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 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, 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:

near zero, large, it has the asymptotic expansion

is the double factorial.

satisfies the differential equation with the initial condition resulting in x=±0.92413887... (), F(x)=±0.54104422... ().

resulting in x=±1.50197526... (), F(x)=±0.42768661... (). (Apart from the trivial inflection point)

Relation to Hilbert transform of Gaussian

The Hilbert transform

P.V. denotes the Cauchy principal value as a generalized function

is the Dawson function as defined above.

can be calculated using the recurrence relation

See also

References

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 the MIT Faddeeva Package

• Dawson's Integral (at Mathworld)

• Error functions