GammaRegularized3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

General characteristics

Domain and analyticity

Q(a,(Subscript[z, 1]),(Subscript[z, 2])) is an analytical function of a, (Subscript[z, 1]), (Subscript[z, 2]) which is defined in (ℂ^3). For fixed (Subscript[z, 1]), (Subscript[z, 2]), it is an entire function of a.

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to (Subscript[z, k])

For fixed a, the function Q(a,(Subscript[z, 1]),(Subscript[z, 2])) has an essential singularity at (Subscript[z, 1])==(OverTilde[∞]) (for fixed (Subscript[z, 2])) and at (Subscript[z, 2])==(OverTilde[∞]) (for fixed (Subscript[z, 1])).  At the same time, the points (Subscript[z, k])==(OverTilde[∞])/;k==1,2 are a branch points for generic a.

With respect to a

For fixed (Subscript[z, 1]), (Subscript[z, 2]), the function Q(a,(Subscript[z, 1]),(Subscript[z, 2])) has only one singular point at a =(OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to (Subscript[z, k])

With respect to a

For fixed (Subscript[z, 1]), (Subscript[z, 2]), the function Q(a,(Subscript[z, 1]),(Subscript[z, 2])) does not have branch points.

Branch cuts

With respect to (Subscript[z, 2])

With respect to (Subscript[z, 1])

With respect to a

For fixed (Subscript[z, 1]), (Subscript[z, 2]), the function Q(a,(Subscript[z, 1]),(Subscript[z, 2])) does not have branch cuts.

Series representations

Generalized power series

Expansions of GammaRegularized[-n+ϵ,(Subscript[z, 1]),(Subscript[z, 2])] at ϵ==0

Expansions at {(Subscript[z, 1]),(Subscript[z, 2])}=={0,0}

Integral representations

On the real axis

Of the direct function

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

Functional identities

Differentiation

Low-order differentiation

With respect to a

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Symbolic differentiation

With respect to a

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Fractional integro-differentiation

With respect to a

With respect to (Subscript[z, 1])

With respect to (Subscript[z, 2])

Integration

Indefinite integration

Involving only one direct function with respect to a

Involving only one direct function with respect to (Subscript[z, 1])

Involving one direct function and elementary functions with respect to (Subscript[z, 1])

Involving only one direct function with respect to (Subscript[z, 2])

Involving one direct function and elementary functions with respect to (Subscript[z, 2])

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

Generalized cases for the direct function itself

Representations through equivalent functions

With inverse function

With related functions

Copyright