# §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 $\displaystyle\mathop{\mathrm{erf}\/}\nolimits z$ $\displaystyle=\frac{1}{\sqrt{\pi}}\mathop{\gamma\/}\nolimits\!\left(\tfrac{1}{% 2},z^{2}\right),$ 7.11.2 $\displaystyle\mathop{\mathrm{erfc}\/}\nolimits z$ $\displaystyle=\frac{1}{\sqrt{\pi}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{% 2},z^{2}\right),$ 7.11.3 $\displaystyle\mathop{\mathrm{erfc}\/}\nolimits z$ $\displaystyle=\frac{z}{\sqrt{\pi}}\mathop{E_{\frac{1}{2}}\/}\nolimits\!\left(z% ^{2}\right).$

## Confluent Hypergeometric Functions

For the notation see §13.2(i).

 7.11.4 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2z}{\sqrt{\pi}}\mathop{M\/}\nolimits% \!\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}% }\mathop{M\/}\nolimits\!\left(1,\tfrac{3}{2},z^{2}\right),$
 7.11.5 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{1}{\sqrt{\pi}}e^{-z^{2}}\mathop{U\/}% \nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^% {-z^{2}}\mathop{U\/}\nolimits\!\left(1,\tfrac{3}{2},z^{2}\right).$
 7.11.6 $\mathop{C\/}\nolimits\!\left(z\right)+i\mathop{S\/}\nolimits\!\left(z\right)=z% \mathop{M\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}% \right)=ze^{\pi iz^{2}/2}\mathop{M\/}\nolimits\!\left(1,\tfrac{3}{2},-\tfrac{1% }{2}\pi iz^{2}\right).$

## Generalized Hypergeometric Functions

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

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