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Faddeeva (or Faddeyeva) function

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1: 7.21 Physical Applications
§7.21 Physical Applications
Fried and Conte (1961) mentions the role of w ( z ) in the theory of linearized waves or oscillations in a hot plasma; w ( z ) is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …Efficient algorithms for computing the Faddeeva (or Faddeyeva) function are discussed in Wells (1999), a paper frequently cited in the astrophysics literature. …
2: 7.10 Derivatives
7.10.2 w ( z ) = - 2 z w ( z ) + ( 2 i / π ) ,
7.10.3 w ( n + 2 ) ( z ) + 2 z w ( n + 1 ) ( z ) + 2 ( n + 1 ) w ( n ) ( z ) = 0 , n = 0 , 1 , 2 , .
3: 7.2 Definitions
§7.2(i) Error Functions
7.2.3 w ( z ) = e - z 2 ( 1 + 2 i π 0 z e t 2 d t ) = e - z 2 erfc ( - i z ) .
4: 7.9 Continued Fractions
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .
5: Bibliography Z
  • M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
  • M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.
  • M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.
  • 6: 7.4 Symmetry
    7.4.3 w ( - z ) = 2 e - z 2 - w ( z ) .
    7: 7.5 Interrelations
    7.5.1 F ( z ) = 1 2 i π ( e - z 2 - w ( z ) ) = - 1 2 i π e - z 2 erf ( i z ) .
    7.5.9 C ( z ) ± i S ( z ) = 1 2 ( 1 ± i ) ( 1 - e ± 1 2 π i z 2 w ( i ζ ) ) .
    8: Bibliography F
  • V. N. Faddeeva and N. M. Terent’ev (1954) Tablicy značeniĭ funkcii w ( z ) = e - z 2 ( 1 + 2 i π 0 z e t 2 d t ) ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
  • V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function w ( z ) = e - z 2 ( 1 + 2 i π - 1 / 2 0 z e t 2 d t ) for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
  • 9: 7.6 Series Expansions
    7.6.3 w ( z ) = n = 0 ( i z ) n Γ ( 1 2 n + 1 ) .
    10: 7.7 Integral Representations
    7.7.2 w ( z ) = 1 π i - e - t 2 d t t - z = 2 z π i 0 e - t 2 d t t 2 - z 2 , z > 0 .