§13.14 Definitions and Basic Properties

Referenced by:: §18.15(i)
Permalink:: http://dlmf.nist.gov/13.14

§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

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).
Keywords:: Whittaker functions
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

¶ Whittaker’s Equation

Keywords:: Kummer’s equation, 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

Keywords:: Kummer functions, Whittaker functions, Whittaker’s equation, power series

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E6
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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 $\lim _{{2\mu\to-n-1}}\frac{\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)}{\mathop{\Gamma\/}\nolimits\!\left(2\mu+1\right)}=\frac{\left(-\frac{1}{2}n-\kappa\right)_{{n+1}}}{(n+1)!}\mathop{M_{{\kappa,\frac{1}{2}(n+1)}}\/}\nolimits\!\left(z\right)=e^{{-\frac{1}{2}z}}z^{{-\frac{1}{2}n}}\sum _{{s=n+1}}^{{\infty}}\frac{\left(-\frac{1}{2}n-\kappa\right)_{{s}}}{\mathop{\Gamma\/}\nolimits\!\left(s-n\right)s!}z^{{s}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E7
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\binom{m}{n}$ : binomial coefficient, $e$ : base of exponential function, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.14(i)
Permalink:: http://dlmf.nist.gov/13.14.E9
Encodings:: TeX, pMML, png

§13.14(ii) Analytic Continuation

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

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

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E10
Encodings:: TeX, pMML, png

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

13.14.11 $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(ze^{{2m\pi i}}\right)=(-1)^{m}e^{{2m\mu\pi i}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $m$ : integer and $z$ : complex variable
Referenced by:: §13.14(ii)
Permalink:: http://dlmf.nist.gov/13.14.E11
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E12
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : 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

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 $z\to 0$

Notes:: Combine §13.2(iii) and (13.14.4), (13.14.5).
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.14.iii

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E14
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E15
Encodings:: TeX, pMML, png

In all other cases

13.14.16 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}z^{{\frac{1}{2}-\mu}}+\mathop{O\/}\nolimits\!\left(z^{{\frac{3}{2}-\realpart{\mu}}}\right),$ $\realpart{\mu}\geq\frac{1}{2}$ , $\mu\not=\frac{1}{2}$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E16
Encodings:: TeX, pMML, png

13.14.17 $\mathop{W_{{\kappa,\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\kappa\right)}+\mathop{O\/}\nolimits\!\left(z\mathop{\ln\/}\nolimits z\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E17
Encodings:: TeX, pMML, png

13.14.18 $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)}z^{{\frac{1}{2}-\mu}}+\frac{\mathop{\Gamma\/}\nolimits\!\left(-2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)}z^{{\frac{1}{2}+\mu}}+\mathop{O\/}\nolimits\!\left(z^{{\frac{3}{2}-\realpart{\mu}}}\right),$ $0\leq\realpart{\mu}<\tfrac{1}{2}$ , $\mu\not=0$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E18
Encodings:: TeX, pMML, png

13.14.19 $\mathop{W_{{\kappa,0}}\/}\nolimits\!\left(z\right)=-\frac{\sqrt{z}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\kappa\right)}\left(\mathop{\ln\/}\nolimits z+\mathop{\psi\/}\nolimits\!\left(\tfrac{1}{2}-\kappa\right)+2\gamma\right)+\mathop{O\/}\nolimits\!\left(z^{{\ifrac{3}{2}}}\mathop{\ln\/}\nolimits z\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E19
Encodings:: TeX, pMML, png

For $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ with $\realpart{\mu}<0$ use (13.14.31).

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

Notes:: See (13.19.2) and (13.19.3).
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.14.iv

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/13.14.E20
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable and $\delta$ : small positive constant
Referenced by:: §33.21(i)
Permalink:: http://dlmf.nist.gov/13.14.E21
Encodings:: TeX, pMML, png

§13.14(v) Numerically Satisfactory Solutions

Notes:: See Olver (1997b, Chapter 7, §§9–11).
Keywords:: Whittaker’s equation
Permalink:: http://dlmf.nist.gov/13.14.v

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

13.14.22

$\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),$

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

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E22
Encodings:: TeX, TeX, pMML, pMML, png, png

13.14.23

$\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),$

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

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E23
Encodings:: TeX, TeX, pMML, pMML, png, png

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

$\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),$

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

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\Integer$ : 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

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

Notes:: See Slater (1960, §2.4.2).
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.14.vi

13.14.25 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),\mathop{M_{{\kappa,-\mu}}\/}\nolimits\!\left(z\right)\right\}=-2\mu,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E25
Encodings:: TeX, pMML, png

13.14.26 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right\}=-\frac{\mathop{\Gamma\/}\nolimits\!\left(1+2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\mu-\kappa\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E26
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E27
Encodings:: TeX, pMML, png

13.14.28 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{M_{{\kappa,-\mu}}\/}\nolimits\!\left(z\right),\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right\}=-\frac{\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu-\kappa\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E28
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E29
Encodings:: TeX, pMML, png

13.14.30 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right),\mathop{W_{{-\kappa,\mu}}\/}\nolimits\!\left(e^{{\pm\pi i}}z\right)\right\}=e^{{\mp\kappa\pi i}}.$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.14.E30
Encodings:: TeX, pMML, png

§13.14(vii) Connection Formulas

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).
Keywords:: Whittaker functions
Referenced by:: §13.29(ii)
Permalink:: http://dlmf.nist.gov/13.14.vii

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: §13.14(vii)
Permalink:: http://dlmf.nist.gov/13.14.E32
Encodings:: TeX, pMML, png

When $2\mu$ is not an integer

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : 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