JacobiZeta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed m

Values at infinities

General characteristics

Domain and analyticity

Ζ(z|m) is an analytical function of z and m which is defined over (ℂ^2).

Symmetries and periodicities

Parity

Ζ(z|m) is an odd function with respect to z.

Mirror symmetry

Periodicity

Ζ(z|m) is a periodic function with respect to z with period π .

Poles and essential singularities

With respect to z

The function Ζ(z|m) does not have poles and essential singularities with respect to z.

With respect to m

The function Ζ(z|m) does not have poles and essential singularities with respect to m.

Branch points

With respect to z

Branch cuts

Branch cut locations: complicated

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at z==0

Expansions at z==(csc^(-1))((Sqrt[m]))+π u/;u∈ℤ

Expansions at z==-(csc^(-1))((Sqrt[m]))+π u/;u∈ℤ

Expansions at z==π/2+π u/;u∈ℤ∧m>1

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

Expansions at m==0

Other series representations

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Sums of the direct function

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to z

With respect to m

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Representations through more general functions

Through hypergeometric functions of two variables

Through Meijer G

Classical cases involving ArcTan in the arguments

Classical cases involving ArcCot in the arguments

Through other functions

Involving some elliptic-type functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

History

– C. G. J. Jacobi (1829)

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