§13.18 Relations to Other Functions

§13.18(i) Elementary Functions

 13.18.1 $\mathop{M_{0,\frac{1}{2}}\/}\nolimits\!\left(2z\right)=2\mathop{\sinh\/}% \nolimits z,$
 13.18.2 $\mathop{M_{\kappa,\kappa-\frac{1}{2}}\/}\nolimits\!\left(z\right)=\mathop{W_{% \kappa,\kappa-\frac{1}{2}}\/}\nolimits\!\left(z\right)=\mathop{W_{\kappa,-% \kappa+\frac{1}{2}}\/}\nolimits\!\left(z\right)=e^{-\frac{1}{2}z}z^{\kappa},$
 13.18.3 $\mathop{M_{\kappa,-\kappa-\frac{1}{2}}\/}\nolimits\!\left(z\right)=e^{\frac{1}% {2}z}z^{-\kappa}.$

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

Special cases are the error functions

 13.18.6 $\mathop{M_{-\frac{1}{4},\frac{1}{4}}\/}\nolimits\!\left(z^{2}\right)=\tfrac{1}% {2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\mathop{\mathrm{erf}\/}\nolimits\!\left(z% \right),$
 13.18.7 $\mathop{W_{-\frac{1}{4},\pm\frac{1}{4}}\/}\nolimits\!\left(z^{2}\right)=e^{% \frac{1}{2}z^{2}}\sqrt{\pi z}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right).$ Symbols: $\mathop{W_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{erfc}\/}\nolimits\NVar{z}$: complementary error function, $\mathrm{e}$: base of exponential function and $z$: complex variable Referenced by: 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 $\mathop{W_{-\frac{1}{4},-\frac{1}{4}}\/}\nolimits\!\left(z^{2}\right)$; the equation is true also for $\mathop{W_{-\frac{1}{4},+\frac{1}{4}}\/}\nolimits\!\left(z^{2}\right)$. Reported 2011-08-21 See also: Annotations for 13.18(ii)

§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 $\mathop{M_{0,\nu}\/}\nolimits\!\left(2z\right)=2^{2\nu+\frac{1}{2}}\mathop{% \Gamma\/}\nolimits\!\left(1+\nu\right)\sqrt{z}\mathop{I_{\nu}\/}\nolimits\!% \left(z\right),$
 13.18.9 $\mathop{W_{0,\nu}\/}\nolimits\!\left(2z\right)=\sqrt{\ifrac{2z}{\pi}}\mathop{K% _{\nu}\/}\nolimits\!\left(z\right),$
 13.18.10 $\mathop{W_{0,\frac{1}{3}}\/}\nolimits\!\left(\tfrac{4}{3}z^{\frac{3}{2}}\right% )=2\sqrt{\pi}z^{\frac{1}{4}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right).$

§13.18(iv) Parabolic Cylinder Functions

For the notation see §12.2.

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

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

Laguerre Polynomials

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

§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)