Faddeeva function

The Faddeeva function (also Kramp function or relativistic plasma dispersion function ) is a scaled complex complementary error function ,

{\ displaystyle w (z): = e ^ {- z ^ {2}} \ operatorname {erfc} (-iz) = e ^ {- z ^ {2}} \ left (1 + {\ frac {2i} {\ sqrt {\ pi}}} \ int _ {0} ^ {z} e ^ {t ^ {2}} {\ text {d}} t \ right).}

It is related to the Fresnel integrals , the Dawson integrals and the Voigt profile . The function is named after Vera Nikolaevna Faddejewa .

history

The function was tabulated in 1954 by Vera Faddejewa and Terentjew. It appears as a nameless function in the standard work by Abramowitz-Stegun (1964), Formula 7.1.3. The name Faddeeva function appears to have been introduced by Poppe and Wijers in 1990. ${\ displaystyle w (z)}$

literature

W. Gautschi ACM Transactions on Mathematical Software (1969?): ACM Algorithmus 363.
W. Gautschi SIAM J. Numer. Anal. 7: 187 (1970).
GPM Poppe, CMJ Wijers, ACM Transactions on Mathematical Software 16, 38-46 (1990): ACM Algorithm 680.
JAC Weideman, SIAM J. Numer. Anal. 31, 1497-1518 (1994): Particularly compact algorithm in 8 lines Matlab.
MR Zaghloul and AN Ali, ACM Transactions on Mathematical Software 38, 15 (2011): ACM Algorithm 916.
SM Abrarov and BM Quine, Appl. Math. Comp. 218: 1894-1902 (2011).
SM Abrarov and BM Quine, Arxiv, Preprint 2012

Implementations

Steven G. Johnson has published a free and open software implementation based on a combination of Algorithms 680 and 916. It is based on the function scipy.special.wofzin the Python SciPy library , and it is also available in the form of a C library libcerf .

swell

↑ UN Faddeeva, NN Terent'ev: Tables of values of the function for complex argument. Gosud. Izdat. Teh.-Teor. Lit. , Moscow, 1954; English transl., Pergamon Press, New York, 1961. ${\ displaystyle w (z) = \ exp (-z ^ {2}) (1 + 2i / {\ sqrt {\ pi}} \ int _ {0} ^ {z} \ exp (t ^ {2}) {\ text {d}} t)}$

↑ Google Scholar Research according to Engl. Wikipedia.

^ Faddeeva Package , under MIT license .

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

[1] UN Faddeeva, NN Terent'ev: Tables of values of the function for complex argument. Gosud. Izdat. Teh.-Teor. Lit. , Moscow, 1954; English transl., Pergamon Press, New York, 1961. ${\ displaystyle w (z) = \ exp (-z ^ {2}) (1 + 2i / {\ sqrt {\ pi}} \ int _ {0} ^ {z} \ exp (t ^ {2}) {\ text {d}} t)}$

[2] Google Scholar Research according to Engl. Wikipedia.

[3] Faddeeva Package , under MIT license .

[4] ttp://apps.jcns.fz-juelich.de/libcerf