Hypergeometric1F1Regularized

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a,b

For fixed a,z

For fixed b,z

For fixed z and integer parameters

For fixed z

For fixed z and a=-(11/2)

For fixed z and a=-(9/2)

For fixed z and a=-(7/2)

For fixed z and a=-(5/2)

For fixed z and a=-(3/2)

For fixed z and a=-(1/2)

For fixed z and a=(1/2)

For fixed z and a=1

For fixed z and a=(3/2)

For fixed z and a=2

For fixed z and a=(5/2)

For fixed z and a=3

For fixed z and a=(7/2)

For fixed z and a=4

For fixed z and a=(9/2)

For fixed z and a=5

For fixed z and a=(11/2)

For fixed z and a=6

General characteristics

Domain and analyticity

(Subscript[, 1])(Subscript[(OverTilde[F]), 1])(a;b;z) is an analytical entire function of a,  b and z which is defined in (ℂ^3). For fixed a, b, it is an entire function of z. For fixed a, z, it is an entire function of b. For fixed b, z, it is an entire function of a. For negative integer a, (Subscript[, 1])(Subscript[(OverTilde[F]), 1])(a;b;z) degenerates to a polynomial in z of order -a.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to b

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

With respect to a

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

Branch points

With respect to z

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

With respect to b

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

With respect to a

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

Branch cuts

With respect to z

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

With respect to b

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

With respect to a

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

Series representations

Generalized power series

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

Expansions at z==0

Asymptotic series expansions

Residue representations

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Major general cases

Eight relations between contiguous functions:

Differentiation

Low-order differentiation

With respect to a

With respect to b

With respect to element of parameters |||  With respect to element of parameters

With respect to z

Symbolic differentiation

With respect to a

With respect to b

With respect to element of parameters |||  With respect to element of parameters

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

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving HypergeometricPFQRegularized

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving  Exp and Cosh

Classical cases involving  Exp and Sinh

Classical cases for products of Hypergeometric1F1Regularized

Classical cases involving  Exp and  products of Hypergeometric1F1Regularized

Classical cases involving Hypergeometric1F1

Classical cases involving  Exp and Hypergeometric1F1

Classical cases involving hypergeometric HypergeometricU

Classical cases involving  Exp and hypergeometric HypergeometricU

Classical cases involving Laguerre LaguerreL

Classical cases involving Exp and Laguerre LaguerreL

Generalized cases for the direct function itself

Generalized cases for products of Hypergeometric1F1Regularized

Generalized cases involving  Exp and  products of Hypergeometric1F1Regularized

Generalized cases involving Hypergeometric1F1

Generalized cases involving  Exp and Hypergeometric1F1

Generalized cases involving hypergeometric HypergeometricU

Generalized cases involving  Exp and hypergeometric HypergeometricU

Generalized cases involving Laguerre LaguerreL

Generalized cases involving Exp and Laguerre LaguerreL

Representations through equivalent functions

With related functions

Copyright