7 Error Functions, Dawson’s and Fresnel Integrals Computation 7.21 Physical Applications 7.23 Tables

§7.22 Methods of Computation

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Permalink:: http://dlmf.nist.gov/7.22
See also:: Annotations for Ch.7

§7.22(i) Main Functions

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Keywords:: Dawson’s integral, Fresnel integrals, auxiliary functions for Fresnel integrals, computation, error functions, inverse functions
Referenced by:: §3.10(iii)
Permalink:: http://dlmf.nist.gov/7.22.i
See also:: Annotations for §7.22 and Ch.7

The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions.

§7.22(ii) Goodwin–Staton Integral

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Keywords:: Goodwin–Staton integral, computation
Permalink:: http://dlmf.nist.gov/7.22.ii
See also:: Annotations for §7.22 and Ch.7

See Goodwin and Staton (1948).

§7.22(iii) Repeated Integrals of the Complementary Error Function

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Keywords:: computation, repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.22.iii
See also:: Annotations for §7.22 and Ch.7

The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$ . See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961).

§7.22(iv) Voigt Functions

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Keywords:: Voigt functions, computation
Permalink:: http://dlmf.nist.gov/7.22.iv
See also:: Annotations for §7.22 and Ch.7

The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3).

§7.22(v) Other References

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Permalink:: http://dlmf.nist.gov/7.22.v
See also:: Annotations for §7.22 and Ch.7

For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).

7.21 Physical Applications 7.23 Tables

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