§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 $U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$

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

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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, : complex variable and : 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 $V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$ $n=0,1,2,\dots$ .

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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, : complex variable and : 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 $V\left(-\tfrac{1}{2},z\right)=(\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}F% \left(z/\sqrt{2}\right),$

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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 : 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 $U\left(\tfrac{1}{2},z\right)=D_{-1}\left(z\right)=\sqrt{\tfrac{1}{2}\pi}\,e^{% \frac{1}{4}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right),$

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

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : complex variable and : 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 $U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),$ $n=-1,0,1,\dots$ .

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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, : complex variable and : 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 $U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{% \frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),$

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

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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 : 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 $U\left(0,z\right)=\sqrt{\frac{z}{2\pi}}K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}% \right),$

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

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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 : 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 , $\pm 3$ , $-\tfrac{1}{2}$ , $-\tfrac{3}{2}$ , $-\tfrac{5}{2}$ , and the corresponding results for $V\left(a,z\right)$ with , $\pm 1$ , $\pm 2$ , $\pm 3$ , $\tfrac{1}{2}$ , $\tfrac{3}{2}$ , $\tfrac{5}{2}$ , 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 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2% },\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1% }{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),$

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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, : complex variable and : 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 $u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{% 2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac% {3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).$

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

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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, : complex variable and : 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, cannot be replaced simply by .)

§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