build::libcerf::src::libcerf-v2.4::man::voigt(3) libcerf manual build::libcerf::src::libcerf-v2.4::man::voigt(3)

voigt - Voigt's function, convolution of Gaussian and Lorentzian

#include <cerf.h>

double voigt ( double x, double sigma, double gamma );

The function voigt returns Voigt's convolution

voigt(x,sigma,gamma) = integral G(t,sigma) L(x-t,gamma) dt

of a Gaussian

G(x,sigma) = 1/sqrt(2*pi)/|sigma| * exp(-x^2/2/sigma^2)

and a Lorentzian

L(x,gamma) = |gamma| / pi / ( x^2 + gamma^2 ),

with the integral extending from -infinity to +infinity.

If sigma=0, L(x,gamma) is returned. Conversely, if gamma=0, G(x,sigma) is returned.

If sigma=gamma=0, the return value is Inf for x=0, and 0 for all other x. It is advisable to test input arguments to exclude this irregular case.

Formula (7.4.13) in Abramowitz & Stegun (1964) relates Voigt's convolution integral to Faddeeva's function w_of_z, upon which this implementation is based:

voigt(x,sigma,gamma) = Re[w(z)] / sqrt(2*pi) / |sigma|


z = (x+i*|gamma|) / sqrt(2) / |sigma|.


Related complex error functions: w_of_z(3), dawson(3), cerf(3), erfcx(3), erfi(3).


Joachim Wuttke <>, Forschungszentrum Juelich,
based on the w_of_z implementation by Steven G. Johnson,, Massachusetts Institute of Technology.

Please report bugs to the authors.

Copyright (c) 2013 Forschungszentrum Juelich GmbH

Software: MIT License.

This documentation: Creative Commons Attribution Share Alike.

2023-08-02 perl v5.36.1