WeierstrassZeta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Special notations for this file:

Specific values

Specialized values

For fixed z

For fixed {(Subscript[g, 2]),(Subscript[g, 3])}

Values at fixed points

Equianharmonic case {(Subscript[g, 2]),(Subscript[g, 3])}=={0,1}

Lemniscatic case {(Subscript[g, 2]),(Subscript[g, 3])}=={1,0}

Values at infinities

General characteristics

Domain and analyticity

ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) is an analytical function of z, (Subscript[g, 2]), and (Subscript[g, 3]), which is defined in (ℂ^3).

Symmetries and periodicities

Parity

ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) is an odd function with respect to z.

Mirror symmetry

Periodicity

ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) is a quasi-periodic function with respect to z.

Transformation of half-periods

Homogeneity

Poles and essential singularities

With respect to z

For fixed (Subscript[g, 2]),  (Subscript[g, 3]), the function ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) has an infinite set of singular points:
a) z==2m (Subscript[ω, 1])((Subscript[g, 2]),(Subscript[g, 3]))+2n (Subscript[ω, 3])((Subscript[g, 2]),(Subscript[g, 3])),{m,n}∈ℤ, are the simple poles with residues 1;
b)  z== (OverTilde[∞]) is an essential singular point.

Branch points

With respect to z

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the function ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have branch points.

Branch cuts

With respect to z

For fixed (Subscript[g, 2]), (Subscript[g, 3]), the function ζ(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

q-series

Other series representations

Integral representations

On the real axis

Of the direct function

Involving related functions

Differential equations

Ordinary nonlinear differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Translation by half-periods

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Multiple angle formula:

Related transformations

Halving half-period

Third of half-period

General fractions of half-periods

Differentiation

Low-order differentiation

With respect to z

With respect to (Subscript[g, 2])

With respect to (Subscript[g, 3])

With respect to (Subscript[ω, 1])

With respect to (Subscript[ω, 3])

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

For the direct function itself

Summation

Finite summation

Representations through equivalent functions

With related functions

Involving other Weierstrass functions

Involving theta functions

Involving elliptic integrals and modular functions

History

– K. Weierstrass (1862)

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