Hypergeometric0F1Regularized

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed b

For fixed z and symbolic parameter

For fixed z

For fixed z and b=(m/2)

For fixed z and b=(m/3)

Values at infinities

General characteristics

Domain and analyticity

(Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) is an analytical entire function of b and z which is defined in (ℂ^2). For fixed b, it is an entire function of z.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed b, the function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

With respect to b

For fixed z, the function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) has only one singular point at b = (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

The function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) does not have branch points with respect to z.

With respect to b

The function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) does not have branch points with respect to b.

Branch cuts

With respect to z

The function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) does not have branch cuts with respect to z.

With respect to b

The function (Subscript[, 0])(Subscript[(OverTilde[F]), 1])(;b;z) does not have branch cuts with respect to b.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Asymptotic series expansions

Expansions for Abs[Arg[z]]<π

The general formulas

Expansions for any z  in exponential form

Expansions for any z  in trigonometric form

Residue representations

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Relations of special kind

Division on even and odd parts and generalization

Differentiation

Low-order differentiation

With respect to b

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

Brychkov Yu.A. (2007)

With respect to z

Symbolic differentiation

With respect to b

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function and elementary functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Involving Hypergeometric0F1

Involving Hypergeometric1F1Regularized

Involving Hypergeometric1F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving Cos

Classical cases involving Sin

Classical cases involving Cosh

Classical cases involving Sinh

Classical cases involving AiryAi

Classical cases involving AiryAiPrime

Classical cases involving AiryBi

Classical cases involving AiryBiPrime

Classical cases for powers of Hypergeometric0FRegularized

Classical cases for products of Hypergeometric0F1Regularized

Classical cases involving Bessel BesselJ

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselY

Classical cases involving Bessel BesselK

Classical cases involving Hypergeometric0F1

Generalized cases involving Cos

Generalized cases involving Sin

Generalized cases involving Cosh

Generalized cases involving Sinh

Generalized cases involving AiryAi

Generalized cases involving AiryAiPrime

Generalized cases involving AiryBi

Generalized cases involving AiryBiPrime

Generalized cases for products of Hypergeometric0F1Regularized

Generalized cases involving Bessel BesselJ

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselY

Generalized cases involving Bessel BesselK

Generalized cases involving Hypergeometric0F1

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Copyright