13 Confluent Hypergeometric Functions Whittaker Functions 13.13 Addition and Multiplication Theorems 13.15 Recurrence Relations and Derivatives

§13.14 Definitions and Basic Properties

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Referenced by:: §18.15(i)
Permalink:: http://dlmf.nist.gov/13.14
See also:: Annotations for Ch.13

§13.14(i) Differential Equation
§13.14(ii) Analytic Continuation
§13.14(iii) Limiting Forms as $z\to 0$
§13.14(iv) Limiting Forms as $z\to\infty$
§13.14(v) Numerically Satisfactory Solutions
§13.14(vi) Wronskians
§13.14(vii) Connection Formulas

§13.14(i) Differential Equation

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Keywords:: Whittaker functions, differential equation
Notes:: See Olver (1997b, Chapter 7, §§9–11) and Buchholz (1969, §2.3a). For (13.14.8) and (13.14.9) take limiting values in (13.14.33), using (13.14.2) and (13.2.2). See also Slater (1960, §1.7.1).
Referenced by:: §10.16, §10.39, §12.7(iv), §18.11(i), §2.8(vi), §32.10(v), §33.14(ii), §33.2(ii), §33.2(iii), §8.5, §9.6(iii)
Permalink:: http://dlmf.nist.gov/13.14.i
See also:: Annotations for §13.14 and Ch.13

Whittaker’s Equation

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Keywords:: Kummer’s equation, Whittaker’s equation, relation to Kummer’s equation, relation to Whittaker’s equation
See also:: Annotations for §13.14(i), §13.14 and Ch.13

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 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 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

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Keywords:: Kummer functions, Whittaker functions, Whittaker’s equation, analytical properties, definitions, power, power series, principal branches (or values), relations to other functions, series expansions, standard solutions
See also:: Annotations for §13.14(i), §13.14 and Ch.13

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), §18.34(i), §18.39(i) 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		$M\left(a,b,z\right)=e^{\frac{1}{2}z}z^{-\frac{1}{2}b}M_{\frac{1}{2}b-a,\frac{1% }{2}b-\frac{1}{2}}\left(z\right),$
ⓘ 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, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Referenced by: §10.16, (13.11.2), §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.25, §13.8(ii), §13.8(ii), §13.8(iii) Permalink: http://dlmf.nist.gov/13.14.E4 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

13.14.5		$U\left(a,b,z\right)=e^{\frac{1}{2}z}z^{-\frac{1}{2}b}W_{\frac{1}{2}b-a,\frac{1% }{2}b-\frac{1}{2}}\left(z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Referenced by: §10.16, §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.8(ii), §13.8(ii), §13.8(iii) Permalink: http://dlmf.nist.gov/13.14.E5 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

The series

13.14.6		$M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}\sum_{s=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}}{{\left(1+2\mu\right)_{% s}}s!}z^{s}=z^{\frac{1}{2}+\mu}\sum_{n=0}^{\infty}{{}_{2}F_{1}}\left({-n,% \tfrac{1}{2}+\mu-\kappa\atop 1+2\mu};2\right)\frac{\left(-\tfrac{1}{2}z\right)% ^{n}}{n!},$
$2\mu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.14.E6 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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		$\lim_{2\mu\to-n-1}\frac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right% )}=\frac{{\left(-\frac{1}{2}n-\kappa\right)_{n+1}}}{(n+1)!}M_{\kappa,\frac{1}{% 2}(n+1)}\left(z\right)=e^{-\frac{1}{2}z}z^{-\frac{1}{2}n}\sum_{s=n+1}^{\infty}% \frac{{\left(-\frac{1}{2}n-\kappa\right)_{s}}}{\Gamma\left(s-n\right)s!}z^{s}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.14.E7 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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

13.14.8		$W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=\frac{(-1)^{n}e^{-\frac{1}{2}z}z^{% \frac{1}{2}n+\frac{1}{2}}}{n!\Gamma\left(\frac{1}{2}-\frac{1}{2}n-\kappa\right% )}\left(\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(\kappa+\frac{1}{2}-\frac{1}% {2}n\right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{\left(\frac{1}{2}n+\frac{1}{% 2}-\kappa\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi\left(% \tfrac{1}{2}n+\tfrac{1}{2}-\kappa+k\right)-\psi\left(1+k\right)-\psi\left(n+1+% k\right)\right)\right),$
$\kappa-\frac{1}{2}n-\frac{1}{2}\neq 0,1,2,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.14(i) Permalink: http://dlmf.nist.gov/13.14.E8 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

13.14.9		$W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=(-1)^{\kappa-\frac{1}{2}n-\frac{1}{2}% }e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}\sum_{k=0}^{\kappa-\frac{1}{2}n-% \frac{1}{2}}\genfrac{(}{)}{0.0pt}{}{\kappa-\frac{1}{2}n-\frac{1}{2}}{k}{\left(% n+1+k\right)_{\kappa-k-\frac{1}{2}n-\frac{1}{2}}}(-z)^{k},$
$\kappa-\frac{1}{2}n-\frac{1}{2}=0,1,2,\dots$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.14(i) Permalink: http://dlmf.nist.gov/13.14.E9 Encodings: TeX, pMML, png See also: Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

