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[[Image:Dawson+ Function.svg|thumb|300px|right|The Dawson function, <math>F(x) = D_+(x)</math>, around the origin]]
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[[Image:Dawson- Function.svg|thumb|300px|right|The Dawson function, <math>D_-(x)</math>, around the origin]]


In [[mathematics]], the '''Dawson function''' or '''Dawson integral''' (named for [[John M. Dawson]]) is either
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:<math>F(x) = D_+(x) = e^{-x^2} \int_0^x e^{t^2}\,dt</math>,
also denoted as ''F''(''x'') or ''D''(''x''), or alternatively
:<math>D_-(x)  = e^{x^2} \int_0^x e^{-t^2}\,dt\!</math>.
 
The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function,
:<math>D_+(x) = \frac12 \int_0^\infty e^{-t^2/4}\,\sin{(xt)}\,dt.</math>
It is closely related to the [[error function]] erf, as
:<math> D_+(x) = {\sqrt{\pi} \over 2}  e^{-x^2}  \mathrm{erfi} (x) = - {i \sqrt{\pi} \over 2}  e^{-x^2}  \mathrm{erf} (ix) </math>
where erfi is the imaginary error function, {{nowrap|1=erfi(''x'') = &minus;''i'' erf(''ix'').}} Similarly,
:<math>D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \mathrm{erf}(x)</math>
in terms of the real error function, erf.
 
In terms of either erfi or the [[Faddeeva function]] ''w''(''z''), the Dawson function can be extended to the entire [[complex plane]]:<ref>Mofreh R. Zaghloul and Ahmed N. Ali, "[http://dx.doi.org/10.1145/2049673.2049679 Algorithm 916: Computing the Faddeyeva and Voigt Functions]," ''ACM Trans. Math. Soft.'' '''38''' (2), 15 (2011).  Preprint available at [http://arxiv.org/abs/1106.0151 arXiv:1106.0151].</ref>
:<math>F(z) = {\sqrt{\pi} \over 2}  e^{-z^2}  \mathrm{erfi} (z) = \frac{i\sqrt{\pi}}{2} \left[ e^{-z^2} - w(z) \right]</math>,
which simplifies to
:<math>D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[ w(x) ]</math>
:<math>D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right]</math>
for real ''x''.
 
For |''x''| near zero, {{nowrap|1=''F''(''x'') ≈ ''x'',}}
and for |''x''| large, {{nowrap|1=''F''(''x'') ≈ 1/(2''x'').}}
More specifically, near the origin it has the series expansion
 
:<math> 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</math>
 
''F''(''x'') satisfies the differential equation
 
:<math> \frac{dF}{dx} + 2xF=1\,\!</math>
 
with the initial condition&nbsp;''F''(0)&nbsp;=&nbsp;0.
 
==References==
*{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.9. Dawson's Integral | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=302}}
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=N. M. |last=Temme}}
 
<references />
 
== External links ==
* [http://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the [[GNU Scientific Library]]
* [http://www.moshier.net/#Cephes Cephes] &ndash; C and C++ language special functions math library
* [http://ab-initio.mit.edu/Faddeeva Faddeeva Package] &ndash; C++ code for the Dawson function of both real and complex arguments, via the [[Faddeeva function]]
* [http://mathworld.wolfram.com/DawsonsIntegral.html Dawson's Integral] ''(at Mathworld)''
* [http://nlpc.stanford.edu/nleht/Science/reference/errorfun.pdf Error functions]
 
[[Category:Special functions]]
[[Category:Gaussian function]]
 
 
{{mathapplied-stub}}

Latest revision as of 14:02, 28 November 2014

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