# §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 $U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{% \frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),$ ⓘ 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 Ch.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 Ch.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 Ch.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 Ch.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 natural logarithm, $z$: 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 Ch.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 natural logarithm, $z$: 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 Ch.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 $-z$.)