Editing Dawson function

{{Short description|Mathematical function}}
[[File:Plot of the Dawson integral function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Dawson integral function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Dawson integral function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
In [[mathematics]], the '''Dawson function''' or '''Dawson integral'''<ref>{{dlmf|id=7|title=Error Functions, Dawson's and Fresnel Integrals|first=N. M. |last=Temme}}</ref>
(named after [[H. G. Dawson]]<ref>{{cite journal
|  author = Dawson, H. G. 
| title = On the Numerical Value of <math>\textstyle\int_0^h \exp(x^2) \, dx</math>
| volume = s1-29 | number = 1 | pages = 519–522 | year = 1897 | doi=10.1112/plms/s1-29.1.519
| journal = Proceedings of the London Mathematical Society | url = https://zenodo.org/record/1433401
}}</ref>) 
is the one-sided Fourier–Laplace [[sine transform]] of the Gaussian function.

==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>)

==Relation to Hilbert transform of Gaussian==

The [[Hilbert transform]] of the Gaussian is defined as
<math display=block>H(y) = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{e^{-x^2}}{y-x} \, dx</math>

P.V. denotes the [[Cauchy principal value]], and we restrict ourselves to real <math>y.</math> <math>H(y)</math> can be related to the Dawson function as follows. Inside a principal value integral, we can treat <math>1/u</math> as a [[generalized function]] or distribution, and use the Fourier representation
<math display=block>{1 \over u} = \int_0^\infty dk \, \sin ku = \int_0^\infty dk \, \operatorname{Im} e^{iku}.</math>

With <math>1/u = 1/(y-x),</math> we use the exponential representation of <math>\sin(ku)</math> and complete the square with respect to <math>x</math> to find
<math display=block>\pi H(y) = \operatorname{Im} \int_0^\infty dk \,\exp[-k^2/4+iky] \int_{-\infty}^\infty dx \, \exp[-(x+ik/2)^2].</math>

We can shift the integral over <math>x</math> to the real axis, and it gives <math>\pi^{1/2}.</math>
Thus
<math display=block>\pi^{1/2} H(y) = \operatorname{Im} \int_0^\infty dk \, \exp[-k^2/4+iky].</math>

We complete the square with respect to <math>k</math> and obtain
<math display=block>\pi^{1/2}H(y) = e^{-y^2} \operatorname{Im} \int_0^\infty dk \, \exp[-(k/2-iy)^2].</math>

We change variables to <math>u = ik/2+y:</math>
<math display=block>\pi^{1/2}H(y) = -2e^{-y^2} \operatorname{Im} i \int_y^{i\infty+y} du\ e^{u^2}.</math>

The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives
<math display=block>H(y) = 2\pi^{-1/2} F(y)</math>
where <math>F(y)</math> is the Dawson function as defined above.

The Hilbert transform of <math>x^{2n}e^{-x^2}</math> is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let
<math display=block>H_n = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{x^{2n}e^{-x^2}}{y-x} \, dx.</math>

Introduce
<math display=block>H_a = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty {e^{-ax^2} \over y-x} \, dx.</math>

The <math>n</math>th derivative is
<math display=block>{\partial^nH_a \over \partial a^n} = (-1)^n\pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{x^{2n}e^{-ax^2}}{y-x} \, dx.</math>

We thus find
<math display=block>\left . H_n = (-1)^n \frac{\partial^nH_a}{\partial a^n} \right|_{a=1}.</math>

The derivatives are performed first, then the result evaluated at <math>a = 1.</math> A change of variable also gives <math>H_a = 2\pi^{-1/2}F(y\sqrt a).</math> Since <math>F'(y) = 1-2yF(y),</math> we can write <math>H_n = P_1(y)+P_2(y)F(y)</math> where <math>P_1</math> and <math>P_2</math> are polynomials. For example, <math>H_1 = -\pi^{-1/2}y + 2\pi^{-1/2}y^2F(y).</math> Alternatively, <math>H_n</math> can be calculated using the [[recurrence relation]] (for <math>n \geq 0</math>)
<math display=block>H_{n+1}(y) = y^2 H_n(y) - \frac{(2n-1)!!}{\sqrt{\pi} 2^n} y.</math>

==See also==

* {{annotated link|List of mathematical functions}}

==References==

{{reflist}}

==External links==

* [https://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the [[GNU Scientific Library]]
* [https://jugit.fz-juelich.de/mlz/libcerf libcerf], numeric C library for complex error functions, provides a function ''voigt(x, sigma, gamma)'' with approximately 13–14 digits precision. It is based on the [[Faddeeva function]] as implemented in the [http://ab-initio.mit.edu/Faddeeva MIT Faddeeva Package]
* [http://mathworld.wolfram.com/DawsonsIntegral.html Dawson's Integral] ''(at Mathworld)''
* [http://nlpc.stanford.edu/nleht/Science/reference/errorfun.pdf Error functions]

[[Category:Gaussian function]]
[[Category:Special functions]]