7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.11 Relations to Other Functions7.13 Zeros

- §7.12(i) Complementary Error Function
- §7.12(ii) Fresnel Integrals
- §7.12(iii) Goodwin–Staton Integral

As $z\to \mathrm{\infty}$

7.12.1 | $\mathrm{erfc}z$ | $\sim {\displaystyle \frac{{\mathrm{e}}^{-{z}^{2}}}{\sqrt{\pi}}}{\displaystyle \sum _{m=0}^{\mathrm{\infty}}}{(-1)}^{m}{\displaystyle \frac{{\left(\frac{1}{2}\right)}_{m}}{{z}^{2m+1}}},$ | ||

$\mathrm{erfc}\left(-z\right)$ | $\sim 2-{\displaystyle \frac{{\mathrm{e}}^{-{z}^{2}}}{\sqrt{\pi}}}{\displaystyle \sum _{m=0}^{\mathrm{\infty}}}{(-1)}^{m}{\displaystyle \frac{{\left(\frac{1}{2}\right)}_{m}}{{z}^{2m+1}}},$ | |||

both expansions being valid when $|\mathrm{ph}z|\le \frac{3}{4}\pi -\delta $ ($$).

When $|\mathrm{ph}z|\le \frac{1}{4}\pi $ the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when $\mathrm{ph}z=0$. When $$ the remainder terms are bounded in magnitude by $\mathrm{csc}\left(2|\mathrm{ph}z|\right)$ times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by ${z}^{2}$; compare (7.11.2).

The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and

7.12.2 | $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty}}{(-1)}^{m}\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi {z}^{2}/2)}^{2m}},$$ | ||

7.12.3 | $$\mathrm{g}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty}}{(-1)}^{m}\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi {z}^{2}/2)}^{2m+1}},$$ | ||

as $z\to \mathrm{\infty}$ in $$. The remainder terms are given by

7.12.4 | $$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^{m}\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi {z}^{2}/2)}^{2m}}+{R}_{n}^{(\mathrm{f})}(z),$$ | ||

7.12.5 | $$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^{m}\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi {z}^{2}/2)}^{2m+1}},+{R}_{n}^{(\mathrm{g})}(z),$$ | ||

where, for $n=0,1,2,\mathrm{\dots}$ and $$,

7.12.6 | $${R}_{n}^{(\mathrm{f})}(z)=\frac{{(-1)}^{n}}{\pi \sqrt{2}}{\int}_{0}^{\mathrm{\infty}}\frac{{\mathrm{e}}^{-\pi {z}^{2}t/2}{t}^{2n-(1/2)}}{{t}^{2}+1}dt,$$ | ||

7.12.7 | $${R}_{n}^{(\mathrm{g})}(z)=\frac{{(-1)}^{n}}{\pi \sqrt{2}}{\int}_{0}^{\mathrm{\infty}}\frac{{\mathrm{e}}^{-\pi {z}^{2}t/2}{t}^{2n+(1/2)}}{{t}^{2}+1}dt.$$ | ||

When $|\mathrm{ph}z|\le \frac{1}{8}\pi $, ${R}_{n}^{(\mathrm{f})}(z)$ and ${R}_{n}^{(\mathrm{g})}(z)$ are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. They are bounded by $|\mathrm{csc}\left(4\mathrm{ph}z\right)|$ times the first neglected terms when $$.

See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.