13 Confluent Hypergeometric Functions Whittaker Functions 13.17 Continued Fractions 13.19 Asymptotic Expansions for Large Argument

§13.18 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/13.18
Addition (effective with 1.0.14):: A subsection §13.18(vi) was added relating Whittaker functions to the Coulomb functions.
See also:: Annotations for Ch.13

§13.18(i) Elementary Functions

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Keywords:: Whittaker functions, elementary functions, relations to other functions
Notes:: Combine §13.6(i) with (13.14.4) and (13.14.5).
Permalink:: http://dlmf.nist.gov/13.18.i
See also:: Annotations for §13.18 and Ch.13

13.18.1		$M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E1 Encodings: TeX, pMML, png See also: Annotations for §13.18(i), §13.18 and Ch.13

13.18.2		$M_{\kappa,\kappa-\frac{1}{2}}\left(z\right)=W_{\kappa,\kappa-\frac{1}{2}}\left% (z\right)=W_{\kappa,-\kappa+\frac{1}{2}}\left(z\right)=e^{-\frac{1}{2}z}z^{% \kappa},$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E2 Encodings: TeX, pMML, png See also: Annotations for §13.18(i), §13.18 and Ch.13

13.18.3		$M_{\kappa,-\kappa-\frac{1}{2}}\left(z\right)=e^{\frac{1}{2}z}z^{-\kappa}.$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E3 Encodings: TeX, pMML, png See also: Annotations for §13.18(i), §13.18 and Ch.13

§13.18(ii) Incomplete Gamma Functions

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Keywords:: Whittaker functions, confluent hypergeometric functions, error functions, incomplete gamma functions, relations to other functions
Notes:: Combine §13.6(ii) with (13.14.4) and (13.14.5).
Permalink:: http://dlmf.nist.gov/13.18.ii
See also:: Annotations for §13.18 and Ch.13

For the notation see §§6.2(i), 7.2(i), and 8.2(i). When $\tfrac{1}{2}-\kappa\pm\mu$ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). For example,

13.18.4		$M_{\mu-\frac{1}{2},\mu}\left(z\right)=2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}% \gamma\left(2\mu,z\right),$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable Referenced by: §8.5 Permalink: http://dlmf.nist.gov/13.18.E4 Encodings: TeX, pMML, png See also: Annotations for §13.18(ii), §13.18 and Ch.13

13.18.5		$W_{\mu-\frac{1}{2},\mu}\left(z\right)=e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}% \Gamma\left(2\mu,z\right).$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E5 Encodings: TeX, pMML, png See also: Annotations for §13.18(ii), §13.18 and Ch.13

Special cases are the error functions

13.18.6		$M_{-\frac{1}{4},\frac{1}{4}}\left(z^{2}\right)=\tfrac{1}{2}e^{\frac{1}{2}z^{2}% }\sqrt{\pi z}\operatorname{erf}\left(z\right),$
ⓘ 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, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E6 Encodings: TeX, pMML, png See also: Annotations for §13.18(ii), §13.18 and Ch.13

13.18.7		$W_{-\frac{1}{4},\pm\frac{1}{4}}\left(z^{2}\right)=e^{\frac{1}{2}z^{2}}\sqrt{% \pi z}\operatorname{erfc}\left(z\right).$
ⓘ 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, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Referenced by: Erratum (V1.0.3) for Equation (13.18.7) Permalink: http://dlmf.nist.gov/13.18.E7 Encodings: TeX, pMML, png Clarification (effective with 1.0.3): Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^{2}\right)$ ; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^{2}\right)$ . See also: Annotations for §13.18(ii), §13.18 and Ch.13

§13.18(iii) Modified Bessel Functions

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Keywords:: Airy functions, Whittaker functions, confluent hypergeometric functions, modified Bessel functions, relations to other functions
Notes:: Combine §13.6(iii) with (13.14.4) and (13.14.5).
Permalink:: http://dlmf.nist.gov/13.18.iii
See also:: Annotations for §13.18 and Ch.13

When $\kappa=0$ the Whittaker functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

13.18.8		$M_{0,\nu}\left(2z\right)=2^{2\nu+\frac{1}{2}}\Gamma\left(1+\nu\right)\sqrt{z}I% _{\nu}\left(z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : complex variable Referenced by: §13.24(ii) Permalink: http://dlmf.nist.gov/13.18.E8 Encodings: TeX, pMML, png See also: Annotations for §13.18(iii), §13.18 and Ch.13

