GammaRegularized

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a

For fixed z

Values at infinities

General characteristics

Domain and analyticity

Q(a,z) is an analytical function of a and z which is defined in (ℂ^2). For fixed z, it is an entire function of a.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed a, the function Q(a,z) has an essential singularity at z==(OverTilde[∞]). At the same time, the point z==(OverTilde[∞]) is a branch point for generic a.

With respect to a

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

Branch points

With respect to z

With respect to a

For fixed z, the function Q(a,z) does not have branch points.

Branch cuts

With respect to z

With respect to a

For fixed z, the function Q(a,z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions of GammaRegularized[-n+ϵ,z] at ϵ==0

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

Expansions on branch cuts

Expansions at z==0

Asymptotic series expansions

Expansions at a==∞

Expansions at z==∞

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral 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

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Differentiation

Low-order differentiation

With respect to a

With respect to z

Symbolic differentiation

With respect to a

With respect to z

Fractional integro-differentiation

With respect to a

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving only one direct function with respect to a

Integral transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1FRegularized

Involving Hypergeometric1F1

Involving hypergeometric HypergeometricU

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases for products of incomplete gamma functions ||| Classical cases for products of incomplete gamma functions

Classical cases involving gamma Gamma

Generalized cases for the direct function itself

Generalized cases involving Exp

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

With related functions

Copyright