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31: 33.14 Definitions and Basic Properties

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ►

33.14.4 $$f(ϵ,\mathrm{\ell };r)={\kappa }^{\mathrm{\ell }+1}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)/(2\mathrm{\ell }+1)!,$$

ⓘ

Defines:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.18(vi), (33.14.14)

Permalink:

http://dlmf.nist.gov/33.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

… ►

33.14.5 $$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.6(vii)

Permalink:

http://dlmf.nist.gov/33.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

… ►For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by ►

33.14.7 $$h(ϵ,\mathrm{\ell };r)=\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1-\kappa \right)}{\pi {\kappa }^{\mathrm{\ell }}}\left(W_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)+{(-1)}^{\mathrm{\ell }}S(ϵ,r)\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\kappa \right)}{2(2\mathrm{\ell }+1)!}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)\right),$$

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Defines:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity and $S(ϵ,r)$: function

Referenced by:

§13.18(vi)

Permalink:

http://dlmf.nist.gov/33.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(iii), §33.14 and Ch.33

…

32: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

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13.21.1 $$M_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(2\mu +1\right){\kappa }^{-\mu }\left(J_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{J}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►

13.21.6 $$M_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{2\mathrm{\Gamma }\left(2\mu +1\right)}{{\kappa }^{\mu -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{I}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►

13.21.15 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(1)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

►

13.21.16 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)={\mathrm{e}}^{-2\mu \pi \mathrm{i}}c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(2)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

… ►For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). …

33: 13.13 Addition and Multiplication Theorems

… ►

§13.13(i) Addition Theorems for $M(a,b,z)$

►The function $M(a,b,x+y)$ has the following expansions: … ►The function $U(a,b,x+y)$ has the following expansions: … ►

13.13.10 $${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!}U(a,b+n,x),$$ ,

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

… ►

§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

…

34: 13.24 Series

… ►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). … ►

13.24.1 $$M_{\kappa ,\mu }\left(z\right)=\mathrm{\Gamma }\left(\kappa +\mu \right)2^{2\kappa +2\mu }{z}^{\frac{1}{2}-\kappa }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2\kappa +2\mu \right)}_s{\left(2\kappa \right)}_s}{{\left(1+2\mu \right)}_{s}s!}\left(\kappa +\mu +s\right)I_{\kappa +\mu +s}\left(\frac{1}{2}z\right),$$ $2\mu ,\kappa +\mu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.11.2), §13.24(ii)

Permalink:

http://dlmf.nist.gov/13.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

… ►

13.24.2 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=2^{2\mu }{z}^{\mu +\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(2\sqrt{\kappa z}\right)}^{-2\mu -s}{J}_{2\mu +s}\left(2\sqrt{\kappa z}\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

…

35: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the generalized hypergeometric function ${}_2{}^{}F_0^{}$, the Laguerre polynomial $L_n^{(\alpha )}$ and the Whittaker function $W_{\kappa ,\mu }$ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. ►

18.34.1 $$y_n(x;a)={}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_n{\left(\frac{x}{2}\right)}^n{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})=n!{\left(-\frac{1}{2}x\right)}^n{L}_n^{(1-a-2n)}\left(2x^{-1}\right)={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).$$

ⓘ

Defines:

$y_n(x;a)$: Bessel polynomial

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$(proved)

Referenced by:

(18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1)

Permalink:

http://dlmf.nist.gov/18.34.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluenthypergeometricfunction.

See also:

Annotations for §18.34(i), §18.34 and Ch.18

… ►

18.34.7_1 $${\varphi }_n(x;\lambda )={\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -\frac{1}{2}}{y}_n({\lambda }^{-1}{\mathrm{e}}^x;2-2\lambda )/n!={(-1)}^n{\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -n-\frac{1}{2}}{L}_n^{(2\lambda -2n-1)}\left(2\lambda {\mathrm{e}}^{-x}\right)={(2\lambda )}^{-\frac{1}{2}}{\mathrm{e}}^{x/2}{W}_{\lambda ,n+\frac{1}{2}-\lambda }\left(2\lambda {\mathrm{e}}^{-x}\right)/n!,$$ $n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$, $\lambda >\frac{1}{2}$,

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Symbols:

$y_n(x;a)$: Bessel polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\lceil x\rceil$: ceiling of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Proof sketch:

The second and third equality follow from (18.34.1).

