multivalued%20function

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11: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables

x

and

y

cannot be expressed as a product of two

{{}_{2}F_{1}}

functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

16.13.1

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},

\max\left(|x|,|y|\right)<1

ⓘ

Defines:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

… ►

16.13.4

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},

\sqrt{|x|}+\sqrt{|y|}<1

ⓘ

Defines:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

… ► …

12: 5.12 Beta Function

§5.12 Beta Function

… ►

Euler’s Beta Integral

… ►

See accompanying text — Figure 5.12.1: $t$ -plane. Contour for first loop integral for the beta function. Magnify

… ►

►

Pochhammer’s Integral

…

13: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

►

§14.20(i) Definitions and Wronskians

… ► … ►

§14.20(ii) Graphics

… ►

§14.20(x) Zeros and Integrals

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14: 4.2 Definitions

… ►The general logarithm function

\operatorname{Ln}z

is defined by …This is a multivalued function of

z

with branch point at

z=0

. … ►Most texts extend the definition of the principal value to include the branch cut … ►

§4.2(iii) The Exponential Function

… ►

§4.2(iv) Powers

…

15: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order

n

is a nonnegative integer. For the other functions when the order

\nu

is replaced by

n

, it can be any integer. For the Kelvin functions the order

\nu

is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

16: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►Each of the six functions is a multivalued function of

z

\operatorname{Arcsinh}z

and

\operatorname{Arccsch}z

have branch points at

z=\pm i

; the other four functions have branch points at

z=\pm 1

. … ►

Other Inverse Functions

… ►

§4.37(vi) Interrelations

…

17: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►Each of the six functions is a multivalued function of

z

\operatorname{Arctan}z

and

\operatorname{Arccot}z

have branch points at

z=\pm\mathrm{i}

; the other four functions have branch points at

z=\pm 1

. … ►

Other Inverse Functions

… ►

§4.23(viii) Gudermannian Function

…

18: 1.10 Functions of a Complex Variable

… ►

§1.10(vi) Multivalued Functions

… ►Let

F(z)

be a multivalued function and

D

be a domain. … ► ►

Example

… ►

§1.10(xi) Generating Functions

…

19: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ► … ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at

z=0,1

, and

\infty

. … ►

Polynomials

… ►

§16.2(v) Behavior with Respect to Parameters

…

20: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►

§8.17(ii) Hypergeometric Representations

… ►

§8.17(iii) Integral Representation

… ►

§8.17(iv) Recurrence Relations

… ►

§8.17(vi) Sums

…