Editing Dawson function (section)

==Definition==

[[Image:DawsonDp.svg|thumb|300px|right|The Dawson function, <math>F(x) = D_+(x),</math> around the origin]]
[[Image:DawsonDm.svg|thumb|300px|right|The Dawson function, <math>D_-(x),</math> around the origin]]
The Dawson function is defined as either:
<math display=block>D_+(x) = e^{-x^2} \int_0^x e^{t^2}\,dt,</math>
also denoted as <math>F(x)</math> or <math>D(x),</math>
or alternatively
<math display=block>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 display=block>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 id="exp(-x^2) was downstairs, should be upstairs"> D_+(x) = {\sqrt{\pi} \over 2} e^{-x^2} \operatorname{erfi} (x) = - {i \sqrt{\pi} \over 2 }e^{-x^2} \operatorname{erf} (ix) </math>
where erfi is the imaginary error function, {{nowrap|1=erfi(''x'') = −''i'' erf(''ix'').}}
<br>
Similarly,
<math display="block">D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \operatorname{erf}(x)</math>
in terms of the real error function, erf.

In terms of either erfi or the [[Faddeeva function]] <math>w(z),</math> the Dawson function can be extended to the entire [[complex plane]]:<ref>Mofreh R. Zaghloul and Ahmed N. Ali, "[https://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 [https://arxiv.org/abs/1106.0151 arXiv:1106.0151].</ref>
<math display=block>F(z) = {\sqrt{\pi} \over 2} e^{-z^2} \operatorname{erfi} (z) = \frac{i\sqrt{\pi}}{2} \left[ e^{-z^2} - w(z) \right],</math>
which simplifies to
<math display=block>D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[w(x)]</math>
<math display=block>D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right]</math>
for real <math>x.</math>

For <math>|x|</math> near zero, {{nowrap|1=''F''(''x'') ≈ ''x''.}}
For <math>|x|</math> large, {{nowrap|1=''F''(''x'') ≈ 1/(2''x'').}}
More specifically, near the origin it has the series expansion
<math display=block>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>
while for large <math>x</math> it has the asymptotic expansion
<math display=block>F(x) = \frac{1}{2 x} + \frac{1}{4 x^3} + \frac{3}{8 x^5} + \cdots.</math>

More precisely 
<math display=block>\left|F(x) - \sum_{k=0}^{N} \frac{(2k-1)!!}{2^{k+1} x^{2k+1}}\right| \leq \frac{C_N}{x^{2N+3}}.</math>
where <math>n!!</math> is the [[double factorial]]. 

<math>F(x)</math> satisfies the differential equation
<math display=block>\frac{dF}{dx} + 2xF = 1\,\!</math>
with the initial condition <math>F(0) = 0.</math> Consequently, it has extrema for
<math display=block>F(x) = \frac{1}{2 x},</math>
resulting in ''x''&nbsp;=&nbsp;±0.92413887... ({{OEIS2C|id=A133841}}), ''F''(''x'')&nbsp;=&nbsp;±0.54104422... ({{OEIS2C|id=A133842}}).

Inflection points follow for
<math display=block>F(x) = \frac{x}{2 x^2 - 1},</math>
resulting in ''x''&nbsp;=&nbsp;±1.50197526... ({{OEIS2C|id=A133843}}), ''F''(''x'')&nbsp;=&nbsp;±0.42768661... ({{OEIS2C|id=A245262}}).
(Apart from the trivial [[inflection point]] at <math>x = 0,</math> <math>F(x) = 0.</math>)