13.18.9		$W_{0,\nu}\left(2z\right)=\sqrt{\ifrac{2z}{\pi}}K_{\nu}\left(z\right),$
ⓘ 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, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $z$ : complex variable Referenced by: (18.34.2) Permalink: http://dlmf.nist.gov/13.18.E9 Encodings: TeX, pMML, png See also: Annotations for §13.18(iii), §13.18 and Ch.13

13.18.10		$W_{0,\frac{1}{3}}\left(\tfrac{4}{3}z^{\frac{3}{2}}\right)=2\sqrt{\pi}z^{\frac{% 1}{4}}\operatorname{Ai}\left(z\right).$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy 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 and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E10 Encodings: TeX, pMML, png See also: Annotations for §13.18(iii), §13.18 and Ch.13

§13.18(iv) Parabolic Cylinder Functions

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Keywords:: Whittaker functions, confluent hypergeometric functions, parabolic cylinder functions, relations to other functions
Notes:: Combine §13.6(iv) with (13.14.4) and (13.14.5).
Referenced by:: §13.18(v)
Permalink:: http://dlmf.nist.gov/13.18.iv
See also:: Annotations for §13.18 and Ch.13

For the notation see §12.2.

13.18.11	$\displaystyle W_{-\frac{1}{2}a,\pm\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$	$\displaystyle=2^{\frac{1}{2}a}\sqrt{z}U(a,z),$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E11 Encodings: TeX, pMML, png See also: Annotations for §13.18(iv), §13.18 and Ch.13
13.18.12	$\displaystyle M_{-\frac{1}{2}a,-\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$	$\displaystyle=2^{\frac{1}{2}a-1}\Gamma\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,z\right)+U\left(a,-z\right)\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E12 Encodings: TeX, pMML, png See also: Annotations for §13.18(iv), §13.18 and Ch.13
13.18.13	$\displaystyle M_{-\frac{1}{2}a,\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$	$\displaystyle=2^{\frac{1}{2}a-2}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,-z\right)-U\left(a,z\right)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E13 Encodings: TeX, pMML, png See also: Annotations for §13.18(iv), §13.18 and Ch.13

§13.18(v) Orthogonal Polynomials

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Keywords:: Whittaker functions, orthogonal polynomials, relations to confluent hypergeometric functions, relations to other functions
Notes:: Combine §13.6(v) with (13.14.4) and (13.14.5).
Permalink:: http://dlmf.nist.gov/13.18.v
See also:: Annotations for §13.18 and Ch.13

Special cases of §13.18(iv) are as follows. For the notation see §18.3.

Hermite Polynomials

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Keywords:: Hermite polynomials, confluent hypergeometric functions, relations to other functions
See also:: Annotations for §13.18(v), §13.18 and Ch.13

13.18.14		$M_{\frac{1}{4}+n,-\frac{1}{4}}\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n)!}e^{-% \frac{1}{2}z^{2}}\sqrt{z}H_{2n}\left(z\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E14 Encodings: TeX, pMML, png See also: Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

13.18.15		$M_{\frac{3}{4}+n,\frac{1}{4}}\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n+1)!}% \frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}H_{2n+1}\left(z\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E15 Encodings: TeX, pMML, png See also: Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

13.18.16		$W_{\frac{1}{4}+\frac{1}{2}n,\frac{1}{4}}\left(z^{2}\right)=2^{-n}e^{-\frac{1}{% 2}z^{2}}\sqrt{z}H_{n}\left(z\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.18.E16 Encodings: TeX, pMML, png See also: Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

Laguerre Polynomials

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Keywords:: Laguerre polynomials, relation to confluent hypergeometric functions
See also:: Annotations for §13.18(v), §13.18 and Ch.13

13.18.17		$W_{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha}\left(z\right)=(-1)^{n}{% \left(\alpha+1\right)_{n}}M_{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha% }\left(z\right)=(-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}L^% {(\alpha)}_{n}\left(z\right).$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable Referenced by: (33.14.14) Permalink: http://dlmf.nist.gov/13.18.E17 Encodings: TeX, pMML, png See also: Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

§13.18(vi) Coulomb Functions

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Referenced by:: §13.18, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/13.18.vi
Addition (effective with 1.0.14):: This subsection has been added.
See also:: Annotations for §13.18 and Ch.13

For representations of Coulomb functions in terms of Whittaker functions see (33.2.3), (33.2.7), (33.14.4) and (33.14.7)

13.17 Continued Fractions 13.19 Asymptotic Expansions for Large Argument

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