home::buildo...::man::w_of_z(3)

📦
version 3.3 · 12 versions
, in alpine 3.24

💖 NAME 💖

okay so bestie, welcome to home::buildo...::man::w_of_z(3) — the page for w_of_z, im_w_of_x, aka Faddeeva's rescaled complex error function. no cap, this is the math icon of the group chat. 💅

✨ SYNOPSIS ✨

the literal include + signatures, copy-paste energy, zero edits:

#include <cerf.h>

double _Complex w_of_z ( double _Complex z );

double im_w_of_x ( double x );

🦋 DESCRIPTION 🦋

Faddeeva's rescaled complex error function w(z) — also called the plasma dispersion function — is the main character here. 🫶

w_of_z returns w(z) = exp(-z^2) * erfc(-i*z).

im_w_of_x returns Im[w(x)].

bestie, those are the exact returns; do not freestyle the formula, iykyk. 💖

🌸 REFERENCES 🌸

To compute w(z), a combination of two algorithms is used:

  • For sufficiently large |z|, a continued-fraction expansion similar to those described by Gautschi (1970) and Poppe & Wijers (1990).
  • Otherwise, Algorithm 916 by Zaghloul & Ali (2011), which is generally competitive at small |z|, and more accurate than the Poppe & Wijers expansion in some regions, e.g. in the vicinity of z=1+i.
  • To compute Im[w(x)], Chebyshev polynomials and continous fractions are used.

the source stack, with every citation keeping its exact little name tag:

  • Milton Abramowitz and Irene M. Stegun, "Handbook of Mathematical Functions", National Bureau of Standards (1964): Formula (7.1.3) introduces the nameless function w(z).
  • Walter Gautschi, "Efficient computation of the complex error function," SIAM J. Numer. Anal. 7, 187 (1970).
  • G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Soft. 16, 38 (1990).
  • Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38, 15 (2011).
  • Steven G. Johnson, http://ab-initio.mit.edu/Faddeeva

The implementation of im_w_of_x is self-contained. It has been entirely rewritten for libcerf-2.5, and now guarantees a relative accuracy of 2.37e-16 for |x| < 12.

The piecewise polynomial approximation for intermediate values of |x| is explained in Wuttke and Kleinknecht, "Code generation for piecewise Chebyshev approximation" (submitted; preprint available from the authors).

🎀 SEE ALSO 🎀

This function is used to compute several other complex error functions: dawson(3), voigt(3), cerf(3), erfcx(3), erfi(3). obsessed with the crossover. ✨

Homepage: http://apps.jcns.fz-juelich.de/libcerf

👑 AUTHORS 👑

  • Steven G. Johnson, http://math.mit.edu/~stevenj, Massachusetts Institute of Technology, researched the numerics, and implemented the Faddeeva function.
  • Joachim Wuttke <j.wuttke@fz-juelich.de>, Forschungszentrum Juelich, reorganized the code into a library, and wrote this man page.

Please report bugs to the authors.

💜 COPYING 💜

Copyright (c) 2012 Massachusetts Institute of Technology

Copyright (c) 2013-25 Forschungszentrum Juelich GmbH

Software: MIT License.

This documentation: Creative Commons Attribution Share Alike.

that's the whole page, bestie 💖 footer tea: perl v5.42.2, dated 2026-02-15, manual slot home::buildo...::man::w_of_z(3). mwah 🫶

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