Dawson function

Image:DawsonDp.svg

Image:DawsonDm.svg

The Dawson function is defined as either:

$$D\_+(x)\; =\; e^\{-x^2\}\; \backslash int\_0^x\; e^\{t^2\}\backslash ,dt,$$

also denoted as $F(x)$ or $D(x),$

or alternatively

$$D\_-(x)\; =\; e^\{x^2\}\; \backslash int\_0^x\; e^\{-t^2\}\backslash ,dt.\backslash !$$

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

$$D\_+(x)\; =\; \backslash frac12\; \backslash int\_0^\backslash infty\; e^\{-t^2/4\}\backslash ,\backslash sin(xt)\backslash ,dt.$$

It is closely related to the error function erf, as

:$D\_+(x)\; =\; \{\backslash sqrt\{\backslash pi\}\; \backslash over\; 2\}\; e^\{-x^2\}\; \backslash operatorname\{erfi\}\; (x)\; =\; -\; \{i\; \backslash sqrt\{\backslash pi\}\; \backslash over\; 2\; \}e^\{-x^2\}\; \backslash operatorname\{erf\}\; (ix)$

where erfi is the imaginary error function, {{nowrap|1=erfi(x) = −i erf(ix).}}

Similarly,

$$D\_-(x)\; =\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}\; e^\{x^2\}\; \backslash operatorname\{erf\}(x)$$

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: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].

$$F(z)\; =\; \{\backslash sqrt\{\backslash pi\}\; \backslash over\; 2\}\; e^\{-z^2\}\; \backslash operatorname\{erfi\}\; (z)\; =\; \backslash frac\{i\backslash sqrt\{\backslash pi\}\}\{2\}\; \backslash left[\; e^\{-z^2\}\; -\; w(z)\; \backslash right],$$

which simplifies to

$$D\_+(x)\; =\; F(x)\; =\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}\; \backslash operatorname\{Im\}[w(x)]$$

$$D\_-(x)\; =\; i\; F(-ix)\; =\; -\backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}\; \backslash left[\; e^\{x^2\}\; -\; w(-ix)\; \backslash right]$$

for real $x.$

For $|x|$ near zero, {{nowrap|1=F(x) ≈ x.}}

For $|x|$ large, {{nowrap|1=F(x) ≈ 1/(2x).}}

More specifically, near the origin it has the series expansion

$$F(x)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^k\; \backslash ,\; 2^k\}\{(2k+1)!!\}\; \backslash ,\; x^\{2k+1\}$$

= x - \frac{2}{3} x^3 + \frac{4}{15} x^5 - \cdots,

while for large $x$ it has the asymptotic expansion

$$F(x)\; =\; \backslash frac\{1\}\{2\; x\}\; +\; \backslash frac\{1\}\{4\; x^3\}\; +\; \backslash frac\{3\}\{8\; x^5\}\; +\; \backslash cdots.$$

More precisely

$$\backslash left|F(x)\; -\; \backslash sum\_\{k=0\}^\{N\}\; \backslash frac\{(2k-1)!!\}\{2^\{k+1\}\; x^\{2k+1\}\}\backslash right|\; \backslash leq\; \backslash frac\{C\_N\}\{x^\{2N+3\}\}.$$

where $n!!$ is the double factorial.

$F(x)$ satisfies the differential equation

$$\backslash frac\{dF\}\{dx\}\; +\; 2xF\; =\; 1\backslash ,\backslash !$$

with the initial condition $F(0)\; =\; 0.$ Consequently, it has extrema for

$$F(x)\; =\; \backslash frac\{1\}\{2\; x\},$$

resulting in x = ±0.92413887... ({{OEIS2C|id=A133841}}), F(x) = ±0.54104422... ({{OEIS2C|id=A133842}}).

Inflection points follow for

$$F(x)\; =\; \backslash frac\{x\}\{2\; x^2\; -\; 1\},$$

resulting in x = ±1.50197526... ({{OEIS2C|id=A133843}}), F(x) = ±0.42768661... ({{OEIS2C|id=A245262}}).

(Apart from the trivial inflection point at $x\; =\; 0,$ $F(x)\; =\; 0.$)

The Hilbert transform of the Gaussian is defined as

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

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

$$\{1\; \backslash over\; u\}\; =\; \backslash int\_0^\backslash infty\; dk\; \backslash ,\; \backslash sin\; ku\; =\; \backslash int\_0^\backslash infty\; dk\; \backslash ,\; \backslash operatorname\{Im\}\; e^\{iku\}.$$

With $1/u\; =\; 1/(y-x),$ we use the exponential representation of $\backslash sin(ku)$ and complete the square with respect to $x$ to find

$$\backslash pi\; H(y)\; =\; \backslash operatorname\{Im\}\; \backslash int\_0^\backslash infty\; dk\; \backslash ,\backslash exp[-k^2/4+iky]\; \backslash int\_\{-\backslash infty\}^\backslash infty\; dx\; \backslash ,\; \backslash exp[-(x+ik/2)^2].$$

We can shift the integral over $x$ to the real axis, and it gives $\backslash pi^\{1/2\}.$

Thus

$$\backslash pi^\{1/2\}\; H(y)\; =\; \backslash operatorname\{Im\}\; \backslash int\_0^\backslash infty\; dk\; \backslash ,\; \backslash exp[-k^2/4+iky].$$

We complete the square with respect to $k$ and obtain

$$\backslash pi^\{1/2\}H(y)\; =\; e^\{-y^2\}\; \backslash operatorname\{Im\}\; \backslash int\_0^\backslash infty\; dk\; \backslash ,\; \backslash exp[-(k/2-iy)^2].$$

We change variables to $u\; =\; ik/2+y:$

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

The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives

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

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

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

Introduce

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

The $n$th derivative is

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

We thus find

$$\backslash left\; .\; H\_n\; =\; (-1)^n\; \backslash frac\{\backslash partial^nH\_a\}\{\backslash partial\; a^n\}\; \backslash right|\_\{a=1\}.$$

The derivatives are performed first, then the result evaluated at $a\; =\; 1.$ A change of variable also gives $H\_a\; =\; 2\backslash pi^\{-1/2\}F(y\backslash sqrt\; a).$ Since $F\text{'}(y)\; =\; 1-2yF(y),$ we can write $H\_n\; =\; P\_1(y)+P\_2(y)F(y)$ where $P\_1$ and $P\_2$ are polynomials. For example, $H\_1\; =\; -\backslash pi^\{-1/2\}y\; +\; 2\backslash pi^\{-1/2\}y^2F(y).$ Alternatively, $H\_n$ can be calculated using the recurrence relation (for $n\; \backslash geq\; 0$)

$$H\_\{n+1\}(y)\; =\; y^2\; H\_n(y)\; -\; \backslash frac\{(2n-1)!!\}\{\backslash sqrt\{\backslash pi\}\; 2^n\}\; y.$$

• {{annotated link|List of mathematical functions}}

{{reflist}}

• [https://www.gnu.org/software/gsl/manual/html_node/Dawson-Function.html gsl_sf_dawson] in the GNU Scientific Library
• [https://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:Gaussian function

Category:Special functions