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11: 16.2 Definition and Analytic Properties

… ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. … ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

12: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …

13: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. …

14: 10.61 Definitions and Basic Properties

… ►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. The branch point is absent, however, in the case of ${\mathrm{ber}}_{\nu }$ and ${\mathrm{bei}}_{\nu }$ when $\nu$ is an integer. …

15: 32.2 Differential Equations

… ►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …

16: 10.25 Definitions

… ►It has a branch point at $z=0$ for all $\nu \in ℂ$. …

17: 4.23 Inverse Trigonometric Functions

… ►

4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

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Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

… ► $\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …

18: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. … ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. …

19: Bibliography H

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The eigenvalues of Mathieu’s equation and their branch points. Stud. Appl. Math. 64 (2), pp. 113–141.

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External Links:

ISSN 0022-2526, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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20: Bibliography C

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The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point. Proc. Roy. Soc. London Ser. A 426, pp. 273–286.

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External Links:

ISSN 0080-4630, FullText, MathReview, ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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