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7 Error Functions, Dawson’s and Fresnel IntegralsNotation

§7.1 Special Notation

(For other notation see Notation for the Special Functions.)

x real variable.
z complex variable.
n nonnegative integer.
δ arbitrary small positive constant.
γ Euler’s constant (§5.2(ii)).

Unless otherwise noted, primes indicate derivatives with respect to the argument.

The main functions treated in this chapter are the error function erfz; the complementary error functions erfcz and w(z); Dawson’s integral F(z); the Fresnel integrals (z), C(z), and S(z); the Goodwin–Staton integral G(z); the repeated integrals of the complementary error function inerfc(z); the Voigt functions U(x,t) and V(x,t).

Alternative notations are Q(z)=12erfc(z/2), P(z)=Φ(z)=12erfc(-z/2), Erfz=12πerfz, Erfiz=ez2F(z), C1(z)=C(2/πz), S1(z)=S(2/πz), C2(z)=C(2z/π), S2(z)=S(2z/π).

The notations P(z), Q(z), and Φ(z) are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions.