§7.1 Special Notation

Keywords:: Dawson’s integral, Fresnel integrals, Gaussian, Gaussian probability functions, complementary error function, error functions, normal probability functions, probability functions
Referenced by:: Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/7.1
Clarification (effective with 1.0.1):: Originally the alternative notation $\Phi(z)$ was associated with $Q(z)$ ; it is more properly associated with $P(z)$ .
Reported 2010-10-25 by Ingemar Nasell

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

$x$	real variable.
$z$	complex variable.
$n$	nonnegative integer.
$\delta$	arbitrary small positive constant.
$\gamma$	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.