathop{\rm erfi}\nolimits (x)=0,$$ (5)

giving

$$D_+(0.9241388730)=0.5410442246,$$ (6)

and an inflection at $D_+''(x)=0$, or

$$-2x+\sqrt{\pi}\,e^{-x^2}(2x^2-1)\mathop{\rm erfi}\nolimits (x)=0,$$ (7)

giving

$$D_+(1.5019752683)=0.4276866160.$$ (8)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 298, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Dawson's Integrals.'' §6.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 252-254, 1992.

Spanier, J. and Oldham, K. B. ``Dawson's Integral.'' Ch. 42 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 405-410, 1987.