# §7.11 Relations to Other Functions

## Incomplete Gamma Functions and Generalized Exponential Integral

For the notation see §§8.2(i) and 8.19(i).

 7.11.1 $\displaystyle\operatorname{erf}z$ $\displaystyle=\frac{1}{\sqrt{\pi}}\gamma\left(\tfrac{1}{2},z^{2}\right),$ 7.11.2 $\displaystyle\operatorname{erfc}z$ $\displaystyle=\frac{1}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2},z^{2}\right),$ 7.11.3 $\displaystyle\operatorname{erfc}z$ $\displaystyle=\frac{z}{\sqrt{\pi}}E_{\frac{1}{2}}\left(z^{2}\right).$

## Confluent Hypergeometric Functions

For the notation see §13.2(i).

 7.11.4 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{% 2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}\right),$
 7.11.5 $\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},\tfrac{% 1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},z^{2}% \right).$
 7.11.6 $C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2% }\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^% {2}\right).$

## Generalized Hypergeometric Functions

For the notation see §§16.2(i) and 16.2(ii).

 7.11.7 $\displaystyle C\left(z\right)$ $\displaystyle=z{{}_{1}F_{2}}\left(\tfrac{1}{4};\tfrac{5}{4},\tfrac{1}{2};-% \tfrac{1}{16}\pi^{2}z^{4}\right),$ 7.11.8 $\displaystyle S\left(z\right)$ $\displaystyle=\tfrac{1}{6}\pi z^{3}{{}_{1}F_{2}}\left(\tfrac{3}{4};\tfrac{7}{4% },\tfrac{3}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right).$