Dawson function
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In mathematics, the Dawson function or Dawson integral<ref>Template:Dlmf</ref> (named after H. G. Dawson<ref>Template:Cite journal</ref>) is the one-sided Fourier–Laplace sine transform of the Gaussian function.
DefinitionEdit
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, Template:Nowrap
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, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38 (2), 15 (2011). Preprint available at 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, Template:Nowrap For <math>|x|</math> large, Template:Nowrap 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 = ±0.92413887... (Template:OEIS2C), F(x) = ±0.54104422... (Template:OEIS2C).
Inflection points follow for <math display=block>F(x) = \frac{x}{2 x^2 - 1},</math> resulting in x = ±1.50197526... (Template:OEIS2C), F(x) = ±0.42768661... (Template:OEIS2C). (Apart from the trivial inflection point at <math>x = 0,</math> <math>F(x) = 0.</math>)
Relation to Hilbert transform of GaussianEdit
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 alsoEdit
ReferencesEdit
External linksEdit
- gsl_sf_dawson in the GNU Scientific Library
- 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 MIT Faddeeva Package
- Dawson's Integral (at Mathworld)
- Error functions