# §12.7 Relations to Other Functions

## §12.7(i) Hermite Polynomials

For the notation see §18.3.

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

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

For the notation see §§7.2 and 7.18.

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

## §12.7(iii) Modified Bessel Functions

For the notation see §10.25(ii).

 12.7.8 $\mathop{U\/}\nolimits\!\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\!% \mathop{K_{\frac{1}{4}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)+3\!\mathop% {K_{\frac{3}{4}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)-\mathop{K_{\frac{% 5}{4}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)\right),$
 12.7.9 $\displaystyle\mathop{U\/}\nolimits\!\left(-1,z\right)$ $\displaystyle=\frac{z^{3/2}}{2\sqrt{2\pi}}\left(\mathop{K_{\frac{1}{4}}\/}% \nolimits\!\left(\tfrac{1}{4}z^{2}\right)+\mathop{K_{\frac{3}{4}}\/}\nolimits% \!\left(\tfrac{1}{4}z^{2}\right)\right),$ 12.7.10 $\displaystyle\mathop{U\/}\nolimits\!\left(0,z\right)$ $\displaystyle=\sqrt{\frac{z}{2\pi}}\mathop{K_{\frac{1}{4}}\/}\nolimits\!\left(% \tfrac{1}{4}z^{2}\right),$ 12.7.11 $\displaystyle\mathop{U\/}\nolimits\!\left(1,z\right)$ $\displaystyle=\frac{z^{3/2}}{\sqrt{2\pi}}\left(\mathop{K_{\frac{3}{4}}\/}% \nolimits\!\left(\tfrac{1}{4}z^{2}\right)-\mathop{K_{\frac{1}{4}}\/}\nolimits% \!\left(\tfrac{1}{4}z^{2}\right)\right).$

For these, the corresponding results for $\mathop{U\/}\nolimits\!\left(a,z\right)$ with $a=2$, $\pm 3$, $-\tfrac{1}{2}$, $-\tfrac{3}{2}$, $-\tfrac{5}{2}$, and the corresponding results for $\mathop{V\/}\nolimits\!\left(a,z\right)$ 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 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}a+% \tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}% \mathop{M\/}\nolimits\!\left(-\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},-\tfrac{% 1}{2}z^{2}\right),$ Defines: $u_{1}(a,z)$: solution (locally) Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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)
 12.7.13 $u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}a+% \tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}% \mathop{M\/}\nolimits\!\left(-\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},-\tfrac{% 1}{2}z^{2}\right).$ Defines: $u_{2}(a,z)$: solution (locally) Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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)

Also,

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

(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 $-z$.)