# §8.5 Confluent Hypergeometric Representations

For the confluent hypergeometric functions $\mathop{M\/}\nolimits$, $\mathop{{\mathbf{M}}\/}\nolimits$, $\mathop{U\/}\nolimits$, and the Whittaker functions $\mathop{M_{\kappa,\mu}\/}\nolimits$ and $\mathop{W_{\kappa,\mu}\/}\nolimits$, see §§13.2(i) and 13.14(i).

 8.5.1 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=a^{-1}z^{a}e^{-z}\mathop{M\/}% \nolimits\!\left(1,1+a,z\right)=a^{-1}z^{a}\mathop{M\/}\nolimits\!\left(a,1+a,% -z\right),$ $a\neq 0,-1,-2,\dots$.
 8.5.2 $\mathop{\gamma^{*}\/}\nolimits\!\left(a,z\right)=e^{-z}\mathop{{\mathbf{M}}\/}% \nolimits\!\left(1,1+a,z\right)=\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,1+a,% -z\right).$
 8.5.3 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=e^{-z}\mathop{U\/}\nolimits\!% \left(1-a,1-a,z\right)=z^{a}e^{-z}\mathop{U\/}\nolimits\!\left(1,1+a,z\right).$
 8.5.4 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac{1}{2}% }e^{-\frac{1}{2}z}\mathop{M_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\/}% \nolimits\!\left(z\right).$
 8.5.5 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-% \frac{1}{2}}\mathop{W_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\/}\nolimits\!% \left(z\right).$