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Dawson's Integral

\begin{figure}\begin{center}\BoxedEPSF{DawsonsIntegral.epsf scaled 830}\end{center}\end{figure}

An Integral which arises in computation of the Voigt lineshape:

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

It is sometimes generalized such that
D_{\pm}(x)\equiv e^{\mp x^2}\int_0^x e^{\pm y^2}\,dy,
\end{displaymath} (2)

$\displaystyle D_+(x)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{\pi}\,e^{-x^2}\mathop{\rm erfi}\nolimits (x)$ (3)
$\displaystyle D_-(x)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{\pi}\,e^{x^2}\mathop{\rm erf}\nolimits (x),$ (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,
\end{displaymath} (5)

\end{displaymath} (6)

and an inflection at $D_+''(x)=0$, or
-2x+\sqrt{\pi}\,e^{-x^2}(2x^2-1)\mathop{\rm erfi}\nolimits (x)=0,
\end{displaymath} (7)

\end{displaymath} (8)


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.

© 1996-9 Eric W. Weisstein