# for confluent hypergeometric functions

(0.015 seconds)

## 1—10 of 94 matching pages

##### 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. …
##### 8: 13.1 Special Notation
The main functions treated in this chapter are the Kummer functions $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$, Olver’s function ${\mathbf{M}}\left(a,b,z\right)$, and the Whittaker functions $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$. …
##### 9: 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: 13.10 Integrals
13.10.6 $\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}\left(a,b,t^{2}\right)% \mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-\tfrac{1}{2}\right)U% \left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),$ $\Re b>\tfrac{1}{2}$, $\Re z>0$,
13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\mathrm{d}t=\frac% {\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a\right)% \Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
13.10.12 $\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}\left(a,b,-t^{2}\right)% \mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{2a-1}e^{-x^{2}}U\left(b% -\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),$ $\Re a>0$.
13.10.14 $\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{\nu}% \left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{1}{2}\nu}e^{-x}}{\Gamma\left% (b-a\right)}U\left(a,a-b+\nu+2,x\right),$ $x>0$, $-1<\Re\nu<2\Re\left(b-a\right)-\tfrac{1}{2}$,
13.10.16 $\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2% \sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)x^{\frac{1}{2}\nu}e^{-x}% {\mathbf{M}}\left(a,a-b+\nu+2,x\right),$ $x>0$, $\max\left(\Re b-2,-1\right)<\Re\nu$.