gives the radial Zernike polynomial .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given when possible.
  • The Zernike polynomials are orthogonal with weight over the unit interval.
  • ZernikeR can be evaluated to arbitrary numerical precision.
  • ZernikeR automatically threads over lists.


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Basic Examples  (4)

Evaluate numerically:

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Scope  (20)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (5)

Values of ZernikeR at fixed points:

Value at zero:

Find the first positive minimum of ZernikeR[7,5,x ]:

Compute the associated ZernikeR[7,3,x] polynomial:

Different ZernikeR types give different symbolic forms:

Visualization  (3)

Plot the ZernikeR function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Function Properties  (4)

Domain of ZernikeR of integer orders:

The range for ZernikeR of integer orders:

The range for complex values:

ZernikeR has the mirror property TemplateBox[{n, m, {z, }}, ZernikeR]=TemplateBox[{n, m, z}, ZernikeR]:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to r:

Higher derivatives with respect to r:

Plot the absolute values of the higher derivatives with respect to r:

Function Identities and Simplifications  (2)

ZernikeR is defined through the identity:

ZernikeR may reduce to simpler form:

Applications  (1)

A function to convert a radial representation to a Cartesian one:

Visualize the combined effect of -astigmatism and -coma aberrations:

Properties & Relations  (2)

Obtain sequence of Zernike polynomials from a generating function:

Compare to the directly computed sequence:

Radial Zernike polynomials are orthogonal on the unit interval with weight function :

Introduced in 2007