§12.7 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/12.7

§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

Notes:: See Miller (1955, pp. 41, 74).
Keywords:: Hermite polynomials, parabolic cylinder functions
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/12.7.i

For the notation see §18.3.

12.7.1 $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2},z\right)=\mathop{D_{{0}}\/}\nolimits\!\left(z\right)=e^{{-\frac{1}{4}z^{2}}},$

Symbols:: $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)$ : parabolic cylinder function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/12.7.E1
Encodings:: TeX, pMML, png

12.7.2 $\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2},z\right)=\mathop{D_{{n}}\/}\nolimits\!\left(z\right)=e^{{-\frac{1}{4}z^{2}}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(z\right)=2^{{-n/2}}e^{{-\frac{1}{4}z^{2}}}\mathop{H_{{n}}\/}\nolimits\!\left(z/\sqrt{2}\right),$ $n=0,1,2,\dots$ ,

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)$ : parabolic cylinder function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $e$ : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

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

Notes:: See Miller (1955, p. 76). For (12.7.7) combine (7.18.11) and (7.18.12).
Keywords:: Dawson’s integral, error functions, parabolic cylinder functions, probability functions, repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/12.7.ii

For the notation see §§7.2 and 7.18.

12.7.4 $\mathop{V\/}\nolimits\!\left(-\tfrac{1}{2},z\right)=(\ifrac{2}{\sqrt{\pi}}\,)e^{{\frac{1}{4}z^{2}}}\mathop{F\/}\nolimits\!\left(z/\sqrt{2}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $e$ : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/12.7.E4
Encodings:: TeX, pMML, png

12.7.5 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2},z\right)=\mathop{D_{{-1}}\/}\nolimits\!\left(z\right)=\sqrt{\tfrac{1}{2}\pi}\, e^{{\frac{1}{4}z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z/\sqrt{2}\right),$

Symbols:: $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)$ : parabolic cylinder function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Referenced by:: §12.11(ii)
Permalink:: http://dlmf.nist.gov/12.7.E5
Encodings:: TeX, pMML, png

12.7.6 $\mathop{U\/}\nolimits\!\left(n+\tfrac{1}{2},z\right)=\mathop{D_{{-n-1}}\/}\nolimits\!\left(z\right)=\sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{{-\frac{1}{4}z^{2}}}\frac{{d}^{n}\left(e^{{\frac{1}{2}z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z/\sqrt{2}\right)\right)}{{dz}^{n}},$ $n=0,1,2,\dots$ ,

Symbols:: $\mathop{D_{{\nu}}\/}\nolimits\!\left(z\right)$ : parabolic cylinder function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/12.7.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathit{Hh}_{{n}}\/}\nolimits\!\left(z\right)$ : probability function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

§12.7(iii) Modified Bessel Functions

Notes:: See Miller (1955, pp. 42–43, 77–79).
Keywords:: modified Bessel functions, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.7.iii

For the notation see §10.25(ii).

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

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.7.9 $\mathop{U\/}\nolimits\!\left(-1,z\right)=\frac{z^{{3/2}}}{2\sqrt{2\pi}}\left(\mathop{K_{{\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)+\mathop{K_{{\frac{3}{4}}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)\right),$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.7.10 $\mathop{U\/}\nolimits\!\left(0,z\right)=\sqrt{\frac{z}{2\pi}}\mathop{K_{{\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right),$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.7.11 $\mathop{U\/}\nolimits\!\left(1,z\right)=\frac{z^{{3/2}}}{\sqrt{2\pi}}\left(\mathop{K_{{\frac{3}{4}}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)-\mathop{K_{{\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{4}z^{2}\right)\right).$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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

Notes:: See Miller (1955, pp. 40, 73).
Keywords:: confluent hypergeometric functions, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.7.iv

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 $u_{1}(a,z)=e^{{-\tfrac{1}{4}z^{2}}}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=e^{{\tfrac{1}{4}z^{2}}}\mathop{M\/}\nolimits\!\left(-\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable, $a$ : real or complex parameter and $u_{1}(a,z)$ : solution
A&S Ref:: 19.2.2 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E12
Encodings:: TeX, pMML, png

12.7.13 $u_{2}(a,z)=ze^{{-\tfrac{1}{4}z^{2}}}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=ze^{{\tfrac{1}{4}z^{2}}}\mathop{M\/}\nolimits\!\left(-\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable, $a$ : real or complex parameter and $u_{2}(a,z)$ : solution
A&S Ref:: 19.2.4 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E13
Encodings:: TeX, pMML, png

Also,

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

(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