removable discontinuity

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11: 10.2 Definitions

… ►Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$; compare §4.2(i). … ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

12: 5.6 Inequalities

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13: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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14: 4.15 Graphics

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Figure 4.15.4: $\arctan x$ and $\mathrm{arccot}x$. …$\mathrm{arccot}x$ is discontinuous at $x=0$. Magnify

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15: 4.24 Inverse Trigonometric Functions: Further Properties

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16: 4.2 Definitions

… ►Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

17: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.13See (5.11.3).

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Referenced by:

§8.18(ii), Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/8.18.E13

Rearrangement (effective with 1.1.8):

This equation has been removed and in its place it is mentioned to see (5.11.3).

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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18: 4.8 Identities

… ►The restriction on $z_1$ can be removed when $z_2$ is an integer.

19: 16.15 Integral Representations and Integrals

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

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20: 5.11 Asymptotic Expansions

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5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

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Defines:

${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.14 $$\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\sim {\left(z+\frac{a+b-1}{2}\right)}^{a-b}\sum _{k=0}^{\mathrm{\infty }}\frac{H_k(a,b)}{{\left(z+\frac{1}{2}(a+b-1)\right)}^{2k}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $z$: complex variable, $a$: real or complex variable, $b$: real or complex variable and $H_k(a,b)$: coefficients

Referenced by:

§5.11(iii), Erratum (V1.0.21) for Equation (5.11.14)

Permalink:

http://dlmf.nist.gov/5.11.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $\mathrm{\Re }\left(b-a\right)>0$ was removed, see Fields (1966, (3)).

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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