# Whittaker functions

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## 11—20 of 103 matching pages

##### 11: 13.26 Addition and Multiplication Theorems
###### §13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$
The function $M_{\kappa,\mu}\left(x+y\right)$ has the following expansions: …
###### §13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$
The function $W_{\kappa,\mu}\left(x+y\right)$ has the following expansions: …
##### 12: 33.14 Definitions and Basic Properties
###### §33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$
33.14.4 $f\left(\epsilon,\ell;r\right)=\kappa^{\ell+1}M_{\kappa,\ell+\frac{1}{2}}\left(% 2r/\kappa\right)/(2\ell+1)!,$
###### §33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$
33.14.7 $h\left(\epsilon,\ell;r\right)=\frac{\Gamma\left(\ell+1-\kappa\right)}{\pi% \kappa^{\ell}}\left(W_{\kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)+(-1)^{% \ell}S(\epsilon,r)\frac{\Gamma\left(\ell+1+\kappa\right)}{2(2\ell+1)!}M_{% \kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)\right),$
33.14.14 $\phi_{n,\ell}(r)=(-1)^{\ell+1+n}(2/n^{3})^{1/2}s\left(-1/n^{2},\ell;r\right)=% \frac{(-1)^{\ell+1+n}}{n^{\ell+2}}\left(\frac{(n-\ell-1)!}{(n+\ell)!}\right)^{% 1/2}(2r)^{\ell+1}{\mathrm{e}}^{-r/n}L^{(2\ell+1)}_{n-\ell-1}\left(2r/n\right)$
##### 13: 13.16 Integral Representations
###### §13.16(i) Integrals Along the Real Line
13.16.5 $W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-% \frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\mathrm{d}t,$ $\Re\mu+\tfrac{1}{2}>\Re\kappa$, $|\operatorname{ph}{z}|<\frac{1}{2}\pi$,
##### 14: 33.2 Definitions and Basic Properties
The function $F_{\ell}\left(\eta,\rho\right)$ is recessive (§2.7(iii)) at $\rho=0$, and is defined by
33.2.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)2^{-\ell-1}(\mp\mathrm% {i})^{\ell+1}M_{\pm\mathrm{i}\eta,\ell+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho% \right),$
The functions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ are defined by
##### 15: 12.1 Special Notation
An older notation, due to Whittaker (1902), for $U\left(a,z\right)$ is $D_{\nu}\left(z\right)$. …
##### 17: 13.17 Continued Fractions
###### §13.17 Continued Fractions
13.17.1 $\frac{\sqrt{z}M_{\kappa,\mu}\left(z\right)}{M_{\kappa-\frac{1}{2},\mu+\frac{1}% {2}}\left(z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{u_{2}z}{1+\cdots}},$
13.17.3 $\frac{W_{\kappa,\mu}\left(z\right)}{\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}% {2}}\left(z\right)}=1+\cfrac{v_{1}/z}{1+\cfrac{v_{2}/z}{1+\cdots}},$
##### 18: 13.21 Uniform Asymptotic Approximations for Large $\kappa$
###### §13.21 Uniform Asymptotic Approximations for Large $\kappa$
13.21.1 $M_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(2\mu+1\right)\kappa^{-\mu}\*% \left(J_{2\mu}\left(2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0muJ_{2\mu}% \left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$
13.21.6 $M_{-\kappa,\mu}\left(4\kappa x\right)=\frac{2\Gamma\left(2\mu+1\right)}{\kappa% ^{\mu-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}\right)^{\frac{1}{4}}I_{2\mu}\left(% 4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O\left(\kappa^{-1}\right)\right)},$
For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa,\mu}\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu|$ bounded, and $x\in[\delta,\infty)$ see Olver (1997b, Chapter 11, Ex. 7.3). …
##### 19: 8.5 Confluent Hypergeometric 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). …
8.5.4 $\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}M_{% \frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
8.5.5 $\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}W_{\frac{1% }{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$
##### 20: 33.22 Particle Scattering and Atomic and Molecular Spectra
For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$, or $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta,\ell+\frac{1}{2}}\left(2\rho\right)$. …