ZernikeR

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n,m

For fixed n,z

For fixed z

General characteristics

Domain and analyticity

The function (Subscript[R, n]^m)(z)is defined over ℕ⊗ℕ⊗ℂ. For fixed n,m/;((n-m)/2)∈ℕ, the function (Subscript[R, n]^m)(z) is a polynomial in z of degree n.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed n,m/;((n-m)/2)∈ℕ, the function (Subscript[R, n]^m)(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

Branch points

With respect to z

For fixed n,m/;((n-m)/2)∈ℕ, the function (Subscript[R, n]^m)(z) does not have branch points.

Branch cuts

With respect to z

For fixed n,m/;((n-m)/2)∈ℕ, the function (Subscript[R, n]^m)(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==∞

Integral representations

On the real axis

Integral representations of negative integer order

Rodrigues-type formula.

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Recurrence identities

Consecutive neighbors

Functional identities

Relations between contiguous functions

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Definite integration

Summation

Infinite summation

Operations

Limit operation

Orthogonality, completeness, and Fourier expansions

Representations through more general functions

Through hypergeometric functions

Through other functions

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