EllipticTheta3

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, 3])(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, 3])(z,q) is a periodic function with respect to z with period π and a quasi-period i log(q).

Poles and essential singularities

With respect to q

The function (Subscript[ϑ, 3])(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

For fixed z, the function (Subscript[ϑ, 3])(z,q) does not have branch points.

With respect to z

For fixed q, the function (Subscript[ϑ, 3])(z,q) does not have branch points.

Branch cuts

With respect to q

For fixed z, the function (Subscript[ϑ, 3])(z,q) does not have branch cuts.

With respect to z

For fixed q, the function (Subscript[ϑ, 3])(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.

Series representations

q-series

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

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

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

Derivative 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

Definite integration

For the direct function itself

Representations through equivalent functions

With related functions

Involving theta functions

Involving Jacobi functions

Involving Weierstrass functions

Zeros

Theorems

Green's function for one-dimensional heat equation

The Green's function for the one-dimensional heat equation ((∂/(∂t))-((∂^2)/(∂(x^2))))G(x,t)==δ(x)δ(t) , with conditions  G(a,t)==G(b,t)==0 on the interval a<x<b, has the representation

G(x,t)==(1/(2(b-a)))((Subscript[Θ, 3])(exp(-((π^2)/(((b-a))^2))t),((π(x-y))/(2(b-a))))-(Subscript[Θ, 3])(exp(-((π^2)/(((b-a))^2))t),((π(x+y))/(2(b-a))))).

The number of representations of n as a sum of four squares

The number (Subscript[ν, n]) of representations of n as a sum of four squares (including sign and permutation) is (Subscript[ν, n])==[(q^n)](Subscript[θ, 3]^4)(0,q).

The Bloch functions composed of Gaussian orbitals of standard deviation of a one-dimensional crystal

The Bloch functions (Subscript[ψ, k]^β)(x) composed of Gaussian orbitals of standard deviation (1/(2 (Sqrt[β]))) of a one-dimensional crystal can be expressed in the form

(Subscript[ψ, k]^β)(x)==(1/(Sqrt[2π]))(1/(Sqrt[(Subscript[ϑ, 3])((k/2),((i  β)/(2π)) )]))(e^(-β (x^2)))(Subscript[ϑ, 3])((k/2)-i  β x,((i  β)/π) ),

where k is the wave vector.

History

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

Copyright