§7.20 Mathematical Applications

§7.20(i) Asymptotics

For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951).

The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv).

§7.20(ii) Cornu’s Spiral

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

§7.20(iii) Statistics

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

 7.20.1 $\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}dt=% \frac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\frac{m-x}{\sigma\sqrt{2}}% \right)=Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m}{\sigma}\right).$

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).