InverseEllipticNomeQ

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

(q^(-1))(z) is an analytical function of z which is defined inside the unite circle of the complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

On the boundary of analyticity |q|==1 the function (q^(-1))(z) has a dense set of poles.

Branch points

The function (q^(-1))(z) does not have branch points.

Branch cuts

The function (q^(-1))(z) 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

Generalized power series

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

Expansions at z==0

Expansions at z==1

Product representations

Differential equations

Ordinary nonlinear differential equations

Identities

Functional identities

Differentiation

Low-order differentiation

Operations

Limit operation

Representations through equivalent functions

With inverse function

With related functions

Involving theta functions

Involving Weierstrass functions

Involving other related functions

Zeros

Theorems

The solutions of the quintic (t^5)-t+ρ==0

The solutions of the quintic (t^5)-t+ρ==0 can be expressed in terms of (q^(-1)) (C. Hermite). For one of the roots we have (ρ ∈ ℝ):

(1/(2 (5^(3/4)) (Sqrt[k]) (Sqrt[1-(k^2)])))((((k^2)^(1/8)) sgn(ρ)) ((((-1))^(3/4)) ((((q^(-1))((𝓆^(1/5)) (e^(-(1/5) 2 i π))))^(1/8))+i (((q^(-1))((e^((2 i π)/5)) (𝓆^(1/5))))^(1/8)))  ((((q^(-1))((𝓆^(1/5)) (e^(-(1/5) 4 i π))))^(1/8))+(((q^(-1))((e^((4 i π)/5)) (𝓆^(1/5))))^(1/8))) ((𝓆^(5/8)) (((q^(-1))((𝓆^5)))^(1/8))/((𝓆^5)^(1/8))+(((q^(-1))((𝓆^(1/5))))^(1/8)))))/;

k=tan((1/4) (sin^(-1))((16/(25 (Sqrt[5]) (ρ^2)))))∧𝓆=q((k^2)).

History

– C. Hermite (1858)

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