BellB2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

General characteristics

Domain and analyticity

(Subscript[B, n])(z) is a polynomial of z and as such an analytical function of z. (Subscript[B, n])(z) is defined in the whole complex z -plane and for n∈ℕ.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[B, n])(z) has a pole of order n at z==(OverTilde[∞]).

Branch points

With respect to z

The function (Subscript[B, n])(z) does not have branch points.

Branch cuts

With respect to z

The function (Subscript[B, n])(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Asymptotic series expansions

Other series representations

Generating functions

Transformations

Addition formulas

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Summation

Finite summation

Infinite summation

Representations through more general functions

Through other functions

Zeros

For each n∈(SuperPlus[ℕ]) the equation (Subscript[B, n])(z)==0 has exactly n different nonpositive real roots.

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