# §13.18 Relations to Other Functions

## §13.18(i) Elementary Functions

 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 $M_{\kappa,\kappa-\frac{1}{2}}\left(z\right)=W_{\kappa,\kappa-\frac{1}{2}}\left% (z\right)=W_{\kappa,-\kappa+\frac{1}{2}}\left(z\right)=e^{-\frac{1}{2}z}z^{% \kappa},$
 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

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 $M_{\mu-\frac{1}{2},\mu}\left(z\right)=2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}% \gamma\left(2\mu,z\right),$
 13.18.5 $W_{\mu-\frac{1}{2},\mu}\left(z\right)=e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}% \Gamma\left(2\mu,z\right).$

Special cases are the error functions

 13.18.6 $M_{-\frac{1}{4},\frac{1}{4}}\left(z^{2}\right)=\tfrac{1}{2}e^{\frac{1}{2}z^{2}% }\sqrt{\pi z}\operatorname{erf}\left(z\right),$
 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

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 $M_{0,\nu}\left(2z\right)=2^{2\nu+\frac{1}{2}}\Gamma\left(1+\nu\right)\sqrt{z}I% _{\nu}\left(z\right),$
 13.18.9 $W_{0,\nu}\left(2z\right)=\sqrt{\ifrac{2z}{\pi}}K_{\nu}\left(z\right),$
 13.18.10 $W_{0,\frac{1}{3}}\left(\tfrac{4}{3}z^{\frac{3}{2}}\right)=2\sqrt{\pi}z^{\frac{% 1}{4}}\operatorname{Ai}\left(z\right).$

## §13.18(iv) Parabolic Cylinder Functions

For the notation see §12.2.

 13.18.11 $\displaystyle W_{-\frac{1}{2}a,\pm\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$ $\displaystyle=2^{\frac{1}{2}a}\sqrt{z}U(a,z),$ 13.18.12 $\displaystyle M_{-\frac{1}{2}a,-\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$ $\displaystyle=2^{\frac{1}{2}a-1}\Gamma\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,z\right)+U\left(a,-z\right)\right),$ 13.18.13 $\displaystyle M_{-\frac{1}{2}a,\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right)$ $\displaystyle=2^{\frac{1}{2}a-2}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,-z\right)-U\left(a,z\right)\right).$

## §13.18(v) Orthogonal Polynomials

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

### Hermite Polynomials

 13.18.14 $M_{\frac{1}{4}+n,-\frac{1}{4}}\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n)!}e^{-% \frac{1}{2}z^{2}}\sqrt{z}H_{2n}\left(z\right),$
 13.18.15 $M_{\frac{3}{4}+n,\frac{1}{4}}\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n+1)!}% \frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}H_{2n+1}\left(z\right),$
 13.18.16 $W_{\frac{1}{4}+\frac{1}{2}n,\frac{1}{4}}\left(z^{2}\right)=2^{-n}e^{-\frac{1}{% 2}z^{2}}\sqrt{z}H_{n}\left(z\right).$

### Laguerre Polynomials

 13.18.17 $W_{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha}\left(z\right)=(-1)^{n}{% \left(\alpha+1\right)_{n}}M_{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha% }\left(z\right)=(-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}L^% {(\alpha)}_{n}\left(z\right).$

## §13.18(vi) Coulomb Functions

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