# confluent hypergeometric functions

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##### 2: 6.11 Relations to Other Functions
###### ConfluentHypergeometricFunction
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 3: 13.28 Physical Applications
###### §13.28(i) Exact Solutions of the Wave Equation
For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). …
##### 4: 13.27 Mathematical Applications
###### §13.27 Mathematical Applications
Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … …
##### 5: 12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U\left(a,b,x\right)$ and $M\left(a,b,x\right)$13.2(i)) whose regions of validity include intervals with endpoints $x=\infty$ and $x=0$, respectively. …
##### 6: 8.5 Confluent Hypergeometric Representations
###### §8.5 ConfluentHypergeometric Representations
For the confluent hypergeometric functions $M$, ${\mathbf{M}}$, $U$, and the Whittaker functions $M_{\kappa,\mu}$ and $W_{\kappa,\mu}$, see §§13.2(i) and 13.14(i). …
8.5.2 $\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M}}% \left(a,1+a,-z\right).$
8.5.3 $\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1,1+a,z% \right).$
8.5.5 $\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}W_{\frac{1% }{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
##### 10: 13.2 Definitions and Basic Properties
13.2.4 $M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{M}}\left(a,b,z\right).$
13.2.34 $\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U\left(a,b,z\right)\right\}=-% \ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)},$
13.2.35 $\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e^{z}U\left(b-a,b,e^{\pm\pi% \mathrm{i}}z\right)\right\}=\ifrac{e^{\mp b\pi\mathrm{i}}z^{-b}e^{z}}{\Gamma% \left(b-a\right)},$
13.2.36 $\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),U\left(a,b,z% \right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1\right)},$