# Kummer functions

(0.004 seconds)

## 1—10 of 62 matching pages

##### 1: 13.6 Relations to Other Functions
###### §13.6(vi) Generalized Hypergeometric Functions
For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).
##### 2: 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 $W_{\kappa,\mu}\left(z\right)$. …
##### 4: 13.12 Products
###### §13.12 Products
13.12.1 $M\left(a,b,z\right)M\left(-a,-b,-z\right)+\frac{a(a-b)z^{2}}{b^{2}(1-b^{2})}M% \left(1+a,2+b,z\right)M\left(1-a,2-b,-z\right)=1.$
##### 6: 13.31 Approximations
###### §13.31(i) Chebyshev-Series Expansions
13.31.3 $z^{a}U\left(a,1+a-b,z\right)=\lim_{n\to\infty}\frac{A_{n}(z)}{B_{n}(z)}.$
##### 8: 13.13 Addition and Multiplication Theorems
###### §13.13(i) Addition Theorems for $M\left(a,b,z\right)$
The function $M\left(a,b,x+y\right)$ has the following expansions: … The function $U\left(a,b,x+y\right)$ has the following expansions: …
13.13.12 $e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),$ $|y|<|x|$.
##### 9: 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. …
##### 10: 13.4 Integral Representations
###### §13.4(i) Integrals Along the Real Line
13.4.6 $U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{\Gamma\left(1+a-b\right)}\int_{0}% ^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t\right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t,$ $\left|\operatorname{ph}z\right|<\pi$, $n=0,1,2,\dots$, $-\Re b,