Dawson's Integral

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

Definition

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

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

It is closely related to the error function erf, as
:$D_+(x) = e^ \operatorname (x) = - e^ \operatorname (ix)$
where erfi is the imaginary error function,


Similarly,
$D_-(x) = \frac e^ \operatorname(x)$
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: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 a
arXiv:1106.0151

F(z) = e^ \operatorname (z) = \frac \left e^ - w(z) \right
which simplifies to
D_+(x) = F(x) = \frac \operatorname (x)/math>
D_-(x) = i F(-ix) = -\frac \left e^ - w(-ix) \right/math>
for real $x.$

For $, x,$ near zero,
For $, x,$ large,
More specifically, near the origin it has the series expansion
$F(x) = \sum_^\infty \frac \, x^ = x - \frac x^3 + \frac x^5 - \cdots,$
while for large $x$ it has the asymptotic expansion
$F(x) = \frac + \frac + \frac + \cdots.$

More precisely
$\left, F(x) - \sum_^ \frac\ \leq \frac.$
where $n!!$ is the double factorial.

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

Inflection points follow for
$F(x) = \frac,$
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
$H(y) = \pi^ \operatorname \int_^\infty \frac \, dx$

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
$= \int_0^\infty dk \, \sin ku = \int_0^\infty dk \, \operatorname e^.$

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 \int_0^\infty dk \,\exp k^2/4+iky\int_^\infty dx \, \exp (x+ik/2)^2

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

We complete the square with respect to $k$ and obtain
\pi^H(y) = e^ \operatorname \int_0^\infty dk \, \exp (k/2-iy)^2

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

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

The Hilbert transform of $x^e^$ is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let
$H_n = \pi^ \operatorname \int_^\infty \frac \, dx.$

Introduce
$H_a = \pi^ \operatorname \int_^\infty \, dx.$

The $n$th derivative is
$= (-1)^n\pi^ \operatorname \int_^\infty \frac \, dx.$

We thus find
$\left . H_n = (-1)^n \frac \_.$

The derivatives are performed first, then the result evaluated at $a = 1.$ A change of variable also gives $H_a = 2\pi^F(y\sqrt a).$ 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, $H_1 = -\pi^y + 2\pi^y^2F(y).$ Alternatively, $H_n$ can be calculated using the recurrence relation (for $n \geq 0$)
$H_(y) = y^2 H_n(y) - \frac y.$

See also

*

References

{{reflist

External links

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 th
MIT Faddeeva Package


''(at Mathworld)''

Error functions

Gaussian function
Special functions