Referenced by:

(18.34.7_2), (18.34.7_3), §18.34(iii), (18.39.17), (18.39.18), §18.39(i), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E7_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

…

36: 12.7 Relations to Other Functions

… ►

§12.7(iv) Confluent Hypergeometric Functions

… ►

12.7.12 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)={\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{1}{4},\frac{1}{2},-\frac{1}{2}{z}^2),$$

ⓘ

Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

►

12.7.13 $$u_2(a,z)=z{\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=z{\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{3}{4},\frac{3}{2},-\frac{1}{2}{z}^2).$$

ⓘ

Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

… ►

12.7.14 $$U(a,z)=2^{-\frac{1}{4}-\frac{1}{2}a}{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=2^{-\frac{3}{4}-\frac{1}{2}a}z{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=2^{-\frac{1}{2}a}{z}^{-\frac{1}{2}}{W}_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: 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:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

…

37: 7.18 Repeated Integrals of the Complementary Error Function

… ►

Confluent Hypergeometric Functions

►

7.18.9 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\mathrm{e}}^{-z^2}\left(\frac{1}{2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}M(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2)-\frac{z}{2^{n-1}\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)}M(\frac{1}{2}n+1,\frac{3}{2},z^2)\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.12

Permalink:

http://dlmf.nist.gov/7.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

►

7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors one has to use the analytic continuation formula (13.2.12). …

38: 13.15 Recurrence Relations and Derivatives

… ►

13.15.4 $$2\mu M_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-2\mu M_{\kappa +\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.28

Permalink:

http://dlmf.nist.gov/13.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

… ►

13.15.8 $$W_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)-\sqrt{z}{W}_{\kappa ,\mu }\left(z\right)+(\kappa -\mu -\frac{1}{2})W_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.9 $$W_{\kappa +\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-\sqrt{z}{W}_{\kappa ,\mu }\left(z\right)+(\kappa +\mu -\frac{1}{2})W_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.30 (in different form)

Permalink:

http://dlmf.nist.gov/13.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.10 $$2\mu W_{\kappa ,\mu }\left(z\right)-\sqrt{z}{W}_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+\sqrt{z}{W}_{\kappa +\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.15(i)

Permalink:

http://dlmf.nist.gov/13.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.11 $$W_{\kappa +1,\mu }\left(z\right)+(2\kappa -z)W_{\kappa ,\mu }\left(z\right)+(\kappa -\mu -\frac{1}{2})(\kappa +\mu -\frac{1}{2})W_{\kappa -1,\mu }\left(z\right)=0,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.31

Permalink:

http://dlmf.nist.gov/13.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

…

39: 9.6 Relations to Other Functions

… ►

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

… ►

9.6.21 $$\mathrm{Ai}\left(z\right)=\frac{1}{2}{\pi }^{-1/2}{z}^{-1/4}{W}_{0,1/3}\left(2\zeta \right)=3^{-1/6}{\pi }^{-1/2}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }U(\frac{5}{6},\frac{5}{3},2\zeta ),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.2) with (10.39.8) and (10.39.6).

Referenced by:

(9.5.8)

Permalink:

http://dlmf.nist.gov/9.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

►

9.6.22 $${\mathrm{Ai}}^{\prime }\left(z\right)=-\frac{1}{2}{\pi }^{-1/2}{z}^{1/4}{W}_{0,2/3}\left(2\zeta \right)=-3^{1/6}{\pi }^{-1/2}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }U(\frac{7}{6},\frac{7}{3},2\zeta ),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.3) with (10.39.8) and (10.39.6).

Permalink:

http://dlmf.nist.gov/9.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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9.6.23 $$\mathrm{Bi}\left(z\right)=\frac{1}{2^{1/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{z}^{-1/4}{M}_{0,-1/3}\left(2\zeta \right)+\frac{3}{2^{5/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}{z}^{-1/4}{M}_{0,1/3}\left(2\zeta \right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.4) with (10.39.7).

Referenced by:

(9.6.25)

Permalink:

http://dlmf.nist.gov/9.6.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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9.6.26 $${\mathrm{Bi}}^{\prime }\left(z\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(-\frac{1}{6};-\frac{1}{3};2\zeta )+\frac{3^{7/6}}{2^{7/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};2\zeta ).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.24) with (13.14.2) and refer to §13.1.

Referenced by:

Erratum (V1.0.9) for Equation (9.6.26)

Permalink:

http://dlmf.nist.gov/9.6.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.9):

Originally the second occurrence of the function ${}_1{}^{}F_1^{}$ was given incorrectly as ${}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};\zeta )$.

Reported 2014-05-21 by Hanyou Chu

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

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40: 13.12 Products

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13.12.1 $$M(a,b,z)M(-a,-b,-z)+\frac{a(a-b)z^2}{b^2(1-b^2)}M(1+a,2+b,z)M(1-a,2-b,-z)=1.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.12, §13.25

Permalink:

http://dlmf.nist.gov/13.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.12 and Ch.13

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