§13.14 Definitions and Basic Properties

13.14.10		$$M_{\kappa ,\mu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}}{M}_{-\kappa ,\mu }\left(z\right).$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E10 Encodings: TeX, pMML, png See also: Annotations for §13.14(ii), §13.14 and Ch.13

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

13.14.11		$$M_{\kappa ,\mu }\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)={(-1)}^m{\mathrm{e}}^{2m\mu \pi \mathrm{i}}{M}_{\kappa ,\mu }\left(z\right).$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $z$: complex variable Referenced by: §13.14(ii) Permalink: http://dlmf.nist.gov/13.14.E11 Encodings: TeX, pMML, png See also: Annotations for §13.14(ii), §13.14 and Ch.13

13.14.12		$$W_{\kappa ,\mu }\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=\begin{array}{l}\frac{{(-1)}^{m+1}2\pi \mathrm{i}\sin \left(2\pi \mu m\right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)\mathrm{\Gamma }\left(1+2\mu \right)\sin \left(2\pi \mu \right)}{M}_{\kappa ,\mu }\left(z\right)\\ +{(-1)}^m{\mathrm{e}}^{-2m\mu \pi \mathrm{i}}{W}_{\kappa ,\mu }\left(z\right).\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E12 Encodings: TeX, pMML, png See also: Annotations for §13.14(ii), §13.14 and Ch.13

13.14.13		$${(-1)}^m{W}_{\kappa ,\mu }\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=\begin{array}{l}-\frac{{\mathrm{e}}^{2\kappa \pi \mathrm{i}}\sin \left(2m\mu \pi \right)+\sin \left((2m-2)\mu \pi \right)}{\sin \left(2\mu \pi \right)}{W}_{\kappa ,\mu }\left(z\right)\\ -\frac{\sin \left(2m\mu \pi \right)2\pi \mathrm{i}{\mathrm{e}}^{\kappa \pi \mathrm{i}}}{\sin \left(2\mu \pi \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{W}_{-\kappa ,\mu }\left(z{\mathrm{e}}^{\pi \mathrm{i}}\right).\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer and $z$: complex variable Referenced by: §13.14(ii) Permalink: http://dlmf.nist.gov/13.14.E13 Encodings: TeX, pMML, png See also: Annotations for §13.14(ii), §13.14 and Ch.13

Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ 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.

13.14.14		$$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left(z\right)\right),$$
$2\mu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E14 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

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

13.14.15		$$W_{\frac{1}{2}\pm \mu +n,\mu }\left(z\right)={(-1)}^n{\left(1\pm 2\mu \right)}_n{z}^{\frac{1}{2}\pm \mu }+O\left(z^{\frac{3}{2}\pm \mu }\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E15 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

In all other cases

13.14.16		$$W_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{z}^{\frac{1}{2}-\mu }+O\left(z^{\frac{3}{2}-\mathrm{\Re }\mu }\right),$$
$\mathrm{\Re }\mu \ge \frac{1}{2}$, $\mu \ne \frac{1}{2}$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E16 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

13.14.17		$$W_{\kappa ,\frac{1}{2}}\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(1-\kappa \right)}+O\left(z\ln z\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E17 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

13.14.18		$$W_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{z}^{\frac{1}{2}-\mu }+\frac{\mathrm{\Gamma }\left(-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{z}^{\frac{1}{2}+\mu }+O\left(z^{\frac{3}{2}-\mathrm{\Re }\mu }\right),$$
, $\mu \ne 0$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E18 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

13.14.19		$$W_{\kappa ,0}\left(z\right)=-\frac{\sqrt{z}}{\mathrm{\Gamma }\left(\frac{1}{2}-\kappa \right)}\left(\ln z+\psi \left(\frac{1}{2}-\kappa \right)+2\gamma \right)+O\left(z^{3/2}\ln z\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E19 Encodings: TeX, pMML, png See also: Annotations for §13.14(iii), §13.14 and Ch.13

