§8.5 Confluent Hypergeometric Representations

Notes:: See Slater (1960, §5.6) and §13.2(vii). For (8.5.4) see (13.18.4).
Keywords:: confluent hypergeometric functions, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.5

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 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=a^{{-1}}z^{a}e^{{-z}}\mathop{M\/}\nolimits\!\left(1,1+a,z\right)=a^{{-1}}z^{a}\mathop{M\/}\nolimits\!\left(a,1+a,-z\right),$ $a\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
A&S Ref:: 6.5.12
Referenced by:: §8.11(ii), §8.14
Permalink:: http://dlmf.nist.gov/8.5.E1
Encodings:: TeX, pMML, png

8.5.2 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=e^{{-z}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(1,1+a,z\right)=\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,1+a,-z\right).$

Symbols:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.7
Permalink:: http://dlmf.nist.gov/8.5.E2
Encodings:: TeX, pMML, png

8.5.3 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=e^{{-z}}\mathop{U\/}\nolimits\!\left(1-a,1-a,z\right)=z^{a}e^{{-z}}\mathop{U\/}\nolimits\!\left(1,1+a,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.14, §8.19(vi)
Permalink:: http://dlmf.nist.gov/8.5.E3
Encodings:: TeX, pMML, png

8.5.4 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=a^{{-1}}z^{{\frac{1}{2}a-\frac{1}{2}}}e^{{-\frac{1}{2}z}}\mathop{M_{{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.5
Permalink:: http://dlmf.nist.gov/8.5.E4
Encodings:: TeX, pMML, png

8.5.5 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=e^{{-\frac{1}{2}z}}z^{{\frac{1}{2}a-\frac{1}{2}}}\mathop{W_{{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.5.E5
Encodings:: TeX, pMML, png