JacobiAmplitude

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed m

Other

Derivatives with respect to z

Derivatives with respect to m

General characteristics

Domain and analyticity

am(z|m) is an analytical meromorphic function of z and m which is defined over (ℂ^2).

Symmetries and periodicities

Parity

am(z|m) is an odd function with respect to z.

Mirror symmetry

Periodicity

am(z|m) is a periodic function with respect to z with period 2 i K(1-m) and a pseudo-periodic function with respect to z with period 2K(m).

Poles and essential singularities

With respect to m

The function am(z|m) does not have poles and essential singularities with respect to m.

With respect to z

The function am(z|m) does not have poles and essential singularities with respect to z.

Branch points

Branch points locations: complicated

Branch cuts

Branch cut locations: complicated

Series representations

Generalized power series

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

Expansions at z==0

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

Expansions at m==0

Expansions at m==1

q-series

Other series representations

Integral representations

On the real axis

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Argument involving inverse Jacobi functions

Related transformations

Differentiation

Low-order differentiation

With respect to z

With respect to m

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Representations through equivalent functions

With inverse function

With related functions

Involving one other Jacobi elliptic function

Zeros

History

- C. G. J. Jacobi (1827)
– V. A. Puiseux (1850) showed am is multivalued

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