# §13.14 Definitions and Basic Properties

## §13.14(i) Differential Equation

### Whittaker’s Equation

 13.14.1 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}+\left(-\frac{1}{4}+\frac{\kappa}{z% }+\frac{\frac{1}{4}-\mu^{2}}{z^{2}}\right)W=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative and $z$: complex variable A&S Ref: 13.1.31 Referenced by: §13.14(v), §13.14(v), §13.28(i), §13.29(ii) Permalink: http://dlmf.nist.gov/13.14.E1 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

This equation is obtained from Kummer’s equation (13.2.1) via the substitutions $W=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}w$, $\kappa=\tfrac{1}{2}b-a$, and $\mu=\tfrac{1}{2}b-\tfrac{1}{2}$. It has a regular singularity at the origin with indices $\tfrac{1}{2}\pm\mu$, and an irregular singularity at infinity of rank one.

### Standard Solutions

Standard solutions are:

 13.14.2 $M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}M\left(\tfrac% {1}{2}+\mu-\kappa,1+2\mu,z\right),$ ⓘ Defines: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 13.1.32 Referenced by: §13.14(i), §13.14(i), §13.16(ii), §13.19, §13.22, §13.29(ii), §13.4(iii), (9.6.25), (9.6.26) Permalink: http://dlmf.nist.gov/13.14.E2 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13
 13.14.3 $W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}U\left(\tfrac% {1}{2}+\mu-\kappa,1+2\mu,z\right),$ ⓘ Defines: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 13.1.33 Referenced by: §13.14(i), §13.16(ii), §13.19, §13.20(i), §13.22, §13.4(iii) Permalink: http://dlmf.nist.gov/13.14.E3 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

except that $M_{\kappa,\mu}\left(z\right)$ does not exist when $2\mu=-1,-2,-3,\dots$.

Conversely,

 13.14.4 $M\left(a,b,z\right)=e^{\frac{1}{2}z}z^{-\frac{1}{2}b}M_{\frac{1}{2}b-a,\frac{1% }{2}b-\frac{1}{2}}\left(z\right),$
 13.14.5 $U\left(a,b,z\right)=e^{\frac{1}{2}z}z^{-\frac{1}{2}b}W_{\frac{1}{2}b-a,\frac{1% }{2}b-\frac{1}{2}}\left(z\right).$

The series

 13.14.6 $M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}\sum_{s=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}}{{\left(1+2\mu\right)_{% s}}s!}z^{s}=z^{\frac{1}{2}+\mu}\sum_{n=0}^{\infty}{{}_{2}F_{1}}\left({-n,% \tfrac{1}{2}+\mu-\kappa\atop 1+2\mu};2\right)\frac{\left(-\tfrac{1}{2}z\right)% ^{n}}{n!},$ $2\mu\neq-1,-2,-3,\dots$,

converge for all $z\in\mathbb{C}$.

In general $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\infty$. The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu}$ and $U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).

Although $M_{\kappa,\mu}\left(z\right)$ does not exist when $2\mu=-1,-2,-3,\dots$, many formulas containing $M_{\kappa,\mu}\left(z\right)$ continue to apply in their limiting form. For example, if $n=0,1,2,\dots$, then

 13.14.7 $\lim_{2\mu\to-n-1}\frac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right% )}=\frac{{\left(-\frac{1}{2}n-\kappa\right)_{n+1}}}{(n+1)!}M_{\kappa,\frac{1}{% 2}(n+1)}\left(z\right)=e^{-\frac{1}{2}z}z^{-\frac{1}{2}n}\sum_{s=n+1}^{\infty}% \frac{{\left(-\frac{1}{2}n-\kappa\right)_{s}}}{\Gamma\left(s-n\right)s!}z^{s}.$

