Dawson function

The Dawson function, F(x) = D ₊ (x), around the origin

The Dawson function, D ₋ (x), around the origin

In mathematics, the Dawson function (named for John M. Dawson) is

$D_+(x) = F(x) = e^{-x^2} \int_0^x e^{t^2}\,dt = \frac12 \int_0^\infty e^{-t^2/4}\,\sin{(xt)}\,dt$

The notation D(x) is also in use. The Dawson function is also called the Dawson integral. A variation of this function is given by

$D_-(x) = e^{x^2} \int_0^x e^{-t^2}\,dt.\,\!$

The Dawson function is closely related to the error function erf, as

$F(x) = {\sqrt{\pi} \over 2} e^{-x^2} \mathrm{erfi} (x) = - {i \sqrt{\pi} \over 2} e^{-x^2} \mathrm{erf} (ix)$

where erfi is the imaginary error function, erfi(x) = −i erf(ix).

For |x| near zero, F(x) ≈ x, and for |x| large, F(x) ≈ 1/(2x). More specifically, near the origin it has the series expansion

$F(x) = \sum_{k=0}^{\infty} \frac{(-1)^k \, 2^k}{(2k+1)!!} \, x^{2k+1} = x - \frac{2}{3} x^3 + \frac{4}{15} x^5 - \cdots$

F(x) satisfies the differential equation

$\frac{dF}{dx} + 2xF=1\,\!$

with the initial condition F(0) = 0.

References

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.9. Dawson's Integral", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, http://apps.nrbook.com/empanel/index.html#pg=302 
• Temme, N. M. (2010), "Error Functions, Dawson’s and Fresnel Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/7 

External links

• Cephes – C and C++ language special functions math library
• Dawson's Integral (at Mathworld)
• Error functions