§13.18 Relations to Other Functions

13.18.1		$$M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z,$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sinh z$: 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		$$M_{\kappa ,\kappa -\frac{1}{2}}\left(z\right)=W_{\kappa ,\kappa -\frac{1}{2}}\left(z\right)=W_{\kappa ,-\kappa +\frac{1}{2}}\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa },$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: 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		$$M_{\kappa ,-\kappa -\frac{1}{2}}\left(z\right)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }.$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: 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

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

13.18.4		$$M_{\mu -\frac{1}{2},\mu }\left(z\right)=2\mu {\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{2}-\mu }\gamma (2\mu ,z),$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: 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		$$W_{\mu -\frac{1}{2},\mu }\left(z\right)={\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{2}-\mu }\mathrm{\Gamma }(2\mu ,z).$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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		$$M_{-\frac{1}{4},\frac{1}{4}}\left(z^2\right)=\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erf}\left(z\right),$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: 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		$$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: 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

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		$$M_{0,\nu }\left(2z\right)=2^{2\nu +\frac{1}{2}}\mathrm{\Gamma }\left(1+\nu \right)\sqrt{z}{I}_{\nu }\left(z\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: 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		$$W_{0,\nu }\left(2z\right)=\sqrt{2z/\pi }{K}_{\nu }\left(z\right),$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable 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		$$W_{0,\frac{1}{3}}\left(\frac{4}{3}{z}^{\frac{3}{2}}\right)=2\sqrt{\pi }{z}^{\frac{1}{4}}\mathrm{Ai}\left(z\right).$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $W_{\kappa ,\mu }\left(z\right)$: 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

For the notation see §12.2.

13.18.11		$W_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right)$	$=2^{\frac{1}{2}a}\sqrt{z}U(a,z),$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $U(a,z)$: 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		$M_{-\frac{1}{2}a,-\frac{1}{4}}\left(\frac{1}{2}{z}^2\right)$	$=2^{\frac{1}{2}a-1}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{3}{4}\right)\sqrt{z/\pi }\left(U(a,z)+U(a,-z)\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: 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		$M_{-\frac{1}{2}a,\frac{1}{4}}\left(\frac{1}{2}{z}^2\right)$	$=2^{\frac{1}{2}a-2}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{1}{4}\right)\sqrt{z/\pi }\left(U(a,-z)-U(a,z)\right).$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: 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

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

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