Dawson function

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Mathematical function

title: "Dawson function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["gaussian-function", "special-functions"] description: "Mathematical function" topic_path: "general/gaussian-function" source: "https://en.wikipedia.org/wiki/Dawson_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical function ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/6e/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" caption="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" 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"] ::

In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson{{cite journal | author = Dawson, H. G. | title = On the Numerical Value of \textstyle\int_0^h \exp(x^2) , dx | 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 is the one-sided Fourier–Laplace sine transform of the Gaussian function.

Definition

::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/90/DawsonDp.svg" caption="The Dawson function, F(x) = D_+(x), around the origin"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f0/DawsonDm.svg" caption="The Dawson function, D_-(x), around the origin"] ::

The Dawson function is defined as either: D_+(x) = e^{-x^2} \int_0^x e^{t^2},dt, also denoted as F(x) or D(x), or alternatively D_-(x) = e^{x^2} \int_0^x e^{-t^2},dt.!

The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function, D_+(x) = \frac12 \int_0^\infty e^{-t^2/4},\sin(xt),dt.

It is closely related to the error function erf, as : D_+(x) = {\sqrt{\pi} \over 2} e^{-x^2} \operatorname{erfi} (x) = - {i \sqrt{\pi} \over 2 }e^{-x^2} \operatorname{erf} (ix) where erfi is the imaginary error function, Similarly, D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \operatorname{erf}(x) 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: F(z) = {\sqrt{\pi} \over 2} e^{-z^2} \operatorname{erfi} (z) = \frac{i\sqrt{\pi}}{2} \left[ e^{-z^2} - w(z) \right], which simplifies to D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[w(x)] D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right] for real x.

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

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

F(x) satisfies the differential equation \frac{dF}{dx} + 2xF = 1,! with the initial condition F(0) = 0. Consequently, it has extrema for F(x) = \frac{1}{2 x}, resulting in x = ±0.92413887... (), F(x) = ±0.54104422... ().

Inflection points follow for F(x) = \frac{x}{2 x^2 - 1}, resulting in x = ±1.50197526... (), F(x) = ±0.42768661... (). (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 H(y) = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{e^{-x^2}}{y-x} , dx

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

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

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

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

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

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

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

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

The nth derivative is {\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.

We thus find \left . H_n = (-1)^n \frac{\partial^nH_a}{\partial a^n} \right|_{a=1}.

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

References

Temme, N. M.. "Error Functions, Dawson's and Fresnel Integrals".
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].

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