multivalued

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21: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ► ►►►

$ℂ$	complex plane (excluding infinity).
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$F(z_0{e}^{2k\pi i})$	multivalued functions. More generally, $F((z_0-a)e^{2k\pi i}+a)$. See §1.10(vi).
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22: 7.18 Repeated Integrals of the Complementary Error Function

… ►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). …

23: 19.11 Addition Theorems

… ►Hence, care has to be taken with the multivalued functions in (19.11.5). …

24: 10.18 Modulus and Phase Functions

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$${\theta }_{\nu }\left(x\right)=\mathrm{Arctan}\left(Y_{\nu }\left(x\right)/J_{\nu }\left(x\right)\right),$$

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$${\varphi }_{\nu }\left(x\right)=\mathrm{Arctan}\left(Y_{\nu }^{\prime }\left(x\right)/J_{\nu }^{\prime }\left(x\right)\right).$$

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25: 10.68 Modulus and Phase Functions

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${\theta }_{\nu }\left(x\right)=\mathrm{Arctan}\left({\mathrm{bei}}_{\nu }x/{\mathrm{ber}}_{\nu }x\right),$

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${\varphi }_{\nu }\left(x\right)=\mathrm{Arctan}\left({\mathrm{kei}}_{\nu }x/{\ker }_{\nu }x\right).$

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26: 22.18 Mathematical Applications

… ►This circumvents the cumbersome branch structure of the multivalued functions $x(y)$ or $y(x)$, and constitutes the process of uniformization; see Siegel (1988, Chapter II). …

27: 22.16 Related Functions

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22.16.1 $$\mathrm{am}(x,k)=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right),$$ $x\in ℝ$,

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Defines:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\in$: element of, $\mathrm{Arcsin}z$: general arcsine function, $ℝ$: real line, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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28: 12.7 Relations to Other Functions

… ►(It should be observed that the functions on the right-hand sides of (12.7.14) are multivalued; hence, for example, $z$ cannot be replaced simply by $-z$.)

29: 15.2 Definitions and Analytical Properties

… ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. …

30: 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 $\mathrm{\infty }$. …