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11: 8.4 Special Values

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8.4.2 $${\gamma }^{\ast }(a,0)=\frac{1}{\mathrm{\Gamma }\left(a+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\gamma }^{\ast }(a,z)$: incomplete gamma function and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.4.E2

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.5 $$\mathrm{\Gamma }(1,z)={\mathrm{e}}^{-z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/8.4.E5

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.7 $$\gamma (n+1,z)=n!(1-{\mathrm{e}}^{-z}{e}_n(z)),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.4.E7

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.8 $$\mathrm{\Gamma }(n+1,z)=n!{\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.4.E8

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.12 $${\gamma }^{\ast }(-n,z)=z^n,$$

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Symbols:

${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

6.5.14

Referenced by:

§8.13(i)

Permalink:

http://dlmf.nist.gov/8.4.E12

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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12: Errata

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

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31.11.6 $$K_j=\frac{(j+\alpha -\mu -1)(j+\beta -\mu -1)(j+\gamma -\mu -1)(j-\mu )}{(2j+\lambda -\mu -1)(2j+\lambda -\mu -2)}$$

The sign has been corrected and the final term in the numerator $(j+\lambda -1)$ has been corrected to be $(j-\mu )$.

Suggested by Hans Volkmer on 2022-06-02

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19.20.11 $$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$

as $y\to 0+$, $p$ ($\ne 0$) real, we have added the constant term $\frac{-3}{p}{R}_C(z,p)$ and the order term $O\left(y\ln y\right)$, and hence $\sim$ was replaced by $=$.

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The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}$$

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$.

Reported 2013-08-01 by Gergő Nemes and subsequently by Nick Jones on December 11, 2013.

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13: 32.13 Reductions of Partial Differential Equations

… ►where $\lambda$ is an arbitrary constant and $W(z)$ is expressible in terms of solutions of ${\text{P}}_{\text{I}}$. … ►Depending whether $A=0$ or $A\ne 0$, $v(z)$ is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of ${\text{P}}_{\text{I}}$, respectively. …

14: 22.18 Mathematical Applications

… ►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …

15: 2.5 Mellin Transform Methods

… ►The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985). … ►where $l$ ($\ge 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. The last term is clearly $O\left({\zeta }^{l-\delta -1}\right)$ as $\zeta \to 0+$. … ►where $\gamma$ is Euler’s constant (§5.2(ii)). …

16: 18.38 Mathematical Applications

… ►where $Q_0$ is a constant with explicit expression in terms of $e_1,e_2,e_3,e_4$ and $q$ given in Koornwinder (2007a, (2.8)). …

17: 18.39 Applications in the Physical Sciences

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). …

18: 2.7 Differential Equations

… ►where ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ are constants, and the $J$th remainder terms in the sums are $O\left(\mathrm{\Gamma }\left(s+{\mu }_2-{\mu }_1-J\right)\right)$ and $O\left(\mathrm{\Gamma }\left(s+{\mu }_1-{\mu }_2-J\right)\right)$, respectively (Olver (1994a)). …

19: 18.2 General Orthogonal Polynomials

… ►(i) the traditional OP standardizations of Table 18.3.1, where each is defined in terms of the above constants. …

20: 5.11 Asymptotic Expansions

… ►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If $z$ is complex, then the remainder terms are bounded in magnitude by ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms. … ►For the remainder term in (5.11.3) write … ►For the error term in (5.11.19) in the case $z=x(>0)$ and $c=1$, see Olver (1995). …