Dawson function

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In mathematics, the Dawson function or Dawson integral[1] (named after H. G. Dawson[2]) is the one-sided Fourier–Laplace sine transform of the Gaussian function.

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
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

Definition

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The Dawson function,   around the origin
 
The Dawson function,   around the origin

The Dawson function is defined as either:   also denoted as   or   or alternatively  

The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function,  

It is closely related to the error function erf, as

 

where erfi is the imaginary error function, erfi(x) = −i erf(ix).
Similarly,   in terms of the real error function, erf.

In terms of either erfi or the Faddeeva function   the Dawson function can be extended to the entire complex plane:[3]   which simplifies to     for real  

For   near zero, F(x) ≈ x. For   large, F(x) ≈ 1/(2x). More specifically, near the origin it has the series expansion   while for large   it has the asymptotic expansion  

More precisely   where   is the double factorial.

  satisfies the differential equation   with the initial condition   Consequently, it has extrema for   resulting in x = ±0.92413887... (OEISA133841), F(x) = ±0.54104422... (OEISA133842).

Inflection points follow for   resulting in x = ±1.50197526... (OEISA133843), F(x) = ±0.42768661... (OEISA245262). (Apart from the trivial inflection point at    )

Relation to Hilbert transform of Gaussian

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The Hilbert transform of the Gaussian is defined as  

P.V. denotes the Cauchy principal value, and we restrict ourselves to real     can be related to the Dawson function as follows. Inside a principal value integral, we can treat   as a generalized function or distribution, and use the Fourier representation  

With   we use the exponential representation of   and complete the square with respect to   to find  

We can shift the integral over   to the real axis, and it gives   Thus  

We complete the square with respect to   and obtain  

We change variables to    

The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives   where   is the Dawson function as defined above.

The Hilbert transform of   is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let  

Introduce  

The  th derivative is  

We thus find  

The derivatives are performed first, then the result evaluated at   A change of variable also gives   Since   we can write   where   and   are polynomials. For example,   Alternatively,   can be calculated using the recurrence relation (for  )  

See also

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References

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  1. ^ Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  2. ^ Dawson, H. G. (1897). "On the Numerical Value of  ". Proceedings of the London Mathematical Society. s1-29 (1): 519–522. doi:10.1112/plms/s1-29.1.519.
  3. ^ 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.
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