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$H(y)=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {e^{-x^{2}}}{y-x}}\,dx$

$H(y)$

$1/u=1/(y-x)$

$\pi H(y)=\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-k^{2}/4+iky]\int _{-\infty }^{\infty }dx\,\exp[-(x+ik/2)^{2}]$

$\pi ^{1/2}H(y)=\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-k^{2}/4+iky]$

$\pi ^{1/2}H(y)=e^{-y^{2}}\operatorname {Im} \int _{0}^{\infty }dk\,\exp[-(k/2-iy)^{2}]$

$u=ik/2+y$

$\pi ^{1/2}H(y)=-2e^{-y^{2}}\operatorname {Im} i\int _{y}^{i\infty +y}du\ e^{u^{2}}$

$H(y)=2\pi ^{-1/2}F(y)$

$F(y)$

$H_{n}=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {x^{2n}e^{-x^{2}}}{y-x}}\,dx$

$H_{a}=\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{e^{-ax^{2}} \over y-x}\,dx$

${\partial ^{n}H_{a} \over \partial a^{n}}=(-1)^{n}\pi ^{-1}\operatorname {P.V.} \int _{-\infty }^{\infty }{\frac {x^{2n}e^{-ax^{2}}}{y-x}}\,dx$

$H_{a}=2\pi ^{-1/2}F(y{\sqrt {a}})$

$F'(y)=1-2yF(y)$

$H_{n}=P_{1}(y)+P_{2}(y)F(y)$

$H_{1}=-\pi ^{-1/2}y+2\pi ^{-1/2}y^{2}F(y)$

$H_{n+1}(y)=y^{2}H_{n}(y)-{\frac {(2n-1)!!}{{\sqrt {\pi }}2^{n}}}y$

Description

Dawson function

respect

p.v.

Hilbert

Cauchy principal value

Gaussian

distribution

exponential representation

Fourier representation

function

principal value

real axis

integral

contour

variable

rectangle in the complex plane

derivative

imaginary part of the result

example

change of variable

polynomial

recurrence relation

result

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Wikitext context	[[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]] 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 \| url = http://plms.oxfordjournals.org/content/s1-29/1/519.short \| journal = Proceedings of the London Mathematical Society }}</ref>) is the one-sided Fourier–Laplace [[sine transform]] of the Gaussian function. ==Definition== The Dawson function is defined as either: :<math>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 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'').}} Similarly, :<math>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]] ''w''(''z''), 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>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>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''.}} 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> while for large ''x'' it has the asymptotic expansion :<math> F(x) = \sum_{k=0}^{\infty} \frac{(2k-1)!!}{2^{k+1} x^{2k+1}} = \frac{1}{2 x} + \frac{1}{4 x^3} + \frac{3}{8 x^5} + \cdots,</math> where ''n''!! is the [[double factorial]]. ''F''(''x'') satisfies the differential equation :<math> \frac{dF}{dx} + 2xF=1\,\!</math> with the initial condition ''F''(0) = 0. Consequently, it has extrema for :<math> F(x) = \frac{1}{2 x},</math> resulting in ''x'' = ±0.92413887... ({{OEIS2C\|id=A133841}}), ''F''(''x'') = ±0.54104422... ({{OEIS2C\|id=A133842}}). Inflection points follow for :<math> F(x) = \frac{x}{2 x^2 - 1},</math> resulting in ''x'' = ±1.50197526... ({{OEIS2C\|id=A133843}}), ''F''(''x'') = ±0.42768661... ({{OEIS2C\|id=A245262}}). (Apart from the trivial inflection point at ''x'' = 0, ''F''(''x'') = 0.) == Relation to Hilbert transform of Gaussian == The [[Hilbert transform]] of the Gaussian is defined as :<math> 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> {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> \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> \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> \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> \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> 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>H_n = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{x^{2n}e^{-x^2}}{y-x} \, dx </math> Introduce :<math>H_a = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty {e^{-ax^2} \over y-x} \, dx </math> The ''n''th derivative is :<math>{\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> \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> H_{n+1}(y) = y^2 H_n(y) - \frac{(2n-1)!!}{\sqrt{\pi} 2^n} y. </math> ==References== <references /> == External links == * [https://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the [[GNU Scientific Library]] * [http://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:Special functions]] [[Category:Gaussian function]]
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