# Whittaker functions

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##### 1: 13.27 Mathematical Applications
###### §13.27 Mathematical Applications
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. …
##### 4: 13.18 Relations to Other Functions
###### §13.18(v) Orthogonal Polynomials
For representations of Coulomb functions in terms of Whittaker functions see (33.2.3), (33.2.7), (33.14.4) and (33.14.7)
##### 5: 13.24 Series
###### §13.24(i) Expansions in Series of WhittakerFunctions
For expansions of arbitrary functions in series of $M_{\kappa,\mu}\left(z\right)$ functions see Schäfke (1961b).
###### §13.24(ii) Expansions in Series of Bessel Functions
13.24.2 $\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=2^{2\mu}z^{\mu% +\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^% {-2\mu-s}J_{2\mu+s}\left(2\sqrt{\kappa z}\right),$
##### 6: 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 $W_{\kappa,\mu}\left(z\right)$. …
##### 8: 13.25 Products
###### §13.25 Products
13.25.1 $M_{\kappa,\mu}\left(z\right)M_{\kappa,-\mu-1}\left(z\right)+\frac{(\frac{1}{2}% +\mu+\kappa)(\frac{1}{2}+\mu-\kappa)}{4\mu(1+\mu)(1+2\mu)^{2}}M_{\kappa,\mu+1}% \left(z\right)M_{\kappa,-\mu}\left(z\right)=1.$
##### 9: 13.19 Asymptotic Expansions for Large Argument
###### §13.19 Asymptotic Expansions for Large Argument
13.19.1 $M_{\kappa,\mu}\left(x\right)\sim\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}x}x^{-\kappa}\*\sum_{s=0}^{\infty}% \frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{\left(\frac{1}{2}+\mu+\kappa% \right)_{s}}}{s!}x^{-s},$ $\mu-\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots$.
13.19.2 $M_{\kappa,\mu}\left(z\right)\sim\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}z}z^{-\kappa}\*\sum_{s=0}^{\infty}% \frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{\left(\frac{1}{2}+\mu+\kappa% \right)_{s}}}{s!}z^{-s}+\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{% 2}+\mu+\kappa\right)}e^{-\frac{1}{2}z\pm(\frac{1}{2}+\mu-\kappa)\pi\mathrm{i}}% z^{\kappa}\*\sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}% {\left(\frac{1}{2}-\mu-\kappa\right)_{s}}}{s!}(-z)^{-s},$ $-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$,
13.19.3 $W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa}\sum_{s=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}{\left(\frac{1}{2}-\mu-\kappa% \right)_{s}}}{s!}{(-z)^{-s}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.
##### 10: 13.14 Definitions and Basic Properties
Standard solutions are: … The series …