§13.14(ii) Analytic Continuation

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Keywords:: Whittaker functions, analytic continuation
Notes:: See Temme (1996b, §7.2). See also Olver (1997b, Chapter 7, Ex. 11.2).
Permalink:: http://dlmf.nist.gov/13.14.ii
See also:: Annotations for §13.14 and Ch.13

13.14.10		$M_{\kappa,\mu}\left(ze^{\pm\pi\mathrm{i}}\right)=\pm\mathrm{i}e^{\pm\mu\pi% \mathrm{i}}M_{-\kappa,\mu}\left(z\right).$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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		$M_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=(-1)^{m}e^{2m\mu\pi\mathrm{i}}% M_{\kappa,\mu}\left(z\right).$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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		$W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=\frac{(-1)^{m+1}2\pi\mathrm{i}% \sin\left(2\pi\mu m\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa\right)\Gamma% \left(1+2\mu\right)\sin\left(2\pi\mu\right)}M_{\kappa,\mu}\left(z\right)+(-1)^% {m}e^{-2m\mu\pi\mathrm{i}}W_{\kappa,\mu}\left(z\right).$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : 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		$(-1)^{m}W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=-\frac{e^{2\kappa\pi% \mathrm{i}}\sin\left(2m\mu\pi\right)+\sin\left((2m-2)\mu\pi\right)}{\sin\left(% 2\mu\pi\right)}W_{\kappa,\mu}\left(z\right)-\frac{\sin\left(2m\mu\pi\right)2% \pi\mathrm{i}e^{\kappa\pi\mathrm{i}}}{\sin\left(2\mu\pi\right)\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}W_{-% \kappa,\mu}\left(ze^{\pi\mathrm{i}}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\sin\NVar{z}$ : 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 $\ifrac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right)}$ 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 $z\to 0$

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Keywords:: Whittaker functions, as $z\rightarrow 0$ , limiting forms
Notes:: Combine §13.2(iii) and (13.14.4), (13.14.5).
Permalink:: http://dlmf.nist.gov/13.14.iii
See also:: Annotations for §13.14 and Ch.13

13.14.14		$M_{\kappa,\mu}\left(z\right)=z^{\mu+\frac{1}{2}}\left(1+O\left(z\right)\right),$
$2\mu\neq-1,-2,-3,\dots$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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: $O\left(\NVar{x}\right)$ : order not exceeding, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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		$W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(2\mu\right)}{\Gamma\left(\frac{% 1}{2}+\mu-\kappa\right)}z^{\frac{1}{2}-\mu}+O\left(z^{\frac{3}{2}-\Re\mu}% \right),$
$\Re\mu\geq\frac{1}{2}$ , $\mu\not=\frac{1}{2}$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\Re$ : 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		$W_{\kappa,\frac{1}{2}}\left(z\right)=\frac{1}{\Gamma\left(1-\kappa\right)}+O% \left(z\ln z\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\ln\NVar{z}$ : 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		$W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(2\mu\right)}{\Gamma\left(\frac{% 1}{2}+\mu-\kappa\right)}z^{\frac{1}{2}-\mu}+\frac{\Gamma\left(-2\mu\right)}{% \Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{\frac{1}{2}+\mu}+O\left(z^{\frac{% 3}{2}-\Re\mu}\right),$
$0\leq\Re\mu<\tfrac{1}{2}$ , $\mu\not=0$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\Re$ : 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		$W_{\kappa,0}\left(z\right)=-\frac{\sqrt{z}}{\Gamma\left(\frac{1}{2}-\kappa% \right)}\left(\ln z+\psi\left(\tfrac{1}{2}-\kappa\right)+2\gamma\right)+O\left% (z^{\ifrac{3}{2}}\ln z\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : 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 $\Re\mu<0$ use (13.14.31).

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

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Keywords:: Whittaker functions, as $z\rightarrow\infty$ , limiting forms
Notes:: See (13.19.2) and (13.19.3).
Permalink:: http://dlmf.nist.gov/13.14.iv
See also:: Annotations for §13.14 and Ch.13

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

13.14.20		$M_{\kappa,\mu}\left(z\right)\sim\ifrac{\Gamma\left(1+2\mu\right)e^{\frac{1}{2}% z}z^{-\kappa}}{\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)},$
$\left\|\operatorname{ph}z\right\|\leq\frac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : 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		$W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa},$
$\left\|\operatorname{ph}z\right\|\leq\frac{3}{2}\pi-\delta$ .
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : 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

