7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.7 Integral Representations7.9 Continued Fractions

Let $\U0001d5ac\left(x\right)$ denote *Mills’ ratio*:

7.8.1 | $$\U0001d5ac\left(x\right)=\frac{{\int}_{x}^{\mathrm{\infty}}{\mathrm{e}}^{-{t}^{2}}dt}{{\mathrm{e}}^{-{x}^{2}}}={\mathrm{e}}^{{x}^{2}}{\int}_{x}^{\mathrm{\infty}}{\mathrm{e}}^{-{t}^{2}}dt.$$ | ||

(Other notations are often used.) Then

7.8.2 | $$ | ||

$x\ge 0$, | |||

7.8.3 | $$ | ||

$x\ge 0$, | |||

7.8.4 | $$ | ||

$x>-\frac{1}{2}\sqrt{2}$, | |||

7.8.5 | $$ | ||

$x\ge 0$. | |||

Next,

7.8.6 | $$ | ||

$a,x>0$. | |||

7.8.7 | $$ | ||

$x>0$. | |||

The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2{x}^{2}}}$ is strictly decreasing for $x>0$. For these and similar results for Dawson’s integral $F\left(x\right)$ see Janssen (2021).

7.8.8 | $$ | ||

$x>0$. | |||