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1: 7.21 Physical Applications
§7.21 Physical Applications
Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ 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
7.10.3 ${{w}^{(n+2)}\left(z\right)+2z{w}^{(n+1)}\left(z\right)+2(n+1){w}^{(n)}\left(z% \right)=0},$ $n=0,1,2,\dots$.
3: 7.2 Definitions
§7.2(i) Error Functions
7.2.3 $w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\,% \mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right).$
4: 7.9 Continued Fractions
7.9.3 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
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.
7: 7.5 Interrelations
7.5.1 $F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-w\left(z\right)\right)% =-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{erf}\left(iz\right).$
7.5.9 $C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\left(1-e^{\pm\frac{1}% {2}\pi iz^{2}}w\left(i\zeta\right)\right).$
8: Bibliography F
• V. N. Faddeeva and N. M. Terent’ev (1954) Tablicy značeniĭ funkcii $w(z)=e^{-z^{2}}(1+\frac{2i}{\sqrt{\pi}}\int^{z}_{0}e^{t^{2}}dt)$ 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 $w(z)=e^{-z^{2}}(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}dt)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
10: 7.7 Integral Representations
7.7.2 $w\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,% \mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\,\mathrm{d% }t}{t^{2}-z^{2}},$ $\Im z>0$.