# Whittaker’s Equation

 13.14.1 $\frac{{d}^{2}W}{{dz}^{2}}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{\frac{1}{4% }-\mu^{2}}{z^{2}}\right)W=0.$ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ 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

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 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1% }{2}+\mu}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right),$ Defines: $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$: Whittaker confluent hypergeometric function Symbols: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function 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(iii) Permalink: http://dlmf.nist.gov/13.14.E2 Encodings: TeX, pMML, png
 13.14.3 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1% }{2}+\mu}\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right),$ Defines: $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$: Whittaker confluent hypergeometric function Symbols: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function 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

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

Conversely,

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

The series

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

converge for all $z\in\Complex$.

In general $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ and $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ 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 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).

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

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

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

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

or

 13.14.9 $\mathop{W_{\kappa,\pm\frac{1}{2}n}\/}\nolimits\!\left(z\right)=(-1)^{\kappa-% \frac{1}{2}n-\frac{1}{2}}e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}\sum_{k=% 0}^{\kappa-\frac{1}{2}n-\frac{1}{2}}\binom{\kappa-\frac{1}{2}n-\frac{1}{2}}{k}% \left(n+1+k\right)_{\kappa-k-\frac{1}{2}n-\frac{1}{2}}(-z)^{k},$ $\kappa-\frac{1}{2}n-\frac{1}{2}=0,1,2,\dots$.

# §13.14(ii) Analytic Continuation

 13.14.10 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(ze^{\pm\pi i}\right)=\pm ie^{\pm\mu% \pi i}\mathop{M_{-\kappa,\mu}\/}\nolimits\!\left(z\right).$

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

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

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

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

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

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

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

In all other cases

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

For $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ with $\realpart{\mu}<0$ 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 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)\sim\ifrac{\mathop{\Gamma\/}% \nolimits\!\left(1+2\mu\right)e^{\frac{1}{2}z}z^{-\kappa}}{\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}+\mu-\kappa\right)},$ $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\frac{1}{2}\pi-\delta$,

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

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

# §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 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right),$ $\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(e^{-\pi i}z\right)$, $-\tfrac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits{z}\leq\tfrac{3}{2}\pi$, 13.14.23 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right),$ $\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(e^{\pi i}z\right)$, $-\tfrac{3}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits{z}\leq\tfrac{1}{2}\pi$.

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

 13.14.24 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right),$ $\mathop{M_{\kappa,-\mu}\/}\nolimits\!\left(z\right)$, $2\mu\not\in\Integer$.

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 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{\kappa,\mu}\/}\nolimits\!% \left(z\right),\mathop{M_{\kappa,-\mu}\/}\nolimits\!\left(z\right)\right\}=-2\mu,$
 13.14.26 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{\kappa,\mu}\/}\nolimits\!% \left(z\right),\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)\right\}=-% \frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)},$
 13.14.27 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{\kappa,\mu}\/}\nolimits\!% \left(z\right),\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(e^{\pm\pi i}z\right)% \right\}=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}{\mathop{\Gamma% \/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}e^{\mp(\frac{1}{2}+\mu)\pi i},$
 13.14.28 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{\kappa,-\mu}\/}\nolimits\!% \left(z\right),\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)\right\}=-% \frac{\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)},$
 13.14.29 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{\kappa,-\mu}\/}\nolimits\!% \left(z\right),\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(e^{\pm\pi i}z\right)% \right\}=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)}{\mathop{\Gamma% \/}\nolimits\!\left(\frac{1}{2}-\mu+\kappa\right)}e^{\mp(\frac{1}{2}-\mu)\pi i},$
 13.14.30 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{W_{\kappa,\mu}\/}\nolimits\!% \left(z\right),\mathop{W_{-\kappa,\mu}\/}\nolimits\!\left(e^{\pm\pi i}z\right)% \right\}=e^{\mp\kappa\pi i}.$

# §13.14(vii) Connection Formulas

 13.14.31 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\mathop{W_{\kappa,-\mu}\/}% \nolimits\!\left(z\right).$ Symbols: $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$: 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
 13.14.32 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}\mathop{M_{\kappa,\mu% }\/}\nolimits\!\left(z\right)=\frac{e^{\pm(\kappa-\mu-\frac{1}{2})\pi i}}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu+\kappa\right)}\mathop{W_{% \kappa,\mu}\/}\nolimits\!\left(z\right)+\frac{e^{\pm\kappa\pi i}}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}\mathop{W_{-\kappa,\mu% }\/}\nolimits\!\left(e^{\pm\pi i}z\right).$

When $2\mu$ is not an integer

 13.14.33 $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(-2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-% \mu-\kappa\right)}\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)+\frac{% \mathop{\Gamma\/}\nolimits\!\left(2\mu\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}+\mu-\kappa\right)}\mathop{M_{\kappa,-\mu}\/}\nolimits\!\left% (z\right).$