The Faddeeva function or Kramp function is a scaled complex complementary error function,
formula_1
It is related to the Fresnel integral, to Dawson's integral, and to the Voigt function.
The function arises in various physical problems, typically relating to electromagnetic responses in complicated media.
Properties.
Real and imaginary parts.
The decomposition into real and imaginary parts is usually written
formula_2,
where "V" and "L" are called the real and imaginary "Voigt functions", since "V(x,y)" is the Voigt profile (up to prefactors).
Sign inversion.
For sign-inverted arguments, the following both apply:
formula_3
and
formula_4
where * denotes complex conjugate.
Relation to the complementary error function.
The Faddeeva function evaluated on imaginary arguments equals the scaled complementary error function (formula_5): 
formula_6,
where erfc is the complementary error function. For large real "x":
formula_7
Derivative.
In some applications, it is necessary to know not only the original values of the Faddeeva function, but also its derivative (e.g. in Non-linear least squares regression in spectroscopy). Its derivative is given by:
formula_8
This expression can also be broken down further in terms of changes in the real and imaginary part of the Faddeeva function formula_9 and formula_10. Basically, this requires knowledge about the real and imaginary part of the product formula_11. Making use of the above definition formula_12, the derivative can therefore be split into partial derivatives with respect to formula_13 and formula_14 as follows:
formula_15andformula_16
formula_17andformula_18
A practical example for the use of these partial derivatives can be found here.
Integral representation.
The Faddeeva function occurs as
formula_19
meaning that it is a convolution of a Gaussian with a simple pole.
History.
The function was tabulated by Vera Faddeeva and N. N. Terentyev in 1954. It appears as nameless function "w(z)" in "Abramowitz and Stegun" (1964), formula 7.1.3. The name Faddeeva function was apparently introduced by G. P. M. Poppe and C. M. J. Wijers in 1990; previously, it was known as Kramp's function (probably after Christian Kramp).
Early implementations used methods by Walter Gautschi (1969–70; ACM Algorithm 363) or by J. Humlicek (1982). A more efficient algorithm was proposed by Poppe and Wijers (1990; ACM Algorithm 680). J.A.C. Weideman (1994) proposed a particularly short algorithm that takes no more than eight lines of MATLAB code. Zaghloul and Ali pointed out deficiencies of previous algorithms and proposed a new one (2011; ACM Algorithm 916). Another algorithm has been proposed by M. Abrarov and B.M. Quine (2011/2012).
Implementations.
Two software implementations, which are free for non-commercial use only, were published in ACM Transactions on Mathematical Software (TOMS) as Algorithm 680 (in Fortran, later translated into C) and Algorithm 916 by Zaghloul and Ali (in MATLAB).
A free and open source C or C++ implementation derived from a combination of Algorithm 680 and Algorithm 916 (using different algorithms for different "z") is also available under the MIT License, and is maintained as a library package "libcerf".
This implementation is also available as a plug-in for Matlab, GNU Octave, and in Python via Scipy as codice_1 (which was originally the TOMS 680 code, but was replaced due to copyright concerns).
For fast but still accurate calculation of plasma dispersion function formula_20, the J-pole Pade approximation is found to be useful, i.e.,
formula_21
For J=8, one can use
b1= -0.017340112270401 - 0.046306439626294i;
b2= -0.739917811220052 + 0.839518284620274i;
b3= 5.840632105105495 + 0.953602751322040i;
b4= -5.583374181615043 -11.208550459628098i;
c1= 2.237687725134293 - 1.625941024120362i;
c2= 1.465234091939142 - 1.789620299603315i;
c3= 0.839253966367922 - 1.891995211531426i;
c4= 0.273936218055381 - 1.941787037576095i;
b(5:8)=(b(1:4))*;
c(5:8)=-(c(1:4))*,
where * denotes complex conjugate. The above approximation is valid for the upper plane. For formula_22 weakly close to the real axis, i.e., formula_23 not far from the real axis of the lower plane, the above approximation is also valid. For accurate calculation of 
formula_24, one can use
formula_25
A coding of the above J-pole approach and comparison with an accurate plasma dispersion function can be found. Comphensive coefficients to rapid computation of the plasma dispersion function with rational and multi-pole approximation are also available.
References.
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