§12.7 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/12.7
See also:: Annotations for Ch.12

§12.7(i) Hermite Polynomials
§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
§12.7(iii) Modified Bessel Functions
§12.7(iv) Confluent Hypergeometric Functions

§12.7(i) Hermite Polynomials

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Keywords:: Hermite polynomial case, Hermite polynomials, parabolic cylinder functions, relations to other functions
Notes:: See Miller (1955, pp. 41, 74).
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/12.7.i
See also:: Annotations for §12.7 and Ch.12

For the notation see §18.3.

12.7.1		$U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$
ⓘ Symbols: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/12.7.E1 Encodings: TeX, pMML, png See also: Annotations for §12.7(i), §12.7 and Ch.12

12.7.2		$U\left(-n-\tfrac{1}{2},z\right)=D_{n}\left(z\right)=e^{-\frac{1}{4}z^{2}}% \mathit{He}_{n}\left(z\right)=2^{-n/2}e^{-\frac{1}{4}z^{2}}H_{n}\left(z/\sqrt{% 2}\right),$
$n=0,1,2,\dots$ ,
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 19.13.1 (modification of) Permalink: http://dlmf.nist.gov/12.7.E2 Encodings: TeX, pMML, png See also: Annotations for §12.7(i), §12.7 and Ch.12

12.7.3		$V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$
$n=0,1,2,\dots$ .
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 19.13.2 (corrected) Permalink: http://dlmf.nist.gov/12.7.E3 Encodings: TeX, pMML, png See also: Annotations for §12.7(i), §12.7 and Ch.12

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

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Keywords:: Dawson’s integral, error and related functions, error functions, parabolic cylinder functions, probability functions, relation to parabolic cylinder functions, relations to other functions, repeated integrals of the complementary error function
Notes:: See Miller (1955, p. 76). For (12.7.7) combine (7.18.11) and (7.18.12).
Permalink:: http://dlmf.nist.gov/12.7.ii
See also:: Annotations for §12.7 and Ch.12

For the notation see §§7.2 and 7.18.

12.7.4		$V\left(-\tfrac{1}{2},z\right)=(\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}F% \left(z/\sqrt{2}\right),$
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/12.7.E4 Encodings: TeX, pMML, png See also: Annotations for §12.7(ii), §12.7 and Ch.12

12.7.5		$U\left(\tfrac{1}{2},z\right)=D_{-1}\left(z\right)=\sqrt{\tfrac{1}{2}\pi}\,e^{% \frac{1}{4}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right),$
ⓘ Symbols: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder 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, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Referenced by: §12.11(ii) Permalink: http://dlmf.nist.gov/12.7.E5 Encodings: TeX, pMML, png See also: Annotations for §12.7(ii), §12.7 and Ch.12

12.7.6		$U\left(n+\tfrac{1}{2},z\right)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi}{2}}% \frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\frac{{\mathrm{d}}^{n}\left(e^{\frac{1% }{2}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right)\right)}{{\mathrm{d}z}^{n}},$
$n=0,1,2,\dots$ ,
ⓘ Symbols: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/12.7.E6 Encodings: TeX, pMML, png See also: Annotations for §12.7(ii), §12.7 and Ch.12

12.7.7		$U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),$
$n=-1,0,1,\dots$ .
ⓘ Symbols: $\mathit{Hh}_{\NVar{n}}\left(\NVar{z}\right)$ : probability function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §12.7(ii) Permalink: http://dlmf.nist.gov/12.7.E7 Encodings: TeX, pMML, png See also: Annotations for §12.7(ii), §12.7 and Ch.12

§12.7(iii) Modified Bessel Functions

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Keywords:: modified Bessel functions, parabolic cylinder functions, relations to other functions
Notes:: See Miller (1955, pp. 42–43, 77–79).
Permalink:: http://dlmf.nist.gov/12.7.iii
See also:: Annotations for §12.7 and Ch.12

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

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Keywords:: confluent hypergeometric functions, parabolic cylinder functions, relations to other functions
Notes:: See Miller (1955, pp. 40, 73).
Permalink:: http://dlmf.nist.gov/12.7.iv
See also:: Annotations for §12.7 and Ch.12

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		$U\left(a,z\right)=2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}U\left(% \tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{3}{% 4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac% {3}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{1}{2}a}z^{-\frac{1}{2}}W_{-\frac{1}{% 2}a,\pm\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.12.2 (modification of) Referenced by: §12.12, §12.20, §12.7(iv), §12.9(i), §12.9(ii) Permalink: http://dlmf.nist.gov/12.7.E14 Encodings: TeX, pMML, png See also: Annotations for §12.7(iv), §12.7 and Ch.12

(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 $-z$ .)

§12.7 Relations to Other Functions

Contents

§12.7(i) Hermite Polynomials

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

§12.7(iii) Modified Bessel Functions

§12.7(iv) Confluent Hypergeometric Functions