build::libcerf::src::libcerf::man::dawson(3)

NAME
cdawson, dawson - Dawson's integral

SYNOPSIS
#include <cerf.h>

double _Complex cdawson ( double _Complex z );

double dawson ( double x );

DESCRIPTION
The function cdawson returns Dawson's integral D(z) = exp(-z^2) integral from 0 to z exp(t^2) dt = sqrt(pi)/2 * exp(-z^2) * erfi(z).

For function dawson takes a real argument x, and returns the real result D(x).

ACCURACY
Errors are given in units of eps = 2^-53 = 1.1e-16, as relative deviations from high-precision reference values; for complex results, of the modulus.

dawson(x) = sqrt(pi)/2 Im w(x) inherits the accuracy of im_w_of_x(3): its relative error was found below 2.7 eps at random points with |x| up to 1e12.

cdawson is computed from w_of_z(3) through the factor exp(-z^2), whose argument -z^2 = (y-x)(x+y) - 2ixy is rounded to double before the exponential is taken. The relative error of the modulus therefore grows in proportion to |z|^2; it stays below about 1.6 |z|^2 eps + 5 eps, and errors close to this bound do occur, for instance 1080 eps at |z| = 26. This limit is intrinsic to the evaluation of exp(-z^2) in double precision. Near the zeros of D(z), which are those of erfi, the first of which lie at z = +-1.8809 +- 1.4506i, cancellation amplifies the relative error without bound. Close to the origin, for 0.005 < |z| < 0.1 off the coordinate axes, where the Maclaurin series is not used, up to about 320 eps were found.

SEE ALSO
The computation of D(z) is based on Faddeeva's function w_of_z(3); to compute D(x), the imaginary part im_w_of_x(3) is used.

Other complex error functions: w_of_z(3), voigt(3), cerf(3), erfcx(3), erfi(3).

Homepage: https://jugit.fz-juelich.de/mlz/lib/cerf

AUTHORS
Steven G. Johnson, Massachusetts Institute of Technology,
  wrote this function as part of the MIT Faddeeva package.

Joachim Wuttke, Forschungszentrum Juelich,
  reorganized the code into a library, and wrote this man page.

CONTACT
Please report bugs to the maintainer:

Joachim Wuttke <j.wuttke@fz-juelich.de>

COPYING
Copyright (c) 2012 Massachusetts Institute of Technology

Copyright (c) 2013 Forschungszentrum Juelich GmbH

Software: MIT License.

This documentation: Creative Commons Attribution Share Alike.