# error functions

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## 11—20 of 168 matching pages

##### 11: 7.22 Methods of Computation
###### §7.22(iii) Repeated Integrals of the Complementary ErrorFunction
The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$. … The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …
##### 12: 7.9 Continued Fractions
###### §7.9 Continued Fractions
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
7.9.3 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
##### 13: 7.15 Sums
###### §7.15 Sums
For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).
##### 14: 7.4 Symmetry
7.4.1 $\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right),$
7.4.2 $\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right),$
##### 15: 7.16 Generalized Error Functions
###### §7.16 Generalized ErrorFunctions
Generalizations of the error function and Dawson’s integral are $\int_{0}^{x}e^{-t^{p}}\mathrm{d}t$ and $\int_{0}^{x}e^{t^{p}}\mathrm{d}t$. …
##### 16: 7.11 Relations to Other Functions
###### Incomplete Gamma Functions and Generalized Exponential Integral
7.11.1 $\operatorname{erf}z=\frac{1}{\sqrt{\pi}}\gamma\left(\tfrac{1}{2},z^{2}\right),$
##### 17: 7.5 Interrelations
###### §7.5 Interrelations
7.5.1 $F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-w\left(z\right)\right)% =-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{erf}\left(iz\right).$
##### 19: 7.13 Zeros
###### §7.13(ii) Zeros of $\operatorname{erfc}z$
The other zeros of $\operatorname{erfc}z$ are $\overline{z}_{n}$. …
##### 20: 7.14 Integrals
###### Laplace Transforms
7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\mathrm{d}t=\frac{1}{% a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
7.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$ $\Re a>0$, $\Re b>0$,