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| [[Image:Dawson+ Function.svg|thumb|300px|right|The Dawson function, <math>F(x) = D_+(x)</math>, around the origin]] | | The name of the writer was Adrianne Quesada. Managing men is what she definitely does in her day task but she's always wanted her own business. Her husband doesn't like thought the way she does but what she relatively likes doing is in order to bake but she's dreaming on starting something new. Vermont is just where her house has always been. Her husband and her help keep a website. You may wish to check it out: http://[http://www.Adobe.com/cfusion/search/index.cfm?term=&Circuspartypanama&loc=en_us&siteSection=home Circuspartypanama].com<br><br> |
| [[Image:Dawson- Function.svg|thumb|300px|right|The Dawson function, <math>D_-(x)</math>, around the origin]]
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| 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>,
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| also denoted as ''F''(''x'') or ''D''(''x''), or alternatively
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| :<math>D_-(x) = e^{x^2} \int_0^x e^{-t^2}\,dt\!</math>.
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| The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function,
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| :<math>D_+(x) = \frac12 \int_0^\infty e^{-t^2/4}\,\sin{(xt)}\,dt.</math>
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| It is closely related to the [[error function]] erf, as
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| :<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>
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| where erfi is the imaginary error function, {{nowrap|1=erfi(''x'') = −''i'' erf(''ix'').}} Similarly,
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| :<math>D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \mathrm{erf}(x)</math>
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| in terms of the real error function, erf.
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| 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>
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| :<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>,
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| which simplifies to
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| :<math>D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[ w(x) ]</math>
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| :<math>D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right]</math>
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| for real ''x''.
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| For |''x''| near zero, {{nowrap|1=''F''(''x'') ≈ ''x'',}}
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| and for |''x''| large, {{nowrap|1=''F''(''x'') ≈ 1/(2''x'').}}
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| More specifically, near the origin it has the series expansion
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| :<math> F(x) = \sum_{k=0}^{\infty} \frac{(-1)^k \, 2^k}{(2k+1)!!} \, x^{2k+1}
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| = x - \frac{2}{3} x^3 + \frac{4}{15} x^5 - \cdots</math>
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| ''F''(''x'') satisfies the differential equation
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| :<math> \frac{dF}{dx} + 2xF=1\,\!</math>
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| with the initial condition ''F''(0) = 0.
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| ==References==
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| *{{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}}
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| *{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=N. M. |last=Temme}}
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| <references />
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| == External links ==
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| * [http://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the [[GNU Scientific Library]]
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| * [http://www.moshier.net/#Cephes Cephes] – C and C++ language special functions math library
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| * [http://ab-initio.mit.edu/Faddeeva Faddeeva Package] – C++ code for the Dawson function of both real and complex arguments, via the [[Faddeeva function]]
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| * [http://mathworld.wolfram.com/DawsonsIntegral.html Dawson's Integral] ''(at Mathworld)''
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| * [http://nlpc.stanford.edu/nleht/Science/reference/errorfun.pdf Error functions]
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| [[Category:Special functions]]
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| [[Category:Gaussian function]]
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| {{mathapplied-stub}}
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