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

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

13.14.20		$$M_{\kappa ,\mu }\left(z\right)\sim \mathrm{\Gamma }\left(1+2\mu \right){\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }/\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right),$$
$\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant Permalink: http://dlmf.nist.gov/13.14.E20 Encodings: TeX, pMML, png See also: Annotations for §13.14(iv), §13.14 and Ch.13

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

13.14.21		$$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa },$$
$\left|\mathrm{ph}z\right|\le \frac{3}{2}\pi -\delta$.
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant Referenced by: §33.21(i) Permalink: http://dlmf.nist.gov/13.14.E21 Encodings: TeX, pMML, png See also: Annotations for §13.14(iv), §13.14 and Ch.13

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

13.14.22		$W_{\kappa ,\mu }\left(z\right),$
		$W_{-\kappa ,\mu }\left({\mathrm{e}}^{-\pi \mathrm{i}}z\right)$,
	$-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi$,
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.14(v), §13.14 and Ch.13
13.14.23		$W_{\kappa ,\mu }\left(z\right),$
		$W_{-\kappa ,\mu }\left({\mathrm{e}}^{\pi \mathrm{i}}z\right)$,
	$-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$.
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E23 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.14(v), §13.14 and Ch.13

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

13.14.24		$M_{\kappa ,\mu }\left(z\right),$
		$M_{\kappa ,-\mu }\left(z\right)$,
	$2\mu \notin \mathbb{Z}$.
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathbb{Z}$: set of all integers, $\notin$: not an element of and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.14(v), §13.14 and Ch.13

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

13.14.25		$$𝒲\left\{M_{\kappa ,\mu }\left(z\right),M_{\kappa ,-\mu }\left(z\right)\right\}=-2\mu ,$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: 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		$$𝒲\left\{M_{\kappa ,\mu }\left(z\right),W_{\kappa ,\mu }\left(z\right)\right\}=-\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E26 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13

13.14.27		$$𝒲\left\{M_{\kappa ,\mu }\left(z\right),W_{-\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right)\right\}=\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{\mp (\frac{1}{2}+\mu )\pi \mathrm{i}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E27 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13

13.14.28		$$𝒲\left\{M_{\kappa ,-\mu }\left(z\right),W_{\kappa ,\mu }\left(z\right)\right\}=-\frac{\mathrm{\Gamma }\left(1-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E28 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13

13.14.29		$$𝒲\left\{M_{\kappa ,-\mu }\left(z\right),W_{-\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right)\right\}=\frac{\mathrm{\Gamma }\left(1-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\kappa \right)}{\mathrm{e}}^{\mp (\frac{1}{2}-\mu )\pi \mathrm{i}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E29 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13

13.14.30		$$𝒲\left\{W_{\kappa ,\mu }\left(z\right),W_{-\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right)\right\}={\mathrm{e}}^{\mp \kappa \pi \mathrm{i}}.$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Permalink: http://dlmf.nist.gov/13.14.E30 Encodings: TeX, pMML, png See also: Annotations for §13.14(vi), §13.14 and Ch.13

13.14.31		$$W_{\kappa ,\mu }\left(z\right)=W_{\kappa ,-\mu }\left(z\right).$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: 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 See also: Annotations for §13.14(vii), §13.14 and Ch.13

13.14.32		$$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{\pm (\kappa -\mu -\frac{1}{2})\pi \mathrm{i}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\kappa ,\mu }\left(z\right)+\frac{{\mathrm{e}}^{\pm \kappa \pi \mathrm{i}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{W}_{-\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §13.14(vii) Permalink: http://dlmf.nist.gov/13.14.E32 Encodings: TeX, pMML, png See also: Annotations for §13.14(vii), §13.14 and Ch.13

When $2\mu$ is not an integer

13.14.33		$$W_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(-2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{M}_{\kappa ,\mu }\left(z\right)+\frac{\mathrm{\Gamma }\left(2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{M}_{\kappa ,-\mu }\left(z\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: 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