analytical properties

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31: 11.10 Anger–Weber Functions

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11.10.2 $${𝐄}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\sin \left(\nu \theta -z\sin \theta \right)d\theta .$$

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Defines:

${𝐄}_{\nu }\left(z\right)$: Weber function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.3.2

Referenced by:

§11.10(v), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(i), §11.10 and Ch.11

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32: 13.2 Definitions and Basic Properties

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33: 13.14 Definitions and Basic Properties

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34: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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35: 2.5 Mellin Transform Methods

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2.5.1 $$ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt,$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $dx$: differential of $x$ , $\int$: integral and $f(x)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E1

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

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36: Bibliography H

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Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.

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External Links:

ISSN 0003-4916, Document, MathReview

Referenced by:

§33.10(ii), §33.13, §33.22(vii).

Encodings:

BibTeX

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37: 19.2 Definitions

… ►For more details on the analytical continuation of these complete elliptic integrals see Lawden (1989, §§8.12–8.14). …

38: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. …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. … ►There are fifty equations with the Painlevé property. …in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. …

39: Errata

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The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

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40: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. …Some properties are included as special cases of properties given in §31.15 below.