WeierstrassSigma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Special notations for this file:

Specific values

Specialized values

For fixed z

Degenerate cases:

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

Half-period values:

One-third period values:

Values at fixed points

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

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

Values at infinities

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Parity

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

Mirror symmetry

Periodicity

σ(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 σ(z;(Subscript[g, 2]),(Subscript[g, 3])) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

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

Series representations

Generalized power series

Expansions at z==0

q-series

q -series for logarithms:

Other series representations

Integral representations

On the real axis

Of the direct function

Product representations

Infinite products involving trigonometric functions

Infinite products involving exponentials

Infinite products involving q, trigonometrics and exponentials

q products for half-period values

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Involving related functions

Ordinary nonlinear differential equations

Partial differential equations

Transformations

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

Involving only one direct function

Representations through equivalent functions

With related functions

Involving other Weierstrass functions

Involving theta functions

Zeros

Theorems

The solution to the special Lamé equation

The solution to the special Lamé equation w''(z)==(2℘(z;(Subscript[g, 2]),(Subscript[g, 3]))+α)w(z) is given by

w(z)==(Subscript[c, 1])exp(u ζ((℘^(-1))(α;(Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3]))) ((σ(z-(℘^(-1))(α;(Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3])))/(σ(z;(Subscript[g, 2]),(Subscript[g, 3]))))+(Subscript[c, 2])exp(u ζ(-(℘^(-1))(α;(Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3]))) ((σ(z+(℘^(-1))(α;(Subscript[g, 2]),(Subscript[g, 3]));(Subscript[g, 2]),(Subscript[g, 3])))/(σ(z;(Subscript[g, 2]),(Subscript[g, 3])).))

The equations of motion for a heavy symmetric top

The equations of motion for a heavy symmetric top

(Subscript[ω, x]^′)(t)==-c (Subscript[ω, y])(t)(Subscript[ω, z])(t)-(Subscript[p, y])(t),  (Subscript[ω, y]^′)(t)==c (Subscript[ω, x])(t)(Subscript[ω, z])(t)+(Subscript[p, x])(t), (Subscript[ω, z]^′)(t)==0, (Subscript[p, x]^′)(t)==(Subscript[p, y])(t) (Subscript[ω, z])(t)-(Subscript[p, z])(t) (Subscript[ω, y])(t), (Subscript[p, y]^′)(t)==(Subscript[p, x])(t) (Subscript[ω, z])(t)-(Subscript[p, x])(t) (Subscript[ω, z])(t), (Subscript[p, z]^′)(t)==(Subscript[p, x])(t) (Subscript[ω, y])(t)-(Subscript[p, y])(t) (Subscript[ω, x])(t)

can be solved in closed form using the Weierstrass sigma function.

Expressing the Cartesian coordinates through ellipsoidal coordinates

Using Weierstrass sigma functions it is possible to give single–valued solutions x(λ,μ,ν), y(λ,μ,ν), z(λ,μ,ν) for expressing the Cartesian coordinates {x,y,z} through ellipsoidal coordinates {λ,μ,ν} connected by (x^2)==((((a^2)+λ)((a^2)+μ)((a^2)+ν))/(((a^2)-(b^2))((a^2)-(c^2)))), (y^2)==((((b^2)+λ)((b^2)+μ)((b^2)+ν))/(((b^2)-(c^2))((a^2)-(b^2)))), (z^2)==((((c^2)+λ)((c^2)+μ)((c^2)+ν))/(((c^2)-(a^2))((c^2)-(b^2))))where a>b>c are the semi-axes of the defining ellipsoid.

History

– F. G. Eisenstein (1847)
– K. Weierstrass (1855, 1862, 1895)

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