7 Error Functions, Dawson’s and Fresnel IntegralsApplications7.20 Mathematical Applications7.22 Methods of Computation

The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. Fresnel integrals and Cornu’s spiral occurred originally in the analysis of the diffraction of light; see Born and Wolf (1999, §8.7). More recently, Cornu’s spiral appears in the design of highways and railroad tracks, robot trajectory planning, and computer-aided design; see Meek and Walton (1992).

Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^{n}\mathrm{erfc}\left(z\right)$. 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). Ng and Geller (1969) cites work with applications from atomic physics and astrophysics. Efficient algorithms for computing the Faddeeva (or Faddeyeva) function are discussed in Wells (1999), a paper frequently cited in the astrophysics literature.

Voigt functions $\mathsf{U}(x,t)$, $\mathsf{V}(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. These applications include astrophysics, plasma diagnostics, neutron diffraction, laser spectroscopy, and surface scattering. See Mitchell and Zemansky (1961, §IV.2), Armstrong (1967), and Ahn et al. (2001). Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007).