If $2\mu=\pm n$, where $n=0,1,2,\dots$, then

 13.14.8 $W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{2}z}z^{% \frac{1}{2}n+\frac{1}{2}}}{n!\Gamma\left(\frac{1}{2}-\frac{1}{2}n-\kappa\right% )}\left(\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(\kappa+\frac{1}{2}-\frac{1}% {2}n\right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{\left(\frac{1}{2}n+\frac{1}{% 2}-\kappa\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi\left(% \tfrac{1}{2}n+\tfrac{1}{2}-\kappa+k\right)-\psi\left(1+k\right)-\psi\left(n+1+% k\right)\right)\right),$ $\kappa-\frac{1}{2}n-\frac{1}{2}\neq 0,1,2,\dots$,

or

 13.14.9 $W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=(-1)^{\kappa-\frac{1}{2}n-\frac{1}{2}% }e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}\sum_{k=0}^{\kappa-\frac{1}{2}n-% \frac{1}{2}}\genfrac{(}{)}{0.0pt}{}{\kappa-\frac{1}{2}n-\frac{1}{2}}{k}{\left(% n+1+k\right)_{\kappa-k-\frac{1}{2}n-\frac{1}{2}}}(-z)^{k},$ $\kappa-\frac{1}{2}n-\frac{1}{2}=0,1,2,\dots$.

## §13.14(ii) Analytic Continuation

 13.14.10 $M_{\kappa,\mu}\left(ze^{\pm\pi\mathrm{i}}\right)=\pm\mathrm{i}e^{\pm\mu\pi% \mathrm{i}}M_{-\kappa,\mu}\left(z\right).$

In (13.14.11)–(13.14.13) $m$ is any integer.

 13.14.11 $M_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=(-1)^{m}e^{2m\mu\pi\mathrm{i}}% M_{\kappa,\mu}\left(z\right).$
 13.14.12 $W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=\frac{(-1)^{m+1}2\pi\mathrm{i}% \sin\left(2\pi\mu m\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa\right)\Gamma% \left(1+2\mu\right)\sin\left(2\pi\mu\right)}M_{\kappa,\mu}\left(z\right)+(-1)^% {m}e^{-2m\mu\pi\mathrm{i}}W_{\kappa,\mu}\left(z\right).$
 13.14.13 $(-1)^{m}W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=-\frac{e^{2\kappa\pi% \mathrm{i}}\sin\left(2m\mu\pi\right)+\sin\left((2m-2)\mu\pi\right)}{\sin\left(% 2\mu\pi\right)}W_{\kappa,\mu}\left(z\right)-\frac{\sin\left(2m\mu\pi\right)2% \pi\mathrm{i}e^{\kappa\pi\mathrm{i}}}{\sin\left(2\mu\pi\right)\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}W_{-% \kappa,\mu}\left(ze^{\pi\mathrm{i}}\right).$

Except when $z=0$, each branch of the functions $\ifrac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right)}$ and $W_{\kappa,\mu}\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$ are assumed to have their principal values.

## §13.14(iii) Limiting Forms as $z\to 0$

 13.14.14 $M_{\kappa,\mu}\left(z\right)=z^{\mu+\frac{1}{2}}\left(1+O\left(z\right)\right),$ $2\mu\neq-1,-2,-3,\dots$. ⓘ

In cases when $\frac{1}{2}-\kappa\pm\mu=-n$, where $n$ is a nonnegative integer,

 13.14.15 $W_{\frac{1}{2}\pm\mu+n,\mu}\left(z\right)=(-1)^{n}{\left(1\pm 2\mu\right)_{n}}% z^{\frac{1}{2}\pm\mu}+O\left(z^{\frac{3}{2}\pm\mu}\right).$

In all other cases

 13.14.16 $W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(2\mu\right)}{\Gamma\left(\frac{% 1}{2}+\mu-\kappa\right)}z^{\frac{1}{2}-\mu}+O\left(z^{\frac{3}{2}-\Re\mu}% \right),$ $\Re\mu\geq\frac{1}{2}$, $\mu\not=\frac{1}{2}$,
 13.14.17 $W_{\kappa,\frac{1}{2}}\left(z\right)=\frac{1}{\Gamma\left(1-\kappa\right)}+O% \left(z\ln z\right),$
 13.14.18 $W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(2\mu\right)}{\Gamma\left(\frac{% 1}{2}+\mu-\kappa\right)}z^{\frac{1}{2}-\mu}+\frac{\Gamma\left(-2\mu\right)}{% \Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{\frac{1}{2}+\mu}+O\left(z^{\frac{% 3}{2}-\Re\mu}\right),$ $0\leq\Re\mu<\tfrac{1}{2}$, $\mu\not=0$,
 13.14.19 $W_{\kappa,0}\left(z\right)=-\frac{\sqrt{z}}{\Gamma\left(\frac{1}{2}-\kappa% \right)}\left(\ln z+\psi\left(\tfrac{1}{2}-\kappa\right)+2\gamma\right)+O\left% (z^{\ifrac{3}{2}}\ln z\right).$

For $W_{\kappa,\mu}\left(z\right)$ with $\Re\mu<0$ use (13.14.31).

