EllipticTheta1

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, 1])(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, 1])(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, 1])(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, 1])(z,q) does not have branch points.

Branch cuts

With respect to q

With respect to z

For fixed q, the function (Subscript[ϑ, 1])(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, 1])(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

Ordinary nonlinear differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

multiple angle formulas:

Identities involving the group of functions

Basic Algebraic Identities

Relations involving squares

Relations involving quartic powers

Addition theorems

Relation between the four theta functions with zero argument

Double angle formulas

The 16 fundamental algebraic identities (from Enneper)

Four linear combinations of the fundamental identities

Alternative version of fundamental identities (from Tannery and Molk)

Automatically generated triple addition formulas (using 16 fundamental relations)

Triple addition formulas (from Enneper)

Automatically generated addition formulas with two variables

Automatically generated double angle formulas

Identities involving transformation of nome q

Equations for z->2z, q->(q^4)

Equations for z->2z,  q->(q^2)

Equations for q->(Sqrt[q])

Equations involving integer powers of q

Algebraic addition theorems involving (q^2)

Landen's transformation

Differential 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 Green's function for the Laplace operator in the rectangle

The Green's function  G(x,x',y,y') for the Laplace operator in the rectangle (0,a)×(0,b) is given as:

G(x,x',y,y')=(1/(2π))Re(log((((Subscript[ϑ, 1])(((π (x+i y+x'-i y'))/(2 a)),(e^(- π b/a))) (Subscript[ϑ, 1])(((π (x+i y-x'+i y'))/(2 a)),(e^(- π b/a))))/((Subscript[ϑ, 1])(((π (x+i y-x'-i y'))/(2 a)),(e^(- π b/a))) (Subscript[ϑ, 1])(((π (x+i y+x'+i y'))/(2 a)),(e^(- π b/a)))))))

Zolotarev's problem

Zolotarev's problem,  (min_((Subscript[a, 2]),(Subscript[a, 3]),…,(Subscript[a, n])))(max_(-1≤x≤1))|(x^n)-n σ (t^(n-1))+(Subscript[a, 2])(t^(n-2))+… +(Subscript[a, n])| can be solved in closed form using elliptic theta functions.

History

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

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