EllipticThetaPrime2

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

(Subscript[ϑ, 2]^′)(z,q), considered as a function of z, has a period of 2 π.

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 q==(Subscript[q, 0])

Expansions on branch cuts

Expansions at q==0

Expansions at q==1

Other q-series representations

Other series representations

Differential equations

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

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 Weierstrass functions

Zeros

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