§7.1 Special Notation

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

$x$ real variable. complex variable. nonnegative integer. arbitrary small positive constant. 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 $\mathop{\mathrm{erf}\/}\nolimits z$; the complementary error functions $\mathop{\mathrm{erfc}\/}\nolimits z$ and $\mathop{w\/}\nolimits\!\left(z\right)$; Dawson’s integral $\mathop{F\/}\nolimits\!\left(z\right)$; the Fresnel integrals $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$, $\mathop{C\/}\nolimits\!\left(z\right)$, and $\mathop{S\/}\nolimits\!\left(z\right)$; the Goodwin–Staton integral $\mathop{G\/}\nolimits\!\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}\/}\nolimits\!\left(z\right)$; the Voigt functions $\mathop{\mathsf{U}\/}\nolimits\!\left(x,t\right)$ and $\mathop{\mathsf{V}\/}\nolimits\!\left(x,t\right)$.

Alternative notations are $Q(z)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits z$, $\operatorname{Erfi}z=e^{z^{2}}\mathop{F\/}\nolimits\!\left(z\right)$, $C_{1}(z)=\mathop{C\/}\nolimits\!\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=\mathop{S\/}\nolimits\!\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=\mathop{C\/}\nolimits\!\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=\mathop{S\/}\nolimits\!\left(\sqrt{2z/\pi}\right)$.

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