7 Error Functions, Dawson’s and Fresnel IntegralsApplications7.19 Voigt Functions7.21 Physical Applications

Let the set $\left\{x\left(t\right),y\left(t\right),t\right\}$ be defined by $x\left(t\right)=C\left(t\right)$,
$y\left(t\right)=S\left(t\right)$, $t\ge 0$. Then the set $\left\{x\left(t\right),y\left(t\right)\right\}$ is called
*Cornu’s spiral*: it is the projection of the corkscrew on the
$\left\{x,y\right\}$-plane. See Figure 7.20.1. The spiral has several special
properties (see Temme (1996b, p. 184)). Let $P\left(t\right)=P\left(x\left(t\right),y\left(t\right)\right)$ be
any point on the projected spiral. Then the arc length between the origin and
$P\left(t\right)$ equals $t$, and is directly proportional to the curvature at $P\left(t\right)$,
which equals $\pi t$. Furthermore, because $dy/dx=\mathrm{tan}\left(\frac{1}{2}\pi {t}^{2}\right)$, the angle between the $x$-axis and the tangent to
the spiral at $P\left(t\right)$ is given by $\frac{1}{2}\pi {t}^{2}$.

The normal distribution function with mean $m$ and standard deviation $\sigma $ is given by

7.20.1 | $$\frac{1}{\sigma \sqrt{2\pi}}{\int}_{-\mathrm{\infty}}^{x}{\mathrm{e}}^{-{\left(t-m\right)}^{2}/\left(2{\sigma}^{2}\right)}dt=\frac{1}{2}\mathrm{erfc}\left(\frac{m-x}{\sigma \sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m}{\sigma}\right).$$ | ||

For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).