§12.7 Relations to Other Functions

§12.7(i) Hermite Polynomials

For the notation see §18.3.

 12.7.1 $U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$
 12.7.2 $U\left(-n-\tfrac{1}{2},z\right)=D_{n}\left(z\right)=e^{-\frac{1}{4}z^{2}}% \mathit{He}_{n}\left(z\right)=2^{-n/2}e^{-\frac{1}{4}z^{2}}H_{n}\left(z/\sqrt{% 2}\right),$ $n=0,1,2,\dots$ ,
 12.7.3 $V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$ $n=0,1,2,\dots$.

§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

For the notation see §§7.2 and 7.18.

 12.7.4 $V\left(-\tfrac{1}{2},z\right)=(\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}F% \left(z/\sqrt{2}\right),$
 12.7.5 $U\left(\tfrac{1}{2},z\right)=D_{-1}\left(z\right)=\sqrt{\tfrac{1}{2}\pi}\,e^{% \frac{1}{4}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right),$
 12.7.6 $U\left(n+\tfrac{1}{2},z\right)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi}{2}}% \frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\frac{{\mathrm{d}}^{n}\left(e^{\frac{1% }{2}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right)\right)}{{\mathrm{d}z}^{n}},$ $n=0,1,2,\dots$,
 12.7.7 $U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),$ $n=-1,0,1,\dots$.

§12.7(iii) Modified Bessel Functions

For the notation see §10.25(ii).

 12.7.8 $U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\!K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+3\!K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{% \frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),$ ⓘ Symbols: $\pi$: 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, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function and $z$: complex variable A&S Ref: 19.15.11 (modification of) Permalink: http://dlmf.nist.gov/12.7.E8 Encodings: TeX, pMML, png See also: Annotations for 12.7(iii), 12.7 and 12
 12.7.9 $\displaystyle U\left(-1,z\right)$ $\displaystyle=\frac{z^{3/2}}{2\sqrt{2\pi}}\left(K_{\frac{1}{4}}\left(\tfrac{1}% {4}z^{2}\right)+K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),$ ⓘ Symbols: $\pi$: 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, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function and $z$: complex variable A&S Ref: 19.15.10 (modification of) Permalink: http://dlmf.nist.gov/12.7.E9 Encodings: TeX, pMML, png See also: Annotations for 12.7(iii), 12.7 and 12 12.7.10 $\displaystyle U\left(0,z\right)$ $\displaystyle=\sqrt{\frac{z}{2\pi}}K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}% \right),$ ⓘ Symbols: $\pi$: 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, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function and $z$: complex variable A&S Ref: 19.15.9 (modification of) Permalink: http://dlmf.nist.gov/12.7.E10 Encodings: TeX, pMML, png See also: Annotations for 12.7(iii), 12.7 and 12 12.7.11 $\displaystyle U\left(1,z\right)$ $\displaystyle=\frac{z^{3/2}}{\sqrt{2\pi}}\left(K_{\frac{3}{4}}\left(\tfrac{1}{% 4}z^{2}\right)-K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}\right)\right).$ ⓘ Symbols: $\pi$: 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, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function and $z$: complex variable A&S Ref: 19.15.3 (modification of) Permalink: http://dlmf.nist.gov/12.7.E11 Encodings: TeX, pMML, png See also: Annotations for 12.7(iii), 12.7 and 12

For these, the corresponding results for $U\left(a,z\right)$ with $a=2$, $\pm 3$, $-\tfrac{1}{2}$, $-\tfrac{3}{2}$, $-\tfrac{5}{2}$, and the corresponding results for $V\left(a,z\right)$ with $a=0$, $\pm 1$, $\pm 2$, $\pm 3$, $\tfrac{1}{2}$, $\tfrac{3}{2}$, $\tfrac{5}{2}$, see Miller (1955, pp. 42–43 and 77–79).

§12.7(iv) Confluent Hypergeometric Functions

For the notation see §§13.2(i) and 13.14(i).

The even and odd solutions of (12.2.2) (see (12.4.3)–(12.4.6)) are given by

 12.7.12 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2% },\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1% }{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),$ ⓘ Defines: $u_{1}(a,z)$: solution (locally) Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.2.2 (modification of) Referenced by: §12.20 Permalink: http://dlmf.nist.gov/12.7.E12 Encodings: TeX, pMML, png See also: Annotations for 12.7(iv), 12.7 and 12
 12.7.13 $u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{% 2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac% {3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).$ ⓘ Defines: $u_{2}(a,z)$: solution (locally) Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.2.4 (modification of) Referenced by: §12.20 Permalink: http://dlmf.nist.gov/12.7.E13 Encodings: TeX, pMML, png See also: Annotations for 12.7(iv), 12.7 and 12

Also,

 12.7.14 $U\left(a,z\right)=2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}U\left(% \tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{3}{% 4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac% {3}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{1}{2}a}z^{-\frac{1}{2}}W_{-\frac{1}{% 2}a,\pm\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right).$

(It should be observed that the functions on the right-hand sides of (12.7.14) are multivalued; hence, for example, $z$ cannot be replaced simply by $-z$.)