§13.14(v) Numerically Satisfactory Solutions

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Keywords:: Whittaker’s equation, fundamental solutions, numerically satisfactory solutions
Notes:: See Olver (1997b, Chapter 7, §§9–11).
Permalink:: http://dlmf.nist.gov/13.14.v
See also:: Annotations for §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		$W_{\kappa,\mu}\left(z\right),$
		$W_{-\kappa,\mu}\left(e^{-\pi\mathrm{i}}z\right)$ ,
	$-\tfrac{1}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{3}{2}\pi$ ,
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : 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		$W_{\kappa,\mu}\left(z\right),$
		$W_{-\kappa,\mu}\left(e^{\pi\mathrm{i}}z\right)$ ,
	$-\tfrac{3}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{1}{2}\pi$ .
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : 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 $|\operatorname{ph}{z}|\leq\pi$ near the origin is

13.14.24		$M_{\kappa,\mu}\left(z\right),$
		$M_{\kappa,-\mu}\left(z\right)$ ,
	$2\mu\not\in\mathbb{Z}$ .
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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 $b=2\mu+1$ , $a=\mu-\kappa+\tfrac{1}{2}$ , 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

ⓘ

Keywords:: Whittaker functions, Wronskians
Notes:: See Slater (1960, §2.4.2).
Permalink:: http://dlmf.nist.gov/13.14.vi
See also:: Annotations for §13.14 and Ch.13

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		$\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)},$
ⓘ 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, $\mathscr{W}$ : 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		$\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac% {1}{2}+\mu+\kappa\right)}e^{\mp(\frac{1}{2}+\mu)\pi\mathrm{i}},$
ⓘ 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, $\mathscr{W}$ : 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		$\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},$
ⓘ 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, $\mathscr{W}$ : 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		$\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac% {1}{2}-\mu+\kappa\right)}e^{\mp(\frac{1}{2}-\mu)\pi\mathrm{i}},$
ⓘ 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, $\mathscr{W}$ : 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		$\mathscr{W}\left\{W_{\kappa,\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=e^{\mp\kappa\pi\mathrm{i}}.$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathscr{W}$ : 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(vii) Connection Formulas

ⓘ

Keywords:: Whittaker functions, connection formulas, interrelations
Notes:: For (13.14.32) and (13.14.33) combine (13.2.41) and (13.2.42) with (13.14.4) and (13.14.5). (13.14.31) follows from (13.14.33).
Referenced by:: §13.29(ii)
Permalink:: http://dlmf.nist.gov/13.14.vii
Addition (effective with 1.2.2):: Equations (13.14.34) and (13.14.35) were added.
See also:: Annotations for §13.14 and Ch.13

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		$\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=\frac{e^{\pm(% \kappa-\mu-\frac{1}{2})\pi\mathrm{i}}}{\Gamma\left(\frac{1}{2}+\mu+\kappa% \right)}W_{\kappa,\mu}\left(z\right)+\frac{e^{\pm\kappa\pi\mathrm{i}}}{\Gamma% \left(\frac{1}{2}+\mu-\kappa\right)}W_{-\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z% \right).$
ⓘ 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, $\pi$ : 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		$W_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(-2\mu\right)}{\Gamma\left(\frac% {1}{2}-\mu-\kappa\right)}M_{\kappa,\mu}\left(z\right)+\frac{\Gamma\left(2\mu% \right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}M_{\kappa,-\mu}\left(z% \right).$
ⓘ 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

13.14.34		$\frac{2\pi}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)% }M_{-\kappa,\mu}\left(z\right)={\mathrm{e}}^{\mu\pi\mathrm{i}}W_{\kappa,\mu}% \left({\mathrm{e}}^{\pi\mathrm{i}}z\right)+{\mathrm{e}}^{-\mu\pi\mathrm{i}}W_{% \kappa,\mu}\left({\mathrm{e}}^{-\pi\mathrm{i}}z\right),$
ⓘ 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, $\pi$ : 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 Source: Slater (1960, Eq. (2.5.18)) Referenced by: §13.14(vii), Erratum (V1.2.2) for Additions Permalink: http://dlmf.nist.gov/13.14.E34 Encodings: TeX, pMML, png Addition (effective with 1.2.2): This equation was added. See also: Annotations for §13.14(vii), §13.14 and Ch.13

13.14.35		$\frac{2\pi\mathrm{i}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(% \frac{1}{2}-\mu-\kappa\right)}W_{-\kappa,\mu}\left(z\right)={\mathrm{e}}^{-% \kappa\pi\mathrm{i}}W_{\kappa,\mu}\left({\mathrm{e}}^{\pi\mathrm{i}}z\right)-{% \mathrm{e}}^{\kappa\pi\mathrm{i}}W_{\kappa,\mu}\left({\mathrm{e}}^{-\pi\mathrm% {i}}z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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 Source: Slater (1960, Eq. (2.5.19)) Referenced by: §13.14(vii), Erratum (V1.2.2) for Additions Permalink: http://dlmf.nist.gov/13.14.E35 Encodings: TeX, pMML, png Addition (effective with 1.2.2): This equation was added. See also: Annotations for §13.14(vii), §13.14 and Ch.13

13.13 Addition and Multiplication Theorems 13.15 Recurrence Relations and Derivatives

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