EllipticTheta2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed q

General characteristics

Domain and analyticity

(Subscript[ϑ, 2])(z,q) is an analytic function of z and q for z,q∈ℂ and |q|<1.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

The function (Subscript[ϑ, 2])(z,q) is a periodic function with respect to z with period 2π and a quasi-period i log(q).

Poles and essential singularities

With respect to q

The function (Subscript[ϑ, 2])(z,q) does not have poles and essential singularities inside of the unit circle |q|<1

With respect to z

Branch points

With respect to q

With respect to z

For fixed q, the function (Subscript[ϑ, 2])(z,q) does not have branch points.

Branch cuts

With respect to q

With respect to z

For fixed q, the function (Subscript[ϑ, 2])(z,q) does not have branch cuts.

Natural boundary of analyticity

The unit circle |q|==1 is the natural boundary of the region of analyticity.

Branch cut endpoints

The function (Subscript[ϑ, 2])(z,q) has one branch cut endpoint: q = -1.

Series representations

q-series

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

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

Expansions on branch cuts

Expansions at q==0

Expansions at q==1

Other q-series representations

Other series representations

Product representations

Differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple angle formulas

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to z

With respect to q

Symbolic differentiation

With respect to z

With respect to q

Fractional integro-differentiation

With respect to z

With respect to q

Integration

Indefinite integration

Involving only one direct function

Involving only one direct function with respect to q

Representations through equivalent functions

With related functions

Involving theta functions

Involving Jacobi functions

Involving Weierstrass functions

Zeros

Theorems

The effective potential for the Lagrangian

The effective potential V(T,μ,m;A) for the Lagrangian ℒ=(OverBar[ψ])(γ^ν)((∂_ν)-i e (Subscript[A, ν]))ψ-m(OverBar[ψ])ψ at temperature T and chemical potential μ  in the one loop approximation is given by

V(T,μ,m;A)==-((Tr[1])/(2(Subscript[k, B])T))(∫_0^∞)(1/s)(Subscript[ϑ, 2])(((2s μ)/((Subscript[k, B])T)),((4i π s)/((Subscript[k, B]^2)(T^2))))s I[A]coth(s I[A])(e^(-s((m^2)-(μ^2))))ds,

where I[A]==e(Sqrt[(E^2)-(B^2)]) and Tr[1] is the corresponding gamma trace.

History

– J. Bernoulli (1713)
– L. Euler; J. Fourier
– C. G. J. Jacobi (1827)
– C. W. Borchardt (1838)
– K. Weierstrass (1862–1863)

Copyright