Beta3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Basic definition

The incomplete beta function (Subscript[Β, z])(a,b) is defined by the above definite integral, that under the stated conditions, is convergent. For other values of the arguments z, a, b, the function (Subscript[Β, z])(a,b) is defined as the analytic continuation with respect to its arguments of this integral.

Complete definition

The function (Subscript[Β, z])(a,b) can be equivalently defined through the above generalized hypergeometric function.

For negative integers α==-n and positive integers b==m/;m≤n, the function (Subscript[Β, z])(a,b) cannot be uniquely defined by a limiting procedure based on the above definition because the two variables a, b can approach integers -n, m/;m≤n at different speeds. In the case a==-n,b==m/;m≤n one defines:

Specific values

Specialized values

For fixed z,a

For fixed z,b

For fixed a,b

General characteristics

Domain and analyticity

(Subscript[Β, z])(a,b) is an analytical function of z, a, and b which is defined in (ℂ^3).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to b

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

With respect to a

For fixed z, b, the function (Subscript[Β, z])(a,b) has an infinite set of singular points:
a) a==-k/;k∈ℕ, are the simple poles with residues ((Subscript[(1-b), k])/(k!));
b) a==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to z

For fixed a, b, the function (Subscript[Β, z])(a,b) does not have poles and essential singularities.

Branch points

With respect to b

For fixed z, a, the function (Subscript[Β, z])(a,b) does not have branch points.

With respect to a

For fixed z, b, the function (Subscript[Β, z])(a,b) does not have branch points.

With respect to z

Branch cuts

With respect to b

For fixed z, a, the function (Subscript[Β, z])(a,b) does not have branch cuts.

With respect to a

For fixed z, b, the function (Subscript[Β, z])(a,b) does not have branch cuts.

With respect to z

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at z==0

Expansions at z==1

Expansions at z==∞

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

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

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

Functional identities

Relations between contiguous functions

Additional relations between contiguous functions

Major general cases

Differentiation

Low-order differentiation

With respect to z

With respect to a

With respect to b

Symbolic differentiation

With respect to z

With respect to a

With respect to b

Fractional integro-differentiation

With respect to z

With respect to a

With respect to b

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving only one direct function with respect to a

Involving one direct function and elementary functions with respect to a

Involving only one direct function with respect to b

Involving one direct function and elementary functions with respect to b

Integral transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1Regularized

Involving Hypergeometric2F1

Through hypergeometric functions of two variables

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving algebraic functions in the arguments

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

With related functions

History

–  I. Newton (1676)
– J. Stirling (1730)
– T. Bayes (1763)
– P.-S. Laplace (1778)

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