## §13.14(iv) Limiting Forms as $z\to\infty$

Except when $\mu-\kappa=-\frac{1}{2},-\frac{3}{2},\dots$ (polynomial cases),

 13.14.20 $M_{\kappa,\mu}\left(z\right)\sim\ifrac{\Gamma\left(1+2\mu\right)e^{\frac{1}{2}% z}z^{-\kappa}}{\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)},$ $\left|\operatorname{ph}z\right|\leq\frac{1}{2}\pi-\delta$,

where $\delta$ is an arbitrary small positive constant. Also,

 13.14.21 $W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa},$ $\left|\operatorname{ph}z\right|\leq\frac{3}{2}\pi-\delta$.

## §13.14(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

 13.14.22 $W_{\kappa,\mu}\left(z\right),$ $W_{-\kappa,\mu}\left(e^{-\pi\mathrm{i}}z\right)$, $-\tfrac{1}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{3}{2}\pi$, 13.14.23 $W_{\kappa,\mu}\left(z\right),$ $W_{-\kappa,\mu}\left(e^{\pi\mathrm{i}}z\right)$, $-\tfrac{3}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{1}{2}\pi$.

A fundamental pair of solutions that is numerically satisfactory in the sector $|\operatorname{ph}{z}|\leq\pi$ near the origin is

 13.14.24 $M_{\kappa,\mu}\left(z\right),$ $M_{\kappa,-\mu}\left(z\right)$, $2\mu\not\in\mathbb{Z}$.

When $2\mu$ is an integer we may use the results of §13.2(v) with the substitutions $b=2\mu+1$, $a=\mu-\kappa+\tfrac{1}{2}$, and $W=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}w$, where $W$ is the solution of (13.14.1) corresponding to the solution $w$ of (13.2.1).

## §13.14(vi) Wronskians

 13.14.25 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),M_{\kappa,-\mu}\left(z\right)% \right\}=-2\mu,$ ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\mathscr{W}$: Wronskian and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E25 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13
 13.14.26 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)},$
 13.14.27 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac% {1}{2}+\mu+\kappa\right)}e^{\mp(\frac{1}{2}+\mu)\pi\mathrm{i}},$
 13.14.28 $\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},$
 13.14.29 $\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac% {1}{2}-\mu+\kappa\right)}e^{\mp(\frac{1}{2}-\mu)\pi\mathrm{i}},$
 13.14.30 $\mathscr{W}\left\{W_{\kappa,\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=e^{\mp\kappa\pi\mathrm{i}}.$

## §13.14(vii) Connection Formulas

 13.14.31 $W_{\kappa,\mu}\left(z\right)=W_{\kappa,-\mu}\left(z\right).$ ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function and $z$: complex variable Referenced by: §13.14(iii), §13.14(vii) Permalink: http://dlmf.nist.gov/13.14.E31 Encodings: TeX, pMML, png See also: Annotations for §13.14(vii), §13.14 and Ch.13
 13.14.32 $\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=\frac{e^{\pm(% \kappa-\mu-\frac{1}{2})\pi\mathrm{i}}}{\Gamma\left(\frac{1}{2}+\mu+\kappa% \right)}W_{\kappa,\mu}\left(z\right)+\frac{e^{\pm\kappa\pi\mathrm{i}}}{\Gamma% \left(\frac{1}{2}+\mu-\kappa\right)}W_{-\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z% \right).$

When $2\mu$ is not an integer

 13.14.33 $W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(-2\mu\right)}{\Gamma\left(\frac% {1}{2}-\mu-\kappa\right)}M_{\kappa,\mu}\left(z\right)+\frac{\Gamma\left(2\mu% \right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}M_{\kappa,-\mu}\left(z% \right).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function and $z$: complex variable A&S Ref: 13.1.34 Referenced by: §13.14(i), §13.14(vii) Permalink: http://dlmf.nist.gov/13.14.E33 Encodings: TeX, pMML, png See also: Annotations for §13.14(vii), §13.14 and Ch.13